Research Article Attribute Reduction Based on...

Research ArticleAttribute Reduction Based on Property Pictorial Diagram

Qing Wan and Ling Wei

School of Mathematics Northwest University Xirsquoan Shaanxi 710069 China

Correspondence should be addressed to Ling Wei wlnwueducn

Received 28 June 2014 Revised 23 July 2014 Accepted 23 July 2014 Published 27 August 2014

Academic Editor Yunqiang Yin

Copyright copy 2014 Q Wan and L Wei This is an open access article distributed under the Creative Commons Attribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

This paper mainly studies attribute reduction which keeps the lattice structure in formal contexts based on the property pictorialdiagram Firstly the property pictorial diagram of a formal context is defined Based on such diagram an attribute reductionapproach of concept lattice is achieved Then through the relation between an original formal context and its complementarycontext an attribute reduct of complementary context concept lattice is obtained which is also based on the property pictorialdiagram of the original formal context Finally attribute reducts in property oriented concept lattice and object oriented conceptlattice can be acquired by the relations of attribute reduction between these two lattices and concept lattice of complementarycontext In addition a detailed illustrative example is presented

1 Introduction

Formal concept analysis (FCA) [1 2] introduced by Ger-man mathematician Wille in 1982 has become one of theimportant tools for knowledge discovery and data analysisThe basic notions of FCA are formal context formal conceptand the corresponding concept lattice Another useful toolfor knowledge discovery and data analysis is rough set theory(RST) originally proposed by Pawlak in 1982 [3 4] in whichthe lower and upper approximations of an arbitrary subset ofuniverse are the basics At present FCA and RST have manyimportant applications in various fields respectively andmany efforts have been made to compare and combine themFor example Duntsch and Gediga [5] introduced the notionsof rough set theory into formal concept analysis and proposedproperty oriented concept lattice Based on such idea Yao[6] proposed object oriented concept lattice Shi et al [7] andWang and Zhang [8] studied the relation between RST andFCA Then Wei and Qi [9] discussed the relation betweenconcept lattice reduction and rough set reduction Liu et al[10] studied the reduction of the concept lattices based onrough set theory Wang [11] defined ldquothe notions of attributereduction in concept lattices in a similar way with that inrough set theoryrdquo Poelmans et al [12] gave ldquoa systematicoverviewof themore than 120 papers published between 2003and 2011 on FCAwith fuzzy attributes and rough FCArdquoTheirresearch enriched the FCA and RST

Attribute reduction is one of the key issues of RST andFCA In the case of RST attribute reduction in informationsystems is based on equivalence relation A reduct is a mini-mum subset of attributes that provides the same classificationability as the entire set of attributes [3] Skowron and Rauszer[13] proposed a reduct construction method based on thediscernibility matrix and many researchers improved thismethod [14ndash16]

While there are some differences between the reductionin FCA and in RST In the case of FCA Ganter and Wille[1] proposed the term of reduction from the viewpoint ofdeleting rows or columns In [17] Zhang et al presentedattribute reduction approaches to finding minimal attributesets which can determine all extents and their original hier-archy in a formal context That is to say the reduction theorycan keep the lattice structure of a formal context Based onsuch reduction the authors also studied the reduction theoryof formal decision contexts [18] This approach to attributereduction is based on the discernibility matrix in FCA Forobject oriented concept lattice and property oriented conceptlattice the reduction of keeping their lattice structure wasstudied by Liu and Wei [19]

Besides this kind of reduction that keeps the lattice struc-ture there are other reduction theories about formal contextsFor example Wang and Ma [20] proposed another approachto attribute reduction that can only preserve the extents

Hindawi Publishing Corporatione Scientific World JournalVolume 2014 Article ID 109706 9 pageshttpdxdoiorg1011552014109706

2 The Scientific World Journal

of meet-irreducible elements in the original concept latticeanddescribed attribute characteristics usingmeet-irreducibleelements For the object oriented concept lattice and propertyoriented concept latticeWang andZhang [21] further studiedsuch reduction and described attribute characteristics usingmeet- (join-) irreducible elements Medina [22] obtained therelation of attribute reduction among complementary contextconcept lattice object oriented concept lattice and propertyoriented concept lattice Wu et al [23] proposed the granularreduction from the viewpoint of keeping object conceptsand discussed information granules and their properties ina formal context Li et al [24] constructed ldquoa new frameworkof knowledge reduction in which the capacity of one conceptlattice implying another is defined tomeasure the significanceof the attributes in a consistent decision formal contextrdquo Shaoet al [25] formulated ldquoan approach to attribute reductionin formal decision contexts such that rules extracted fromthe reduced formal decision contexts are identical to thatextracted from the initial formal decision contextsrdquo AswaniKumar and Srinivas [26] proposed ldquoa new method based onfuzzy K-means clustering for reducing the size of the conceptlatticesrdquo

In this paper based on [20 22] we discuss the reductionof a formal context which can keep the lattice structure usingarrow relation defined by Ganter and Wille [1] First of allwe obtain the new approach to acquiring the arrow relation on the basis of a property pictorial diagram defined byus Then combining the relations between arrow relation meet-irreducible elements and attribute characteristics wepresent an approach to construct attribute reducts of conceptlattice complementary context concept lattice object ori-ented concept lattice and property oriented concept lattice

The rest of the paper is organized as follows Section 2reviews the basic notions of FCA Section 3 constructsattribute reducts based on property pictorial diagram of aformal context Section 4 uses anUCI database to explain ourapproach in more detail Section 5 concludes the paper

2 Preliminaries

In this section we recall some basic notions in formal conceptanalysis [1 2]

Definition 1 (see [1]) A formal context (119866119872 119868) consists oftwo sets 119866 and 119872 and a relation 119868 between 119866 and 119872 Theelements of119866 are called the objects and the elements of119872 arecalled the attributes of the context In order to express that anobject 119892 is in a relation 119868with an attribute119898 we write 119892119868119898 or(119892119898) isin 119868 and read it as ldquothe object 119892 has the attribute119898rdquo

Let (119866119872 119868) be a formal context For 119860 sube 119866 119861 sube 119872 twooperators are defined as follows

119860lowast

= 119898 isin 119872 | (119892119898) isin 119868 forall119892 isin 119860

1198611015840


(1)

(119860 119861) is called a formal concept for short a concept if andonly if 119860lowast

= 119861 119860 = 1198611015840 where 119860 is called the extent of

the formal concept and 119861 is called its intent Particularly

(1198981015840

1198981015840lowast

) is a formal concept and is called an attributeconcept and119898

1015840 is called attribute extent [1]The set of all con-cepts of (119866119872 119868) is denoted by 119871(119866119872 119868) For any (119860

1 119861

1)

(1198602 119861

2) isin 119871(119866119872 119868) we have (119860

1 119861

1) ⩽ (119860

2 119861

2) hArr 119860

1sube

1198602(hArr 119861

1supe 119861

2) And the infimum and supremum are given

by

(1198601 119861

1) and (119860

2 119861

2) = (119860

1cap 119860

2 (119861

1cup 119861

2)1015840lowast

)

(1198601 119861

1) or (119860

2 119861

2) = ((119860

1cup 119860

2)lowast1015840

1198611cap 119861

2)

(2)

Thus 119871(119866119872 119868) is a complete lattice and is called theconcept lattice

To simplify for all 119892 isin 119866 for all 119898 isin 119872 119892lowast and 1198981015840

are replaced by 119892lowast and 119898

1015840 respectively If for all 119892 isin 119866119892lowast

= 0 119892lowast

= 119872 and for all 119898 isin 119872 1198981015840

= 0 1198981015840

= 119866then the formal context is called canonical That is to saythere is neither full rowcolumn nor empty rowcolumn ina formal context Noting this an irregular formal context canbe regularized by removing the full rowcolumn and emptyrowcolumn Such way of regularization causes no effect onthe analysis results of the formal context Thus without lossof generality we suppose that all formal contexts are finite andcanonical in this paper

Let (119866119872 119868) be a formal context Denote 119868119888

= (119866 times

119872) 119868 then we call (119866119872 119868119888

) the complementary contextof (119866119872 119868) [1] the mappings defined in (1) on (119866119872 119868

119888

) aredenoted by lowast119888 and 1015840119888

All concepts of (119866119872 119868119888

) are denoted by 119871119862(119866119872 119868)

which is also a complete latticeLet (119866119872 119868) be a formal context For any 119860 sube 119866 119861 sube 119872

Duntsch and Gediga defined a pair of approximate operators ◻ as follows [5]

119860= 119898 isin 119872 | 119898

1015840

cap 119860 = 0

119861= 119892 isin 119866119892

lowast

cap 119861 = 0

119860◻

= 119898 isin 119872 | 1198981015840

sube 119860

119861◻

= 119892 isin 119866 | 119892lowast

sube 119861

(3)

A pair (119860 119861)119860 sube 119866 119861 sube 119872 is called a property orientedconcept if119860

= 119861 and119861◻

= 119860 All property oriented conceptsof (119866119872 119868) are denoted by 119871

119875(119866119872 119868) For any (119860

1 119861

1)

(1198602 119861

2) isin 119871

119875(119866119872 119868) (119860

1 119861

1) ⩽ (119860

2 119861

2) hArr 119860

1sube 119860

2(hArr

1198611sube 119861

2) And the infimum and supremum are given by

(1198601 119861

1) and (119860

2 119861

2) = (119860

1cap 119860

2 (119861

1cap 119861

2)◻

)

(1198601 119861

1) or (119860

2 119861

2) = ((119860

1cup 119860

2)◻

1198611cup 119861

2)

(4)

Thus 119871119875(119866119872 119868) is a complete lattice and is called the

property oriented concept latticeBased on the work of Duntsch and Gediga Yao proposed

the object oriented concept lattice [6]A pair (119860 119861) 119860 sube 119866 119861 sube 119872 is called an object

oriented concept if 119860◻

= 119861 and 119861

= 119860 All object orientedconcepts of (119866119872 119868) are denoted by 119871

119874(119866119872 119868) For any

The Scientific World Journal 3

Table 1 A formal context (119866119872 119868)

119866 119886 119887 119888 119889 119890 119891

1 times times times times

2 times times times

3 times


(1198601 119861

1) (119860

2 119861

2) isin 119871

119874(119866119872 119868) (119860

1 119861

1) ⩽ (119860

2 119861

2) hArr

1198601

sube 1198602(hArr 119861

1sube 119861

2) And the infimum and supremum

are given by

(1198601 119861

1) and (119860

2 119861

2) = ((119860

1cap 119860

2)◻

1198611cap 119861

2)

(1198601 119861

1) or (119860

2 119861

2) = (119860

1cup 119860

2 (119861

1cup 119861

2)◻

)

(5)

Hence 119871119874(119866119872 119868) is a complete lattice and is called the

object oriented concept lattice [6 27]Thus for one formal context (119866119872 119868) we have four

different lattices concept lattice 119871(119866119872 119868) complemen-tary context concept lattice 119871

119862(119866119872 119868) property oriented

concept lattice 119871119875(119866119872 119868) and object oriented concept

lattice 119871119874(119866119872 119868) respectively In [27] Yao studied the

relations among 119871119862(119866119872 119868) 119871

119875(119866119872 119868) and 119871

119874(119866119872 119868)

and proved these three different lattices are isomorphicNamely 119871

119862(119866119872 119868) cong 119871

119875(119866119872 119868) cong 119871

119874(119866119872 119868)

Zhang et al [17] have ever given detailed approach tofind the reduction of a formal context which can keep thestructure of 119871(119866119872 119868) That is if there exists an attributesubset 119863 sube 119872 such that 119871(119866119863 119868

119863) cong 119871(119866119872 119868) then 119863

is called a consistent set of (119866119872 119868) And further if for all119889 isin 119863 119871(119866119863 minus 119889 119868

119863minus119889) ≇ 119871(119866119872 119868) then 119863 is called

a reduct of (119866119872 119868) where 119868119863

= 119868 cap (119866 times 119863) Accordingto this idea the attributes are classified into three typescore attribute relatively necessary attribute and absolutelyunnecessary attribute

In this paper for these four different lattices we stillstudy attribute reduction based on keeping structures ofthe lattices Analogously the attributes are classified intocore attribute relative necessary attribute and absolutelyunnecessary attribute To simplify their attribute reducts aredenote by 119863

119894 The set of core attributes is 119862

119894 that is 119862

119894=

cap119895isin120591

119863119894119895 the set of relatively necessary attributes is 119870

119894 that is

119870119894= cup

119895isin120591119863

119894119895minus cap

119895isin120591119863

119894119895 and the set of absolutely unnecessary

attributes is 119868119894 that is 119868

119894= 119872 minus cup

119895isin120591119863

119894119895 where 120591 is an index

set 119894 isin 119891 119888 119901 119900 which represents 119871(119866119872 119868) 119871119862(119866119872 119868)

119871119875(119866119872 119868) and 119871

119874(119866119872 119868) respectively

An example is given in the following to show the abovedefinitions

Example 2 Table 1 is a formal context (119866119872 119868) 119866 =

1 2 3 4 is an object set and 119872 = 119886 119887 119888 119889 119890 119891 is anattribute set Table 2 is its complementary context (119866119872 119868

119888

)

According to the definitions of formal concept prop-erty oriented concept and object oriented concept we canobtain the corresponding concept lattices The concept lat-tice 119871(119866119872 119868) and complementary context concept lattice119871119862(119866119872 119868) are shown in Figures 1 and 2 The property

Table 2 A formal context (119866119872 119868119888

)

119866 119886 119887 119888 119889 119890 119891

1 times times

2 times times times

3 times times times times times

4 times times

(G 0)

(13 d) (124 ab)

(24 abc)

(4 abcf)

(0M)

(1 abde)

Figure 1 119871(119866119872 119868)

(G 0)

(0M)

(234 e)

(24 de)

(2 def)

(123 f)

(3 abcef)

(13 cf)(23 ef)

Figure 2 119871119862(119866119872 119868)

oriented concept lattice 119871119875(119866119872 119868) and the object oriented

concept lattice 119871119874(119866119872 119868) are shown in Figures 3 and 4

respectively in which every set is denoted directly by listingits elements except 119866119872 and 0

For119871(119866119872 119868)119862119891= 119888 119889 119891119870

119891= 119886 119887 119868

119891= 119890119863

1198911=

119886 119888 119889 119891 and1198631198912

= 119887 119888 119889 119891For 119871

119862(119866119872 119868) 119862

119888= 119888 119889 119890 119891 119870

119888= 0 119868

119888= 119886 119887 and

119863119888= 119888 119889 119890 119891For 119871

119875(119866119872 119868) 119862

119901= 119888 119889 119890 119891119870

119901= 0 119868

119901= 119886 119887 and

119863119901= 119888 119889 119890 119891For 119871

119874(119866119872 119868) 119862

119900= 119888 119889 119890 119891 119870

119900= 0 119868

119900= 119886 119887 and

119863119900= 119888 119889 119890 119891In Example 2 we noticed that if we remove 119886 or 119887 from119872

the structures of four different lattices of the formal context


(234 abcdf)

(GM)

(2 abc)

(123 abcde)

(13 abde)

(3 d)

(0 0)

(23 abcd)(24 abcf)

Figure 3 119871119875(119866119872 119868)

(GM)

(13 de)

(1 e)

(14 ef)

(124 abcef)

(4 f)

(134 def)

(24 cf)

(0 0)

Figure 4 119871119874(119866119872 119868)

will not be changed That is if1198981015840

1= 119898

1015840

2for any119898

1119898

2isin 119872

then1198981 119898

2notin 119862

119894and119898

1 119898

2isin 119870

119894or119898

1 119898

2isin 119868

119894

In order to clarify the situation we presuppose that theformal context we study in this paper does not have the samecolumn Here we delete attribute 119887 from Tables 1 and 2For convenience we still use 119872 as attribute set But 119872 =

119886 119888 119889 119890 119891

3 Attribute Reduction Based onProperty Pictorial Diagram

In this section we mainly propose a method to find attributereducts of four different lattices based on the propertypictorial diagram of a formal context

31 Attribute Reduction of 119871(119866119872 119868) In the following wefirst give the definition of property pictorial diagram

Definition 3 Let (119866119872 119868) be a formal context 119867119898

=

(1198981015840

119898) | 119898 isin 119872 For any 119898119904 119898

119905isin 119872 if 1198981015840

119904sube 119898

1015840

119905 then

one denotes (1198981015840

119904 119898

119904) le (119898

1015840

119905 119898

119905) And (119867

119898 le) is called the

property pictorial diagram of (119866119872 119868)

(124 a)

(24 c)

(4 f)

(13 d)

(1 e)

Figure 5 (119867119898 le) of (119866119872 119868) in Table 1

In fact the Hasse diagram (119867119898 le) gives another expres-

sion of (119866119872 119868) The diagrammatic approach to formalcontext obtains the relations among attribute extents easily

Definition 4 (see [1]) 119886 is called a lower neighbor of 119887 if 119886 lt 119887

and there is no element of 119888 fulfilling 119886 lt 119888 lt 119887 In this case119887 is an upper neighbor of 119886 and one writes 119886 ≺ 119887

Based on this definition we can easily obtain upperneighbors and lower neighbors of each element (1198981015840

119898) in119867

119898 For any 119898 isin 119872 denote 119880

119898= 119898

119905isin 119872 | (119898

1015840

119898) ≺

(1198981015840

119905 119898

119905) and 119871

119898= 119898

119904isin 119872 | (119898

1015840

119904 119898

119904) ≺ (119898

1015840

119898) where119904 isin 119878 119905 isin 119879 (119878 and 119879 are index sets)

Example 5 (continue with Example 2) Consider the formalcontext in Table 1 we have 119886

1015840

= 124 1198881015840 = 24 1198891015840 = 131198901015840

= 1 and 1198911015840

= 4 According to Definition 3 we have119867

119898= (124 119886) (24 119888) (13 119889) (1 119890) (4 119891) and the property

pictorial diagram is shown in Figure 5 Thus we have

119880119886= 0 119880

119888= 119886 119880

119889= 0 119880

119890= 119886 119889 119880

119891= 119888

119871119886= 119888 119890 119871

119888= 119891 119871

119889= 119890 119871

119890= 0 119871

119891= 0

It is easy to see that the maximal elements of 119867119898have

no upper neighbor and the minimal elements of119867119898have no

lower neighbor We denote the set of maximal and minimalelements of119867

119898by Max(119867

119898) and Min(119867

119898) respectively

In [1] the arrow relation on the (119866119872 119868) was definedas follows 119892 119898hArr not(119892119868119898) and if 1198981015840

sube 1198991015840 and 119898

1015840

= 1198991015840

then 119892119868119899 where 119892 isin 119866 119898 119899 isin 119872 In the following we willgive a new method to obtain the arrow relation based onproperty pictorial diagram (119867

119898 le)

Theorem 6 Let (119866119872 119868) be a formal context and let (119867119898 le)

be its property pictorial diagram The following statementshold

(1) If (1198981015840

119898) isin Max(119867119898) then 119892 isin 119866 minus 119898

1015840

hArr 119892 119898(2) If (1198981015840

119898) notin Max(119867119898) then 119892 isin ⋂

119905isin119879119898

1015840

119905minus119898

1015840

hArr 119892

119898 where119898119905isin 119880

119898(119905 isin 119879)

Proof (1) Suppose (1198981015840

119898) isin Max(119867119898) Thus there does not

exist (1198981015840

119905 119898

119905) isin 119867

119898such that 1198981015840

sub 1198981015840

119905 And by 119892 isin 119866 minus 119898

1015840we havenot(119892119868119898) So from the definition of we have 119892 119898Hence for any (119898

1015840

119898) isin Max(119867119898) 119892 isin 119866 minus 119898

1015840

rArr 119892 119898Conversely because the formal context is canonical 119866 minus

1198981015840

= 0 And since (1198981015840

119898) isin Max(119867119898) and 119892 119898 we have


Table 3 The arrow relation of (119866119872 119868)

119866 119886 119888 119889 119890 119891

1 times times times

2 times times

3 times

4 times times times

119892 isin 119866 minus 1198981015840 from the definition of Thus for any (119898

1015840

119898) isin

Max(119867119898) 119892 119898 rArr 119892 isin 119866 minus 119898

1015840(2) Suppose (119898

1015840

119898) notin Max(119867119898) According to the

definition of maximal elements there exists (1198981015840

119905 119898

119905) isin 119867

119898

such that1198981015840

sub 1198981015840

119905 And by 119892 isin ⋂

119905isin119879119898

1015840

119905minus119898

1015840 we have 119892 notin 1198981015840

and 119892 isin 1198981015840

119905 that is not(119892119868119898) 119892119868119898

119905 So we have 119892 119898 from

the definition of Since (1198981015840

119898) notin Max(119867119898) there exist some119898

119905isin 119872 such

that 1198981015840

sub 1198981015840

119905 And by 119892 119898 we have 119892 notin 119898

1015840 and 119892 isin 1198981015840

119905

that is not(119892119868119898) 119892119868119898119905 So 119892 isin ⋂

119905isin119879119898

1015840

119905minus 119898

1015840

Example 7 (continue with Example 2) From Theorem 6 wecan obtain the arrow relation of Table 1 based on (119867

119898 le)

it is illustrated in Table 3

Here we recall an important definition as follows

Definition 8 (see [28]) Let 119871 be a lattice An element 119909 isin 119871 ismeet-irreducible if

(1) 119909 = 1 (in case 119871 has a unit)(2) 119909 = 119886 and 119887 implies 119909 = 119886 or 119909 = 119887 for all 119886 119887 isin 119871

We denote the set of meet-irreducible elements of119871(119866119872 119868) by119872(119871)

Based on the arrow relations Ganter and Wille gavethe method to judge whether an attribute concept is a meet-irreducible element of 119871(119866119872 119868)

Lemma 9 (see [1]) The following statements hold for everycontext (1198981015840

1198981015840lowast

) isin 119872(119871) hArr there is a 119892 isin 119866 with 119892 119898

According to the properties of meet-irreducible elementsof concept lattices Wang and Ma [20] gave the judgementmethod of absolutely unnecessary attributes

Lemma 10 (see [20]) If (119866119872 119868) is a context for any119898 isin 119872one has

119898 isin 119868119891lArrrArr (119898

1015840

1198981015840lowast

) notin 119872 (119871) (6)

Combining these two lemmas we have the followingresult

Theorem 11 Let (119866119872 119868) be a formal context and let (119867119898 le

) be its property pictorial diagram For any 119898 isin 119872 one has119898 isin 119868

119891hArr (119898

1015840

119898) notin Max(119867119898) and ⋂

119905isin119879119898

1015840

119905minus 119898

1015840

= 0 where119898

119905isin 119880

119898(119905 isin 119879)

Proof FromLemmas 9 and 10 it is easy to see that (1198981015840

1198981015840lowast

) notin

119872(119871) hArr there does not exist 119892 isin 119866 with 119892 119898

According to Theorem 6 we obtain that there does notexist119892 isin 119866with119892 119898 hArr (119898

1015840

119898) notin Max(119867119898) and⋂

119905isin119879119898

1015840

119905minus

1198981015840

= 0 Then this theorem is proved

Theorem 11 shows that if |119880119898| le 1 for 119898 isin 119872 then 119898 notin

119868119891Because the formal contexts we study do not have the

same column that is there is no relatively necessary attributewe can get the following statement

Theorem 12 Let (119866119872 119868) be a formal context One has119863119891=

119872 119868119891

By this theorem we can obtain an attribute reduct of119871(119866119872 119868) The steps are as follows

(1) Compute1198981015840 for all119898 isin 119872(2) Draw the property pictorial diagram (119867

119898 le)

(3) Find 119880119898 If |119880

119898| ge 2 and 119898

1015840

= ⋂119898119905isin119880119898

1198981015840

119905 then 119898 isin

119868119891

(4) Obtain an attribute reduct119863119891= 119872 119868

119891

Example 13 (continue with Example 5) Example 5 told usthat |119880

119886| = 0 |119880

119888| = 1 |119880

119889| = 0 |119880

119890| = 2 and |119880

119891| = 1

FromTheorem 11 we only need to check attribute 119890 Because119880119890= 119886 119889 and 119886

1015840

cap 1198891015840

= 1198901015840 we have 119890 isin 119868

119891 Thus 119863

119891=

119886 119888 119889 119891 The result is consistent with Example 2

32 Attribute Reduction of 119871119862(119866119872 119868) 119871

119874(119866119872 119868) and

119871119875(119866119872 119868) For a formal context its complementary con-

text is unique and 119871119862(119866119872 119868) cong 119871

119874(119866119872 119868) cong 119871

119875(119866119872 119868)

Therefore we will discuss the attribute reduction of thesethree different lattices based on the property pictorial dia-gram of original formal context

For the complementary context (119866119872 119868119888

) of (119866119872 119868) wedenote its property pictorial diagram by (119867

119862

119898 le)


be its property pictorial diagram For any (1198981015840

119898) isin 119867119898 one

has

(1) (sim 1198981015840

119898) isin 119867119862

119898

(2) (1198981015840

119898) ≺ (1198981015840

119905 119898

119905) hArr (sim 119898

1015840

119905 119898

119905) ≺ (sim 119898

1015840

119898) (119905 isin 119879)(3) Max(119867119862

119898) = (sim 119898

1015840

119898)(1198981015840

119898) isin Min(119867119898)

(4) 119867119898

cong 119867119862

119898

Proof

(1) From the definition of complementary context weknow that 1198981015840119888

=sim 1198981015840 Thus (sim 119898

1015840

119898) = (1198981015840119888

119898)Hence we have (sim 119898

1015840

119898) isin 119867119862

119898by Definition 4

(2) Consider (1198981015840

119898) ≺ (1198981015840

119905 119898

119905) hArr 119898

1015840

sub 1198981015840

119905hArrsim 119898

1015840

119905subsim

1198981015840

hArr (sim 1198981015840

119905 119898

119905) ≺ (sim 119898

1015840

119898)(3) It is easy to be obtained from (2)(4) It can be proved by (1) and (2)


(123 f)

(13 c)

(3 a)

(234 e)

(24 d)

Figure 6 (119867119862

119898 le) of (119866119872 119868

119888

) in Table 2

Table 4 The arrow relation of (119866119872 119868119888

)

119866 119886 119888 119889 119890 119891

1 times times

2 times times times


4 times times


) we have thefollowing result fromTheorems 6 and 14



(1) If (1198981015840

119898) isin Min(119867119898) then 119892 isin 119898

1015840

hArr 119892 119898 in(119866119872 119868

119888

)(2) If (1198981015840

119898) notin Min(119867119898) then 119892 isin 119898

1015840

minus ⋃119904isin119878

1198981015840

119904hArr 119892

119898 in (119866119872 119868119888

) where119898119904isin 119871

119898(119904 isin 119878)

Example 16 (continue with Example 2) Consider the formalcontext in Table 2 According to Definition 3 we have 119867

119862

119898=

(3 119886) (13 119888) (24 119889) (234 119890) (123 119891) and the property pic-torial diagram is in Figure 6

It is easy to verify Theorem 14 by Figures 5 and 6 ByTheorem 15 the arrow relation of Table 2 can be obtainedas Table 4

Combining Lemmas 9 and 10 we have the followingconclusion similar to Theorem 11


be its property pictorial diagram For any119898 isin 119872 one has119898 isin

119868119888hArr (119898

1015840

119898) notin Min(119867119898) and 119898

1015840

minus ⋃119904isin119878

1198981015840

119904= 0 where 119898

119904isin

119871119898(119904 isin 119878)

This theorem implies that if |119871119898| le 1 for 119898 isin 119872 then

119898 notin 119868119888

Similar to Theorem 12 we have the following result


119872 119868119888

By Theorem 18 we can obtain an attribute reduct of119871119862(119866119872 119868)

In [22] Medina studied attribute reduction of objectoriented concept lattice and property oriented concept latticeusing the relations between these two lattices and comple-mentary context concept lattice in a formal contextThemainconclusions are as follows

Theorem 19 (see [22]) Let (119866119872 119868) be a formal context Forall119898 isin 119872 one has the following

(1) 119898 isin 119868119901hArr 119898 isin 119868

119900hArr 119898 isin 119868

119888

(2) 119898 isin 119870119901hArr 119898 isin 119870

119900hArr 119898 isin 119870

119888

(3) 119898 isin 119862119901hArr 119898 isin 119862

119900hArr 119898 isin 119862

119888

(4) 119863119888= 119863

119901= 119863

119900

Combing Theorems 18 and 19 the corresponding reduc-tion process is as follows


119898 le)

(3) Find 119871119898 If |119871

119898| ge 2 and 119898

1015840

= ⋃119898119904isin119871119898

1198981015840

119904 then 119898 isin

119868119888

(4) Obtain attribute reducts119863119888= 119863

119901= 119863

119900= 119872 119868

119888

Example 20 (continue with Example 5) According toExample 5 we obtain |119871

119886| = 2 |119871

119888| = 1 |119871

119889| = 1 |119871

119890| = 1

and |119871119891| = 1 We only need to check attribute 119886 Because

119871119886= 119888 119890 and 119886

1015840

= 1198881015840

cup 1198901015840 we have 119886 isin 119868

119888by Theorem 17

Thereby 119863119888

= 119863119901

= 119863119900

= 119888 119889 119890 119891 These results areconsistent with Example 2

4 An Illustrated Example

Example 1 To illustrate the application of the methodproposed by this paper we use the data set of bacterialtaxonomy from UCI The data set contains six speciesand 16 phenotypic characters Table 5 shows the formalcontext (119866119872 119868) of the bacterial data set We denote119866 = 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 and119872 = 119886 119887 119888 119889 119890 119891 119892 ℎ 119894 119895 119896 119897 119898 119899 119900 119901 The species areEscherichia coli (1ndash3) Salmonella typhi (4ndash6) Klebsiella pneu-moniae (7ndash11) Proteus vulgaris (12ndash14) Proteus morganii (1516) and Serratia marcesens (17) respectively

First compute attribute extents 1198981015840 for all 119898 isin 119872 as

follows

1198861015840

= 3 6 12 13 141198871015840

= 1 2 3 4 5 6 7 8 9 10 11 171198881015840

= 1 4 5 6 7 8 9 10 11 15 171198891015840

= 1 2 3 7 8 9 10 11 12 13 14 15 161198901015840

= 2 3 15 16 171198911015840

= 7 8 9 10 11 12 171198921015840

= 7 8 9 10 12 14 15ℎ1015840

= 1 2 3 7 8 9 10 11 171198941015840

= 7 8 9 11 17


Table 5 Original formal context (119866119872 119868) from the bacterial data set

119866 119886 119887 119888 119889 119890 119891 119892 ℎ 119894 119895 119896 119897 119898 119899 119900 119901

1 times times times times times times


3 times times times times times times times

4 times times times

5 times times


7 times times times times times times times times times times times


9 times times times times times times times times times times times times




13 times times times





a

l

m

g

d b

o

p

j

h

n

c

i

k

f e


1198951015840

= 7 8 9 10 111198961015840

= 171198971015840

= 12 13 14 15119898

1015840

= 8 9 10 111198991015840

= 7 9 10 111199001015840

= 1 2 3 4 6 7 8 9 10 111199011015840

= 1 3 7 8 9 10 11

Second draw the property pictorial diagram Here forclarification every element of property pictorial diagram isdenoted directly by the corresponding attribute label whichis shown in Figure 7

Third for any119898 isin 119872 compute 119880119898and 119871

119898(Table 6)

According to Theorem 11 we only need to examineattributes 119894 119895 119896 and 119901 We have the following

119880119894= 119888 ℎ 119891 and 119888

1015840

cap ℎ1015840

cap 1198911015840

minus 1198941015840

= 10 119894 notin 119868119891

119880119895= 119888 119891 119901 and 119888

1015840

cap 1198911015840

cap 1199011015840

minus 1198951015840

= 0 119895 isin 119868119891

Table 6

119880119886= 0 119880

119887= 0

119871119886= 0 119871

119887= 119900 ℎ

119880119888= 0 119880

119889= 0

119871119888= 119894 119895 119871

119889= 119892 119897 119901

119880119890= 0 119880

119891= 0

119871119890= 119896 119871

119891= 119894 119895

119880119892= 119889 119880

ℎ= 119887

119871119892= 0 119871

ℎ= 119894 119901

119880119894= 119888 ℎ 119891 119880

119895= 119888 119891 119901

119871119894= 119896 119871

119895= 119898 119899

119880119896= 119890 119894 119880

119897= 119889

119871119896= 0 119871

119897= 0

119880119898= 119895 119880

119899= 119895

119871119898= 0 119871

119899= 0

119880119900= 119887 119880

119901= 119889 ℎ 119900

119871119900= 119901 119871

119901= 119895

119880119896= 119890 119894 and 119890

1015840

cap 1198941015840

minus 1198961015840

= 0 119896 isin 119868119891

119880119901= 119889 ℎ 119900 and 119889

1015840

cap ℎ1015840

cap 1199001015840

minus 1199011015840

= 2 119901 notin 119868119891

According to Theorem 17 we only need to examineattributes 119887 119888 119889 119891 ℎ and 119895 We have the following

119871119887= ℎ 119900 and 119887

1015840

minus ℎ1015840

cup 1199001015840

= 5 119887 notin 119868119888

119871119888= 119894 119895 and 119888

1015840

minus 1198941015840

cup 1198951015840

= 1 4 5 6 15 119888 notin 119868119888

119871119889= 119892 119897 119901 and 119889

1015840

minus 1198951015840

cup 1198971015840

cup 1199011015840

= 2 16 119889 notin 119868119888

119871119891= 119894 119895 and 119891

1015840

minus 1198941015840

cup 1198951015840

= 12 119891 notin 119868119888

119871ℎ= 119894 119901 and ℎ

1015840

minus 1198941015840

cup 1199011015840

= 2 ℎ notin 119868119888

119871119895= 119898 119899 and 119895

1015840

minus 1198981015840

cup 1198991015840

= 0 119895 isin 119868119888


Fourth we obtain119863119891= 119886 119887 119888 119889 119890 119891 119892 ℎ 119894 119897 119898 119899 119900 119901

119863119888= 119863

119900= 119863

119901= 119886 119887 119888 119889 119890 119891 119892 ℎ 119894 119896 119897 119898 119899 119900 119901

5 Conclusion

Attribute reduction to keep the lattice structure is an impor-tant issue in FCA On the basis of equivalent relation thepaper presents a new expression for a formal context which isnamed property pictorial diagram According to the propertypictorial diagram of original formal context we propose amethod to obtain attribute reducts of four different latticesusing the interconnection between arrow relation meet-irreducible elements and absolutely unnecessary attributesBased on the method in this paper we can study other typesof attribute reduction

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgment

The authors gratefully acknowledge the support of the Natu-ral Science Foundation of China (no 11371014 no 11071281and no 61202206)

References

[1] B Ganter and R Wille Formal Concept Analysis MathematicalFoundations Springer Berlin Germany 1999

[2] R Wille ldquoRestructuring lattice theory an approach based onhierarchies of conceptsrdquo in Ordered Sets I Rival Ed pp 445ndash470 Reidel Dordrecht The Netherlands 1982

[3] Z Pawlak ldquoRough setsrdquo International Journal of Computer andInformation Sciences vol 11 no 5 pp 341ndash356 1982

[4] Z Pawlak Rough Sets Theoretical Aspects of Reasoning aboutData Kluwer Academic Publishers Dordrecht The Nether-lands 1991

[5] I Duntsch and G Gediga ldquoModal-style operators in qualitativedata analysisrdquo in Proceedings of the 2nd IEEE International Con-ference on Data Mining (ICDM rsquo02) pp 155ndash162 WashingtonDC USA December 2002

[6] Y Y Yao ldquoA comparative study of formal concept analysisand rough set theory in data analysisrdquo in Proceedings of 4thInternational Conference on Rough Sets and Current Trends inComputing (RSCTC rsquo04) pp 59ndash68 Uppsala Sweden 2004

[7] C Shi Z Niu and T Wang ldquoConsidering the relationshipbetween RST and FCArdquo in Proceedings of the 3rd InternationalConference on Knowledge Discovery and Data Mining (WKDDrsquo10) pp 224ndash227 January 2010

[8] H Wang and W X Zhang ldquoRelationships between con-cept lattice and rough setrdquo in Artificial Intelligence and SoftComputingmdashICAISC 2006 vol 4029 of Lecture Notes in Com-puter Science pp 538ndash547 Springer Berlin Germany 2006

[9] L Wei and J J Qi ldquoRelation between concept lattice reductionand rough set reductionrdquo Knowledge-Based Systems vol 23 no8 pp 934ndash938 2010

[10] M Liu M Shao W Zhang and C Wu ldquoReduction method forconcept lattices based on rough set theory and its applicationrdquo

Computers amp Mathematics with Applications vol 53 no 9 pp1390ndash1410 2007

[11] X Wang ldquoApproaches to attribute reduction in concept latticesbased on rough set theoryrdquo International Journal of HybridInformation Technology vol 5 no 2 pp 67ndash80 2012

[12] J Poelmans D I Ignatov S O Kuznetsov and G DedeneldquoFuzzy and rough formal concept analysis a surveyrdquo Interna-tional Journal of General Systems vol 43 no 2 pp 105ndash1342014

[13] A Skowron and C Rauszer ldquoThe discernibility matrices andfunctions in information systemsrdquo inHandbook of Applicationsand Advances of the Rough SetsTheory R lowinski Ed KluwerDordrecht the Netherlands 1992

[14] D Q Miao Y Zhao Y Y Yao H X Li and F F Xu ldquoRelativereducts in consistent and inconsistent decision tables of thePawlak rough set modelrdquo Information Sciences vol 179 no 24pp 4140ndash4150 2009

[15] D Ye and Z Chen ldquoAn improved discernibility matrix forcomputing all reducts of an inconsistent decision tablerdquo in Pro-ceedings of the 5th IEEE International Conference on CognitiveInformatics (ICCI rsquo06) pp 305ndash308 July 2006

[16] Y Yao and Y Zhao ldquoDiscernibility matrix simplification forconstructing attribute reductsrdquo Information Sciences vol 179no 7 pp 867ndash882 2009

[17] W X Zhang L Wei and J J Qi ldquoAttribute reduction theoryand approach to concept latticerdquo Science in China F InformationSciences vol 48 no 6 pp 713ndash726 2005

[18] L Wei J Qi and W Zhang ldquoAttribute reduction theory ofconcept lattice based on decision formal contextsrdquo Science inChina F Information Sciences vol 51 no 7 pp 910ndash923 2008

[19] M Q Liu and L Wei ldquoThe reduction theory of object orientedconcept lattices and property oriented concept latticesrdquo inProceedings of the 4th International Conference on Rough SetsandKnowledge Tschnology (RSKT rsquo09) vol 5589 ofLectureNotesin Computer Science pp 587ndash593 2009

[20] XWang and J M Ma ldquoA novel approach to attribute reductionin concept latticesrdquo in Proceedings of RSKT vol 4062 of LectureNotes in Artificial Intelligence pp 522ndash529 Springer BerlinGermany 2006

[21] X Wang and W Zhang ldquoRelations of attribute reduc-tion between object and property oriented concept latticesrdquoKnowledge-Based Systems vol 21 no 5 pp 398ndash403 2008

[22] J Medina ldquoRelating attribute reduction in formal object-oriented and property-oriented concept latticesrdquo Computersand Mathematics with Applications vol 64 no 6 pp 1992ndash2002 2012

[23] W Z Wu Y Leung and J S Mi ldquoGranular computing andknowledge reduction in formal contextsrdquo IEEE Transactions onKnowledge and Data Engineering vol 21 no 10 pp 1461ndash14742009

[24] J Li C Mei and Y Lv ldquoA heuristic knowledge-reductionmethod for decision formal contextsrdquo Computers and Mathe-matics with Applications vol 61 no 4 pp 1096ndash1106 2011

[25] M W Shao Y Leung and W Z Wu ldquoRule acquisition andcomplexity reduction in formal decision contextsrdquo InternationalJournal of Approximate Reasoning vol 55 no 1 part 2 pp 259ndash274 2014

[26] C Aswani Kumar and S Srinivas ldquoConcept lattice reductionusing fuzzy k-Means clusteringrdquo Expert Systems with Applica-tions vol 37 no 3 pp 2696ndash2704 2010


[27] Y Y Yao ldquoConcept lattices in rough set theoryrdquo in Proceedings ofthe IEEE Annual Meeting of the North American Fuzzy Informa-tion Processing Society (NAFIPS rsquo04) pp 796ndash801 WashingtonDC USA June 2004

[28] B A Davey and H A Priestley Introduction to Lattices andOrder Cambridge University Press Cambridge UK 2002

Submit your manuscripts athttpwwwhindawicom

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Hindawi Publishing Corporationhttpwwwhindawicom

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Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Algebra

Discrete Dynamics in Nature and Society



Decision SciencesAdvances in

Discrete MathematicsJournal of


Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of


of meet-irreducible elements in the original concept latticeanddescribed attribute characteristics usingmeet-irreducibleelements For the object oriented concept lattice and propertyoriented concept latticeWang andZhang [21] further studiedsuch reduction and described attribute characteristics usingmeet- (join-) irreducible elements Medina [22] obtained therelation of attribute reduction among complementary contextconcept lattice object oriented concept lattice and propertyoriented concept lattice Wu et al [23] proposed the granularreduction from the viewpoint of keeping object conceptsand discussed information granules and their properties ina formal context Li et al [24] constructed ldquoa new frameworkof knowledge reduction in which the capacity of one conceptlattice implying another is defined tomeasure the significanceof the attributes in a consistent decision formal contextrdquo Shaoet al [25] formulated ldquoan approach to attribute reductionin formal decision contexts such that rules extracted fromthe reduced formal decision contexts are identical to thatextracted from the initial formal decision contextsrdquo AswaniKumar and Srinivas [26] proposed ldquoa new method based onfuzzy K-means clustering for reducing the size of the conceptlatticesrdquo

In this paper based on [20 22] we discuss the reductionof a formal context which can keep the lattice structure usingarrow relation defined by Ganter and Wille [1] First of allwe obtain the new approach to acquiring the arrow relation on the basis of a property pictorial diagram defined byus Then combining the relations between arrow relation meet-irreducible elements and attribute characteristics wepresent an approach to construct attribute reducts of conceptlattice complementary context concept lattice object ori-ented concept lattice and property oriented concept lattice

The rest of the paper is organized as follows Section 2reviews the basic notions of FCA Section 3 constructsattribute reducts based on property pictorial diagram of aformal context Section 4 uses anUCI database to explain ourapproach in more detail Section 5 concludes the paper

2 Preliminaries

In this section we recall some basic notions in formal conceptanalysis [1 2]

Definition 1 (see [1]) A formal context (119866119872 119868) consists oftwo sets 119866 and 119872 and a relation 119868 between 119866 and 119872 Theelements of119866 are called the objects and the elements of119872 arecalled the attributes of the context In order to express that anobject 119892 is in a relation 119868with an attribute119898 we write 119892119868119898 or(119892119898) isin 119868 and read it as ldquothe object 119892 has the attribute119898rdquo

Let (119866119872 119868) be a formal context For 119860 sube 119866 119861 sube 119872 twooperators are defined as follows

119860lowast


1198611015840


(1)

(119860 119861) is called a formal concept for short a concept if andonly if 119860lowast

= 119861 119860 = 1198611015840 where 119860 is called the extent of

the formal concept and 119861 is called its intent Particularly

(1198981015840

1198981015840lowast

) is a formal concept and is called an attributeconcept and119898

1015840 is called attribute extent [1]The set of all con-cepts of (119866119872 119868) is denoted by 119871(119866119872 119868) For any (119860

1 119861

1)

(1198602 119861

2) isin 119871(119866119872 119868) we have (119860

1 119861

1) ⩽ (119860

2 119861

2) hArr 119860

1sube

1198602(hArr 119861

1supe 119861

2) And the infimum and supremum are given

by

(1198601 119861

1) and (119860

2 119861

2) = (119860

1cap 119860

2 (119861

1cup 119861

2)1015840lowast

)

(1198601 119861

1) or (119860

2 119861

2) = ((119860

1cup 119860

2)lowast1015840

1198611cap 119861

2)

(2)

Thus 119871(119866119872 119868) is a complete lattice and is called theconcept lattice

To simplify for all 119892 isin 119866 for all 119898 isin 119872 119892lowast and 1198981015840

are replaced by 119892lowast and 119898

1015840 respectively If for all 119892 isin 119866119892lowast

= 0 119892lowast

= 119872 and for all 119898 isin 119872 1198981015840

= 0 1198981015840

= 119866then the formal context is called canonical That is to saythere is neither full rowcolumn nor empty rowcolumn ina formal context Noting this an irregular formal context canbe regularized by removing the full rowcolumn and emptyrowcolumn Such way of regularization causes no effect onthe analysis results of the formal context Thus without lossof generality we suppose that all formal contexts are finite andcanonical in this paper

Let (119866119872 119868) be a formal context Denote 119868119888

= (119866 times

119872) 119868 then we call (119866119872 119868119888

) the complementary contextof (119866119872 119868) [1] the mappings defined in (1) on (119866119872 119868

119888

) aredenoted by lowast119888 and 1015840119888

All concepts of (119866119872 119868119888

) are denoted by 119871119862(119866119872 119868)

which is also a complete latticeLet (119866119872 119868) be a formal context For any 119860 sube 119866 119861 sube 119872

Duntsch and Gediga defined a pair of approximate operators ◻ as follows [5]

119860= 119898 isin 119872 | 119898

1015840

cap 119860 = 0

119861= 119892 isin 119866119892

lowast

cap 119861 = 0

119860◻

= 119898 isin 119872 | 1198981015840

sube 119860

119861◻

= 119892 isin 119866 | 119892lowast

sube 119861

(3)

A pair (119860 119861)119860 sube 119866 119861 sube 119872 is called a property orientedconcept if119860

= 119861 and119861◻

= 119860 All property oriented conceptsof (119866119872 119868) are denoted by 119871

119875(119866119872 119868) For any (119860

1 119861

1)

(1198602 119861

2) isin 119871

119875(119866119872 119868) (119860

1 119861

1) ⩽ (119860

2 119861

2) hArr 119860

1sube 119860

2(hArr

1198611sube 119861

2) And the infimum and supremum are given by

(1198601 119861

1) and (119860

2 119861

2) = (119860

1cap 119860

2 (119861

1cap 119861

2)◻

)

(1198601 119861

1) or (119860

2 119861

2) = ((119860

1cup 119860

2)◻

1198611cup 119861

2)

(4)

Thus 119871119875(119866119872 119868) is a complete lattice and is called the

property oriented concept latticeBased on the work of Duntsch and Gediga Yao proposed

the object oriented concept lattice [6]A pair (119860 119861) 119860 sube 119866 119861 sube 119872 is called an object

oriented concept if 119860◻

= 119861 and 119861

= 119860 All object orientedconcepts of (119866119872 119868) are denoted by 119871

119874(119866119872 119868) For any



119866 119886 119887 119888 119889 119890 119891


2 times times times

3 times


(1198601 119861

1) (119860

2 119861

2) isin 119871

119874(119866119872 119868) (119860

1 119861

1) ⩽ (119860

2 119861

2) hArr

1198601

sube 1198602(hArr 119861

1sube 119861


are given by

(1198601 119861

1) and (119860

2 119861

2) = ((119860

1cap 119860

2)◻

1198611cap 119861

2)

(1198601 119861

1) or (119860

2 119861

2) = (119860

1cup 119860

2 (119861

1cup 119861

2)◻

)

(5)







relations among 119871119862(119866119872 119868) 119871

119875(119866119872 119868) and 119871

119874(119866119872 119868)


119862(119866119872 119868) cong 119871

119875(119866119872 119868) cong 119871

119874(119866119872 119868)


119863) cong 119871(119866119872 119868) then 119863



a reduct of (119866119872 119868) where 119868119863




119894 that is 119862

119894=

cap119895isin120591


119894 that is

119870119894= cup

119895isin120591119863

119894119895minus cap

119895isin120591119863



119894= 119872 minus cup

119895isin120591119863

119894119895 where 120591 is an index


119871119875(119866119872 119868) and 119871

119874(119866119872 119868) respectively




119888

)



)

119866 119886 119887 119888 119889 119890 119891

1 times times

2 times times times


4 times times

(G 0)

(13 d) (124 ab)

(24 abc)

(4 abcf)

(0M)

(1 abde)

Figure 1 119871(119866119872 119868)

(G 0)

(0M)

(234 e)

(24 de)

(2 def)

(123 f)

(3 abcef)

(13 cf)(23 ef)

Figure 2 119871119862(119866119872 119868)




For119871(119866119872 119868)119862119891= 119888 119889 119891119870

119891= 119886 119887 119868

119891= 119890119863

1198911=

119886 119888 119889 119891 and1198631198912

= 119887 119888 119889 119891For 119871

119862(119866119872 119868) 119862

119888= 119888 119889 119890 119891 119870

119888= 0 119868

119888= 119886 119887 and

119863119888= 119888 119889 119890 119891For 119871

119875(119866119872 119868) 119862

119901= 119888 119889 119890 119891119870

119901= 0 119868

119901= 119886 119887 and

119863119901= 119888 119889 119890 119891For 119871

119874(119866119872 119868) 119862

119900= 119888 119889 119890 119891 119870

119900= 0 119868

119900= 119886 119887 and




(234 abcdf)

(GM)

(2 abc)

(123 abcde)

(13 abde)

(3 d)

(0 0)

(23 abcd)(24 abcf)

Figure 3 119871119875(119866119872 119868)

(GM)

(13 de)

(1 e)

(14 ef)

(124 abcef)

(4 f)

(134 def)

(24 cf)

(0 0)

Figure 4 119871119874(119866119872 119868)


1= 119898

1015840

2for any119898

1119898

2isin 119872

then1198981 119898

2notin 119862

119894and119898

1 119898

2isin 119870

119894or119898

1 119898

2isin 119868

119894


119886 119888 119889 119890 119891





=

(1198981015840

119898) | 119898 isin 119872 For any 119898119904 119898

119905isin 119872 if 1198981015840

119904sube 119898

1015840

119905 then

one denotes (1198981015840

119904 119898

119904) le (119898

1015840

119905 119898

119905) And (119867



(124 a)

(24 c)

(4 f)

(13 d)

(1 e)







119898) in119867


119898= 119898

119905isin 119872 | (119898

1015840

119898) ≺

(1198981015840

119905 119898

119905) and 119871

119898= 119898

119904isin 119872 | (119898

1015840

119904 119898

119904) ≺ (119898

1015840



1015840

= 124 1198881015840 = 24 1198891015840 = 131198901015840

= 1 and 1198911015840


119898= (124 119886) (24 119888) (13 119889) (1 119890) (4 119891) and the property


119880119886= 0 119880

119888= 119886 119880

119889= 0 119880

119890= 119886 119889 119880

119891= 119888

119871119886= 119888 119890 119871

119888= 119891 119871

119889= 119890 119871

119890= 0 119871

119891= 0




119898by Max(119867

119898) and Min(119867



sube 1198991015840 and 119898

1015840

= 1198991015840


119898 le)



(1) If (1198981015840


1015840

hArr 119892 119898(2) If (1198981015840


119905isin119879119898

1015840

119905minus119898

1015840

hArr 119892

119898 where119898119905isin 119880

119898(119905 isin 119879)

Proof (1) Suppose (1198981015840


exist (1198981015840

119905 119898

119905) isin 119867

119898such that 1198981015840

sub 1198981015840

119905 And by 119892 isin 119866 minus 119898


1015840


1015840


1198981015840

= 0 And since (1198981015840




119866 119886 119888 119889 119890 119891

1 times times times

2 times times

3 times

4 times times times


1015840

119898) isin


1015840(2) Suppose (119898

1015840



119905 119898

119905) isin 119867

119898

such that1198981015840

sub 1198981015840

119905 And by 119892 isin ⋂

119905isin119879119898

1015840

119905minus119898

1015840 we have 119892 notin 1198981015840

and 119892 isin 1198981015840

119905 that is not(119892119868119898) 119892119868119898

119905 So we have 119892 119898 from



119905isin 119872 such

that 1198981015840

sub 1198981015840


1015840 and 119892 isin 1198981015840

119905


119905isin119879119898

1015840

119905minus 119898

1015840


119898 le)








1198981015840lowast




119898 isin 119868119891lArrrArr (119898

1015840

1198981015840lowast

) notin 119872 (119871) (6)




119891hArr (119898

1015840

119898) notin Max(119867119898) and ⋂

119905isin119879119898

1015840

119905minus 119898

1015840

= 0 where119898

119905isin 119880

119898(119905 isin 119879)


1198981015840lowast

) notin



1015840

119898) notin Max(119867119898) and⋂

119905isin119879119898

1015840

119905minus

1198981015840






119872 119868119891



119898 le)

(3) Find 119880119898 If |119880

119898| ge 2 and 119898

1015840

= ⋂119898119905isin119880119898

1198981015840

119905 then 119898 isin

119868119891


119891


119886| = 0 |119880

119888| = 1 |119880

119889| = 0 |119880

119890| = 2 and |119880

119891| = 1


1015840

cap 1198891015840

= 1198901015840 we have 119890 isin 119868

119891 Thus 119863

119891=



119874(119866119872 119868) and



119874(119866119872 119868) cong 119871

119875(119866119872 119868)




119862

119898 le)



119898) isin 119867119898 one

has

(1) (sim 1198981015840

119898) isin 119867119862

119898

(2) (1198981015840

119898) ≺ (1198981015840

119905 119898

119905) hArr (sim 119898

1015840

119905 119898

119905) ≺ (sim 119898

1015840

119898) (119905 isin 119879)(3) Max(119867119862

119898) = (sim 119898

1015840

119898)(1198981015840

119898) isin Min(119867119898)

(4) 119867119898

cong 119867119862

119898

Proof


=sim 1198981015840 Thus (sim 119898

1015840

119898) = (1198981015840119888


1015840

119898) isin 119867119862


(2) Consider (1198981015840

119898) ≺ (1198981015840

119905 119898

119905) hArr 119898

1015840

sub 1198981015840

119905hArrsim 119898

1015840

119905subsim

1198981015840

hArr (sim 1198981015840

119905 119898

119905) ≺ (sim 119898

1015840



(123 f)

(13 c)

(3 a)

(234 e)

(24 d)

Figure 6 (119867119862

119898 le) of (119866119872 119868

119888

) in Table 2


)

119866 119886 119888 119889 119890 119891

1 times times

2 times times times


4 times times





(1) If (1198981015840


1015840

hArr 119892 119898 in(119866119872 119868

119888

)(2) If (1198981015840


1015840

minus ⋃119904isin119878

1198981015840

119904hArr 119892

119898 in (119866119872 119868119888

) where119898119904isin 119871

119898(119904 isin 119878)


119862

119898=






119868119888hArr (119898

1015840

119898) notin Min(119867119898) and 119898

1015840

minus ⋃119904isin119878

1198981015840

119904= 0 where 119898

119904isin

119871119898(119904 isin 119878)


119898 notin 119868119888



119872 119868119888




(1) 119898 isin 119868119901hArr 119898 isin 119868

119900hArr 119898 isin 119868

119888

(2) 119898 isin 119870119901hArr 119898 isin 119870

119900hArr 119898 isin 119870

119888

(3) 119898 isin 119862119901hArr 119898 isin 119862

119900hArr 119898 isin 119862

119888

(4) 119863119888= 119863

119901= 119863

119900



119898 le)

(3) Find 119871119898 If |119871

119898| ge 2 and 119898

1015840

= ⋃119898119904isin119871119898

1198981015840

119904 then 119898 isin

119868119888


119901= 119863

119900= 119872 119868

119888


119886| = 2 |119871

119888| = 1 |119871

119889| = 1 |119871

119890| = 1


119871119886= 119888 119890 and 119886

1015840

= 1198881015840

cup 1198901015840 we have 119886 isin 119868

119888by Theorem 17

Thereby 119863119888

= 119863119901

= 119863119900





follows

1198861015840

= 3 6 12 13 141198871015840

= 1 2 3 4 5 6 7 8 9 10 11 171198881015840

= 1 4 5 6 7 8 9 10 11 15 171198891015840

= 1 2 3 7 8 9 10 11 12 13 14 15 161198901015840

= 2 3 15 16 171198911015840

= 7 8 9 10 11 12 171198921015840

= 7 8 9 10 12 14 15ℎ1015840

= 1 2 3 7 8 9 10 11 171198941015840

= 7 8 9 11 17



119866 119886 119887 119888 119889 119890 119891 119892 ℎ 119894 119895 119896 119897 119898 119899 119900 119901




4 times times times

5 times times













a

l

m

g

d b

o

p

j

h

n

c

i

k

f e


1198951015840

= 7 8 9 10 111198961015840

= 171198971015840

= 12 13 14 15119898

1015840

= 8 9 10 111198991015840

= 7 9 10 111199001015840

= 1 2 3 4 6 7 8 9 10 111199011015840

= 1 3 7 8 9 10 11



119898(Table 6)


119880119894= 119888 ℎ 119891 and 119888

1015840

cap ℎ1015840

cap 1198911015840

minus 1198941015840

= 10 119894 notin 119868119891

119880119895= 119888 119891 119901 and 119888

1015840

cap 1198911015840

cap 1199011015840

minus 1198951015840

= 0 119895 isin 119868119891

Table 6

119880119886= 0 119880

119887= 0

119871119886= 0 119871

119887= 119900 ℎ

119880119888= 0 119880

119889= 0

119871119888= 119894 119895 119871

119889= 119892 119897 119901

119880119890= 0 119880

119891= 0

119871119890= 119896 119871

119891= 119894 119895

119880119892= 119889 119880

ℎ= 119887

119871119892= 0 119871

ℎ= 119894 119901

119880119894= 119888 ℎ 119891 119880

119895= 119888 119891 119901

119871119894= 119896 119871

119895= 119898 119899

119880119896= 119890 119894 119880

119897= 119889

119871119896= 0 119871

119897= 0

119880119898= 119895 119880

119899= 119895

119871119898= 0 119871

119899= 0

119880119900= 119887 119880

119901= 119889 ℎ 119900

119871119900= 119901 119871

119901= 119895

119880119896= 119890 119894 and 119890

1015840

cap 1198941015840

minus 1198961015840

= 0 119896 isin 119868119891

119880119901= 119889 ℎ 119900 and 119889

1015840

cap ℎ1015840

cap 1199001015840

minus 1199011015840

= 2 119901 notin 119868119891


119871119887= ℎ 119900 and 119887

1015840

minus ℎ1015840

cup 1199001015840

= 5 119887 notin 119868119888

119871119888= 119894 119895 and 119888

1015840

minus 1198941015840

cup 1198951015840

= 1 4 5 6 15 119888 notin 119868119888

119871119889= 119892 119897 119901 and 119889

1015840

minus 1198951015840

cup 1198971015840

cup 1199011015840

= 2 16 119889 notin 119868119888

119871119891= 119894 119895 and 119891

1015840

minus 1198941015840

cup 1198951015840

= 12 119891 notin 119868119888

119871ℎ= 119894 119901 and ℎ

1015840

minus 1198941015840

cup 1199011015840

= 2 ℎ notin 119868119888

119871119895= 119898 119899 and 119895

1015840

minus 1198981015840

cup 1198991015840

= 0 119895 isin 119868119888



119863119888= 119863

119900= 119863

119901= 119886 119887 119888 119889 119890 119891 119892 ℎ 119894 119896 119897 119898 119899 119900 119901

5 Conclusion




Acknowledgment


References






































Volume 2014




Journal of











Journal of


Function Spaces






Algebra











119866 119886 119887 119888 119889 119890 119891


2 times times times

3 times


(1198601 119861

1) (119860

2 119861

2) isin 119871

119874(119866119872 119868) (119860

1 119861

1) ⩽ (119860

2 119861

2) hArr

1198601

sube 1198602(hArr 119861

1sube 119861


are given by

(1198601 119861

1) and (119860

2 119861

2) = ((119860

1cap 119860

2)◻

1198611cap 119861

2)

(1198601 119861

1) or (119860

2 119861

2) = (119860

1cup 119860

2 (119861

1cup 119861

2)◻

)

(5)







relations among 119871119862(119866119872 119868) 119871

119875(119866119872 119868) and 119871

119874(119866119872 119868)


119862(119866119872 119868) cong 119871

119875(119866119872 119868) cong 119871

119874(119866119872 119868)


119863) cong 119871(119866119872 119868) then 119863



a reduct of (119866119872 119868) where 119868119863




119894 that is 119862

119894=

cap119895isin120591


119894 that is

119870119894= cup

119895isin120591119863

119894119895minus cap

119895isin120591119863



119894= 119872 minus cup

119895isin120591119863

119894119895 where 120591 is an index


119871119875(119866119872 119868) and 119871

119874(119866119872 119868) respectively




119888

)



)

119866 119886 119887 119888 119889 119890 119891

1 times times

2 times times times


4 times times

(G 0)

(13 d) (124 ab)

(24 abc)

(4 abcf)

(0M)

(1 abde)

Figure 1 119871(119866119872 119868)

(G 0)

(0M)

(234 e)

(24 de)

(2 def)

(123 f)

(3 abcef)

(13 cf)(23 ef)

Figure 2 119871119862(119866119872 119868)




For119871(119866119872 119868)119862119891= 119888 119889 119891119870

119891= 119886 119887 119868

119891= 119890119863

1198911=

119886 119888 119889 119891 and1198631198912

= 119887 119888 119889 119891For 119871

119862(119866119872 119868) 119862

119888= 119888 119889 119890 119891 119870

119888= 0 119868

119888= 119886 119887 and

119863119888= 119888 119889 119890 119891For 119871

119875(119866119872 119868) 119862

119901= 119888 119889 119890 119891119870

119901= 0 119868

119901= 119886 119887 and

119863119901= 119888 119889 119890 119891For 119871

119874(119866119872 119868) 119862

119900= 119888 119889 119890 119891 119870

119900= 0 119868

119900= 119886 119887 and




(234 abcdf)

(GM)

(2 abc)

(123 abcde)

(13 abde)

(3 d)

(0 0)

(23 abcd)(24 abcf)

Figure 3 119871119875(119866119872 119868)

(GM)

(13 de)

(1 e)

(14 ef)

(124 abcef)

(4 f)

(134 def)

(24 cf)

(0 0)

Figure 4 119871119874(119866119872 119868)


1= 119898

1015840

2for any119898

1119898

2isin 119872

then1198981 119898

2notin 119862

119894and119898

1 119898

2isin 119870

119894or119898

1 119898

2isin 119868

119894


119886 119888 119889 119890 119891





=

(1198981015840

119898) | 119898 isin 119872 For any 119898119904 119898

119905isin 119872 if 1198981015840

119904sube 119898

1015840

119905 then

one denotes (1198981015840

119904 119898

119904) le (119898

1015840

119905 119898

119905) And (119867



(124 a)

(24 c)

(4 f)

(13 d)

(1 e)







119898) in119867


119898= 119898

119905isin 119872 | (119898

1015840

119898) ≺

(1198981015840

119905 119898

119905) and 119871

119898= 119898

119904isin 119872 | (119898

1015840

119904 119898

119904) ≺ (119898

1015840



1015840

= 124 1198881015840 = 24 1198891015840 = 131198901015840

= 1 and 1198911015840


119898= (124 119886) (24 119888) (13 119889) (1 119890) (4 119891) and the property


119880119886= 0 119880

119888= 119886 119880

119889= 0 119880

119890= 119886 119889 119880

119891= 119888

119871119886= 119888 119890 119871

119888= 119891 119871

119889= 119890 119871

119890= 0 119871

119891= 0




119898by Max(119867

119898) and Min(119867



sube 1198991015840 and 119898

1015840

= 1198991015840


119898 le)



(1) If (1198981015840


1015840

hArr 119892 119898(2) If (1198981015840


119905isin119879119898

1015840

119905minus119898

1015840

hArr 119892

119898 where119898119905isin 119880

119898(119905 isin 119879)

Proof (1) Suppose (1198981015840


exist (1198981015840

119905 119898

119905) isin 119867

119898such that 1198981015840

sub 1198981015840

119905 And by 119892 isin 119866 minus 119898


1015840


1015840


1198981015840

= 0 And since (1198981015840




119866 119886 119888 119889 119890 119891

1 times times times

2 times times

3 times

4 times times times


1015840

119898) isin


1015840(2) Suppose (119898

1015840



119905 119898

119905) isin 119867

119898

such that1198981015840

sub 1198981015840

119905 And by 119892 isin ⋂

119905isin119879119898

1015840

119905minus119898

1015840 we have 119892 notin 1198981015840

and 119892 isin 1198981015840

119905 that is not(119892119868119898) 119892119868119898

119905 So we have 119892 119898 from



119905isin 119872 such

that 1198981015840

sub 1198981015840


1015840 and 119892 isin 1198981015840

119905


119905isin119879119898

1015840

119905minus 119898

1015840


119898 le)








1198981015840lowast




119898 isin 119868119891lArrrArr (119898

1015840

1198981015840lowast

) notin 119872 (119871) (6)




119891hArr (119898

1015840

119898) notin Max(119867119898) and ⋂

119905isin119879119898

1015840

119905minus 119898

1015840

= 0 where119898

119905isin 119880

119898(119905 isin 119879)


1198981015840lowast

) notin



1015840

119898) notin Max(119867119898) and⋂

119905isin119879119898

1015840

119905minus

1198981015840






119872 119868119891



119898 le)

(3) Find 119880119898 If |119880

119898| ge 2 and 119898

1015840

= ⋂119898119905isin119880119898

1198981015840

119905 then 119898 isin

119868119891


119891


119886| = 0 |119880

119888| = 1 |119880

119889| = 0 |119880

119890| = 2 and |119880

119891| = 1


1015840

cap 1198891015840

= 1198901015840 we have 119890 isin 119868

119891 Thus 119863

119891=



119874(119866119872 119868) and



119874(119866119872 119868) cong 119871

119875(119866119872 119868)




119862

119898 le)



119898) isin 119867119898 one

has

(1) (sim 1198981015840

119898) isin 119867119862

119898

(2) (1198981015840

119898) ≺ (1198981015840

119905 119898

119905) hArr (sim 119898

1015840

119905 119898

119905) ≺ (sim 119898

1015840

119898) (119905 isin 119879)(3) Max(119867119862

119898) = (sim 119898

1015840

119898)(1198981015840

119898) isin Min(119867119898)

(4) 119867119898

cong 119867119862

119898

Proof


=sim 1198981015840 Thus (sim 119898

1015840

119898) = (1198981015840119888


1015840

119898) isin 119867119862


(2) Consider (1198981015840

119898) ≺ (1198981015840

119905 119898

119905) hArr 119898

1015840

sub 1198981015840

119905hArrsim 119898

1015840

119905subsim

1198981015840

hArr (sim 1198981015840

119905 119898

119905) ≺ (sim 119898

1015840



(123 f)

(13 c)

(3 a)

(234 e)

(24 d)

Figure 6 (119867119862

119898 le) of (119866119872 119868

119888

) in Table 2


)

119866 119886 119888 119889 119890 119891

1 times times

2 times times times


4 times times





(1) If (1198981015840


1015840

hArr 119892 119898 in(119866119872 119868

119888

)(2) If (1198981015840


1015840

minus ⋃119904isin119878

1198981015840

119904hArr 119892

119898 in (119866119872 119868119888

) where119898119904isin 119871

119898(119904 isin 119878)


119862

119898=






119868119888hArr (119898

1015840

119898) notin Min(119867119898) and 119898

1015840

minus ⋃119904isin119878

1198981015840

119904= 0 where 119898

119904isin

119871119898(119904 isin 119878)


119898 notin 119868119888



119872 119868119888




(1) 119898 isin 119868119901hArr 119898 isin 119868

119900hArr 119898 isin 119868

119888

(2) 119898 isin 119870119901hArr 119898 isin 119870

119900hArr 119898 isin 119870

119888

(3) 119898 isin 119862119901hArr 119898 isin 119862

119900hArr 119898 isin 119862

119888

(4) 119863119888= 119863

119901= 119863

119900



119898 le)

(3) Find 119871119898 If |119871

119898| ge 2 and 119898

1015840

= ⋃119898119904isin119871119898

1198981015840

119904 then 119898 isin

119868119888


119901= 119863

119900= 119872 119868

119888


119886| = 2 |119871

119888| = 1 |119871

119889| = 1 |119871

119890| = 1


119871119886= 119888 119890 and 119886

1015840

= 1198881015840

cup 1198901015840 we have 119886 isin 119868

119888by Theorem 17

Thereby 119863119888

= 119863119901

= 119863119900





follows

1198861015840

= 3 6 12 13 141198871015840

= 1 2 3 4 5 6 7 8 9 10 11 171198881015840

= 1 4 5 6 7 8 9 10 11 15 171198891015840

= 1 2 3 7 8 9 10 11 12 13 14 15 161198901015840

= 2 3 15 16 171198911015840

= 7 8 9 10 11 12 171198921015840

= 7 8 9 10 12 14 15ℎ1015840

= 1 2 3 7 8 9 10 11 171198941015840

= 7 8 9 11 17



119866 119886 119887 119888 119889 119890 119891 119892 ℎ 119894 119895 119896 119897 119898 119899 119900 119901




4 times times times

5 times times













a

l

m

g

d b

o

p

j

h

n

c

i

k

f e


1198951015840

= 7 8 9 10 111198961015840

= 171198971015840

= 12 13 14 15119898

1015840

= 8 9 10 111198991015840

= 7 9 10 111199001015840

= 1 2 3 4 6 7 8 9 10 111199011015840

= 1 3 7 8 9 10 11



119898(Table 6)


119880119894= 119888 ℎ 119891 and 119888

1015840

cap ℎ1015840

cap 1198911015840

minus 1198941015840

= 10 119894 notin 119868119891

119880119895= 119888 119891 119901 and 119888

1015840

cap 1198911015840

cap 1199011015840

minus 1198951015840

= 0 119895 isin 119868119891

Table 6

119880119886= 0 119880

119887= 0

119871119886= 0 119871

119887= 119900 ℎ

119880119888= 0 119880

119889= 0

119871119888= 119894 119895 119871

119889= 119892 119897 119901

119880119890= 0 119880

119891= 0

119871119890= 119896 119871

119891= 119894 119895

119880119892= 119889 119880

ℎ= 119887

119871119892= 0 119871

ℎ= 119894 119901

119880119894= 119888 ℎ 119891 119880

119895= 119888 119891 119901

119871119894= 119896 119871

119895= 119898 119899

119880119896= 119890 119894 119880

119897= 119889

119871119896= 0 119871

119897= 0

119880119898= 119895 119880

119899= 119895

119871119898= 0 119871

119899= 0

119880119900= 119887 119880

119901= 119889 ℎ 119900

119871119900= 119901 119871

119901= 119895

119880119896= 119890 119894 and 119890

1015840

cap 1198941015840

minus 1198961015840

= 0 119896 isin 119868119891

119880119901= 119889 ℎ 119900 and 119889

1015840

cap ℎ1015840

cap 1199001015840

minus 1199011015840

= 2 119901 notin 119868119891


119871119887= ℎ 119900 and 119887

1015840

minus ℎ1015840

cup 1199001015840

= 5 119887 notin 119868119888

119871119888= 119894 119895 and 119888

1015840

minus 1198941015840

cup 1198951015840

= 1 4 5 6 15 119888 notin 119868119888

119871119889= 119892 119897 119901 and 119889

1015840

minus 1198951015840

cup 1198971015840

cup 1199011015840

= 2 16 119889 notin 119868119888

119871119891= 119894 119895 and 119891

1015840

minus 1198941015840

cup 1198951015840

= 12 119891 notin 119868119888

119871ℎ= 119894 119901 and ℎ

1015840

minus 1198941015840

cup 1199011015840

= 2 ℎ notin 119868119888

119871119895= 119898 119899 and 119895

1015840

minus 1198981015840

cup 1198991015840

= 0 119895 isin 119868119888



119863119888= 119863

119900= 119863

119901= 119886 119887 119888 119889 119890 119891 119892 ℎ 119894 119896 119897 119898 119899 119900 119901

5 Conclusion




Acknowledgment


References






































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










(234 abcdf)

(GM)

(2 abc)

(123 abcde)

(13 abde)

(3 d)

(0 0)

(23 abcd)(24 abcf)

Figure 3 119871119875(119866119872 119868)

(GM)

(13 de)

(1 e)

(14 ef)

(124 abcef)

(4 f)

(134 def)

(24 cf)

(0 0)

Figure 4 119871119874(119866119872 119868)


1= 119898

1015840

2for any119898

1119898

2isin 119872

then1198981 119898

2notin 119862

119894and119898

1 119898

2isin 119870

119894or119898

1 119898

2isin 119868

119894


119886 119888 119889 119890 119891





=

(1198981015840

119898) | 119898 isin 119872 For any 119898119904 119898

119905isin 119872 if 1198981015840

119904sube 119898

1015840

119905 then

one denotes (1198981015840

119904 119898

119904) le (119898

1015840

119905 119898

119905) And (119867



(124 a)

(24 c)

(4 f)

(13 d)

(1 e)







119898) in119867


119898= 119898

119905isin 119872 | (119898

1015840

119898) ≺

(1198981015840

119905 119898

119905) and 119871

119898= 119898

119904isin 119872 | (119898

1015840

119904 119898

119904) ≺ (119898

1015840



1015840

= 124 1198881015840 = 24 1198891015840 = 131198901015840

= 1 and 1198911015840


119898= (124 119886) (24 119888) (13 119889) (1 119890) (4 119891) and the property


119880119886= 0 119880

119888= 119886 119880

119889= 0 119880

119890= 119886 119889 119880

119891= 119888

119871119886= 119888 119890 119871

119888= 119891 119871

119889= 119890 119871

119890= 0 119871

119891= 0




119898by Max(119867

119898) and Min(119867



sube 1198991015840 and 119898

1015840

= 1198991015840


119898 le)



(1) If (1198981015840


1015840

hArr 119892 119898(2) If (1198981015840


119905isin119879119898

1015840

119905minus119898

1015840

hArr 119892

119898 where119898119905isin 119880

119898(119905 isin 119879)

Proof (1) Suppose (1198981015840


exist (1198981015840

119905 119898

119905) isin 119867

119898such that 1198981015840

sub 1198981015840

119905 And by 119892 isin 119866 minus 119898


1015840


1015840


1198981015840

= 0 And since (1198981015840




119866 119886 119888 119889 119890 119891

1 times times times

2 times times

3 times

4 times times times


1015840

119898) isin


1015840(2) Suppose (119898

1015840



119905 119898

119905) isin 119867

119898

such that1198981015840

sub 1198981015840

119905 And by 119892 isin ⋂

119905isin119879119898

1015840

119905minus119898

1015840 we have 119892 notin 1198981015840

and 119892 isin 1198981015840

119905 that is not(119892119868119898) 119892119868119898

119905 So we have 119892 119898 from



119905isin 119872 such

that 1198981015840

sub 1198981015840


1015840 and 119892 isin 1198981015840

119905


119905isin119879119898

1015840

119905minus 119898

1015840


119898 le)








1198981015840lowast




119898 isin 119868119891lArrrArr (119898

1015840

1198981015840lowast

) notin 119872 (119871) (6)




119891hArr (119898

1015840

119898) notin Max(119867119898) and ⋂

119905isin119879119898

1015840

119905minus 119898

1015840

= 0 where119898

119905isin 119880

119898(119905 isin 119879)


1198981015840lowast

) notin



1015840

119898) notin Max(119867119898) and⋂

119905isin119879119898

1015840

119905minus

1198981015840






119872 119868119891



119898 le)

(3) Find 119880119898 If |119880

119898| ge 2 and 119898

1015840

= ⋂119898119905isin119880119898

1198981015840

119905 then 119898 isin

119868119891


119891


119886| = 0 |119880

119888| = 1 |119880

119889| = 0 |119880

119890| = 2 and |119880

119891| = 1


1015840

cap 1198891015840

= 1198901015840 we have 119890 isin 119868

119891 Thus 119863

119891=



119874(119866119872 119868) and



119874(119866119872 119868) cong 119871

119875(119866119872 119868)




119862

119898 le)



119898) isin 119867119898 one

has

(1) (sim 1198981015840

119898) isin 119867119862

119898

(2) (1198981015840

119898) ≺ (1198981015840

119905 119898

119905) hArr (sim 119898

1015840

119905 119898

119905) ≺ (sim 119898

1015840

119898) (119905 isin 119879)(3) Max(119867119862

119898) = (sim 119898

1015840

119898)(1198981015840

119898) isin Min(119867119898)

(4) 119867119898

cong 119867119862

119898

Proof


=sim 1198981015840 Thus (sim 119898

1015840

119898) = (1198981015840119888


1015840

119898) isin 119867119862


(2) Consider (1198981015840

119898) ≺ (1198981015840

119905 119898

119905) hArr 119898

1015840

sub 1198981015840

119905hArrsim 119898

1015840

119905subsim

1198981015840

hArr (sim 1198981015840

119905 119898

119905) ≺ (sim 119898

1015840



(123 f)

(13 c)

(3 a)

(234 e)

(24 d)

Figure 6 (119867119862

119898 le) of (119866119872 119868

119888

) in Table 2


)

119866 119886 119888 119889 119890 119891

1 times times

2 times times times


4 times times





(1) If (1198981015840


1015840

hArr 119892 119898 in(119866119872 119868

119888

)(2) If (1198981015840


1015840

minus ⋃119904isin119878

1198981015840

119904hArr 119892

119898 in (119866119872 119868119888

) where119898119904isin 119871

119898(119904 isin 119878)


119862

119898=






119868119888hArr (119898

1015840

119898) notin Min(119867119898) and 119898

1015840

minus ⋃119904isin119878

1198981015840

119904= 0 where 119898

119904isin

119871119898(119904 isin 119878)


119898 notin 119868119888



119872 119868119888




(1) 119898 isin 119868119901hArr 119898 isin 119868

119900hArr 119898 isin 119868

119888

(2) 119898 isin 119870119901hArr 119898 isin 119870

119900hArr 119898 isin 119870

119888

(3) 119898 isin 119862119901hArr 119898 isin 119862

119900hArr 119898 isin 119862

119888

(4) 119863119888= 119863

119901= 119863

119900



119898 le)

(3) Find 119871119898 If |119871

119898| ge 2 and 119898

1015840

= ⋃119898119904isin119871119898

1198981015840

119904 then 119898 isin

119868119888


119901= 119863

119900= 119872 119868

119888


119886| = 2 |119871

119888| = 1 |119871

119889| = 1 |119871

119890| = 1


119871119886= 119888 119890 and 119886

1015840

= 1198881015840

cup 1198901015840 we have 119886 isin 119868

119888by Theorem 17

Thereby 119863119888

= 119863119901

= 119863119900





follows

1198861015840

= 3 6 12 13 141198871015840

= 1 2 3 4 5 6 7 8 9 10 11 171198881015840

= 1 4 5 6 7 8 9 10 11 15 171198891015840

= 1 2 3 7 8 9 10 11 12 13 14 15 161198901015840

= 2 3 15 16 171198911015840

= 7 8 9 10 11 12 171198921015840

= 7 8 9 10 12 14 15ℎ1015840

= 1 2 3 7 8 9 10 11 171198941015840

= 7 8 9 11 17



119866 119886 119887 119888 119889 119890 119891 119892 ℎ 119894 119895 119896 119897 119898 119899 119900 119901




4 times times times

5 times times













a

l

m

g

d b

o

p

j

h

n

c

i

k

f e


1198951015840

= 7 8 9 10 111198961015840

= 171198971015840

= 12 13 14 15119898

1015840

= 8 9 10 111198991015840

= 7 9 10 111199001015840

= 1 2 3 4 6 7 8 9 10 111199011015840

= 1 3 7 8 9 10 11



119898(Table 6)


119880119894= 119888 ℎ 119891 and 119888

1015840

cap ℎ1015840

cap 1198911015840

minus 1198941015840

= 10 119894 notin 119868119891

119880119895= 119888 119891 119901 and 119888

1015840

cap 1198911015840

cap 1199011015840

minus 1198951015840

= 0 119895 isin 119868119891

Table 6

119880119886= 0 119880

119887= 0

119871119886= 0 119871

119887= 119900 ℎ

119880119888= 0 119880

119889= 0

119871119888= 119894 119895 119871

119889= 119892 119897 119901

119880119890= 0 119880

119891= 0

119871119890= 119896 119871

119891= 119894 119895

119880119892= 119889 119880

ℎ= 119887

119871119892= 0 119871

ℎ= 119894 119901

119880119894= 119888 ℎ 119891 119880

119895= 119888 119891 119901

119871119894= 119896 119871

119895= 119898 119899

119880119896= 119890 119894 119880

119897= 119889

119871119896= 0 119871

119897= 0

119880119898= 119895 119880

119899= 119895

119871119898= 0 119871

119899= 0

119880119900= 119887 119880

119901= 119889 ℎ 119900

119871119900= 119901 119871

119901= 119895

119880119896= 119890 119894 and 119890

1015840

cap 1198941015840

minus 1198961015840

= 0 119896 isin 119868119891

119880119901= 119889 ℎ 119900 and 119889

1015840

cap ℎ1015840

cap 1199001015840

minus 1199011015840

= 2 119901 notin 119868119891


119871119887= ℎ 119900 and 119887

1015840

minus ℎ1015840

cup 1199001015840

= 5 119887 notin 119868119888

119871119888= 119894 119895 and 119888

1015840

minus 1198941015840

cup 1198951015840

= 1 4 5 6 15 119888 notin 119868119888

119871119889= 119892 119897 119901 and 119889

1015840

minus 1198951015840

cup 1198971015840

cup 1199011015840

= 2 16 119889 notin 119868119888

119871119891= 119894 119895 and 119891

1015840

minus 1198941015840

cup 1198951015840

= 12 119891 notin 119868119888

119871ℎ= 119894 119901 and ℎ

1015840

minus 1198941015840

cup 1199011015840

= 2 ℎ notin 119868119888

119871119895= 119898 119899 and 119895

1015840

minus 1198981015840

cup 1198991015840

= 0 119895 isin 119868119888



119863119888= 119863

119900= 119863

119901= 119886 119887 119888 119889 119890 119891 119892 ℎ 119894 119896 119897 119898 119899 119900 119901

5 Conclusion




Acknowledgment


References






































Volume 2014




Journal of











Journal of


Function Spaces






Algebra











119866 119886 119888 119889 119890 119891

1 times times times

2 times times

3 times

4 times times times


1015840

119898) isin


1015840(2) Suppose (119898

1015840



119905 119898

119905) isin 119867

119898

such that1198981015840

sub 1198981015840

119905 And by 119892 isin ⋂

119905isin119879119898

1015840

119905minus119898

1015840 we have 119892 notin 1198981015840

and 119892 isin 1198981015840

119905 that is not(119892119868119898) 119892119868119898

119905 So we have 119892 119898 from



119905isin 119872 such

that 1198981015840

sub 1198981015840


1015840 and 119892 isin 1198981015840

119905


119905isin119879119898

1015840

119905minus 119898

1015840


119898 le)








1198981015840lowast




119898 isin 119868119891lArrrArr (119898

1015840

1198981015840lowast

) notin 119872 (119871) (6)




119891hArr (119898

1015840

119898) notin Max(119867119898) and ⋂

119905isin119879119898

1015840

119905minus 119898

1015840

= 0 where119898

119905isin 119880

119898(119905 isin 119879)


1198981015840lowast

) notin



1015840

119898) notin Max(119867119898) and⋂

119905isin119879119898

1015840

119905minus

1198981015840






119872 119868119891



119898 le)

(3) Find 119880119898 If |119880

119898| ge 2 and 119898

1015840

= ⋂119898119905isin119880119898

1198981015840

119905 then 119898 isin

119868119891


119891


119886| = 0 |119880

119888| = 1 |119880

119889| = 0 |119880

119890| = 2 and |119880

119891| = 1


1015840

cap 1198891015840

= 1198901015840 we have 119890 isin 119868

119891 Thus 119863

119891=



119874(119866119872 119868) and



119874(119866119872 119868) cong 119871

119875(119866119872 119868)




119862

119898 le)



119898) isin 119867119898 one

has

(1) (sim 1198981015840

119898) isin 119867119862

119898

(2) (1198981015840

119898) ≺ (1198981015840

119905 119898

119905) hArr (sim 119898

1015840

119905 119898

119905) ≺ (sim 119898

1015840

119898) (119905 isin 119879)(3) Max(119867119862

119898) = (sim 119898

1015840

119898)(1198981015840

119898) isin Min(119867119898)

(4) 119867119898

cong 119867119862

119898

Proof


=sim 1198981015840 Thus (sim 119898

1015840

119898) = (1198981015840119888


1015840

119898) isin 119867119862


(2) Consider (1198981015840

119898) ≺ (1198981015840

119905 119898

119905) hArr 119898

1015840

sub 1198981015840

119905hArrsim 119898

1015840

119905subsim

1198981015840

hArr (sim 1198981015840

119905 119898

119905) ≺ (sim 119898

1015840



(123 f)

(13 c)

(3 a)

(234 e)

(24 d)

Figure 6 (119867119862

119898 le) of (119866119872 119868

119888

) in Table 2


)

119866 119886 119888 119889 119890 119891

1 times times

2 times times times


4 times times





(1) If (1198981015840


1015840

hArr 119892 119898 in(119866119872 119868

119888

)(2) If (1198981015840


1015840

minus ⋃119904isin119878

1198981015840

119904hArr 119892

119898 in (119866119872 119868119888

) where119898119904isin 119871

119898(119904 isin 119878)


119862

119898=






119868119888hArr (119898

1015840

119898) notin Min(119867119898) and 119898

1015840

minus ⋃119904isin119878

1198981015840

119904= 0 where 119898

119904isin

119871119898(119904 isin 119878)


119898 notin 119868119888



119872 119868119888




(1) 119898 isin 119868119901hArr 119898 isin 119868

119900hArr 119898 isin 119868

119888

(2) 119898 isin 119870119901hArr 119898 isin 119870

119900hArr 119898 isin 119870

119888

(3) 119898 isin 119862119901hArr 119898 isin 119862

119900hArr 119898 isin 119862

119888

(4) 119863119888= 119863

119901= 119863

119900



119898 le)

(3) Find 119871119898 If |119871

119898| ge 2 and 119898

1015840

= ⋃119898119904isin119871119898

1198981015840

119904 then 119898 isin

119868119888


119901= 119863

119900= 119872 119868

119888


119886| = 2 |119871

119888| = 1 |119871

119889| = 1 |119871

119890| = 1


119871119886= 119888 119890 and 119886

1015840

= 1198881015840

cup 1198901015840 we have 119886 isin 119868

119888by Theorem 17

Thereby 119863119888

= 119863119901

= 119863119900





follows

1198861015840

= 3 6 12 13 141198871015840

= 1 2 3 4 5 6 7 8 9 10 11 171198881015840

= 1 4 5 6 7 8 9 10 11 15 171198891015840

= 1 2 3 7 8 9 10 11 12 13 14 15 161198901015840

= 2 3 15 16 171198911015840

= 7 8 9 10 11 12 171198921015840

= 7 8 9 10 12 14 15ℎ1015840

= 1 2 3 7 8 9 10 11 171198941015840

= 7 8 9 11 17



119866 119886 119887 119888 119889 119890 119891 119892 ℎ 119894 119895 119896 119897 119898 119899 119900 119901




4 times times times

5 times times













a

l

m

g

d b

o

p

j

h

n

c

i

k

f e


1198951015840

= 7 8 9 10 111198961015840

= 171198971015840

= 12 13 14 15119898

1015840

= 8 9 10 111198991015840

= 7 9 10 111199001015840

= 1 2 3 4 6 7 8 9 10 111199011015840

= 1 3 7 8 9 10 11



119898(Table 6)


119880119894= 119888 ℎ 119891 and 119888

1015840

cap ℎ1015840

cap 1198911015840

minus 1198941015840

= 10 119894 notin 119868119891

119880119895= 119888 119891 119901 and 119888

1015840

cap 1198911015840

cap 1199011015840

minus 1198951015840

= 0 119895 isin 119868119891

Table 6

119880119886= 0 119880

119887= 0

119871119886= 0 119871

119887= 119900 ℎ

119880119888= 0 119880

119889= 0

119871119888= 119894 119895 119871

119889= 119892 119897 119901

119880119890= 0 119880

119891= 0

119871119890= 119896 119871

119891= 119894 119895

119880119892= 119889 119880

ℎ= 119887

119871119892= 0 119871

ℎ= 119894 119901

119880119894= 119888 ℎ 119891 119880

119895= 119888 119891 119901

119871119894= 119896 119871

119895= 119898 119899

119880119896= 119890 119894 119880

119897= 119889

119871119896= 0 119871

119897= 0

119880119898= 119895 119880

119899= 119895

119871119898= 0 119871

119899= 0

119880119900= 119887 119880

119901= 119889 ℎ 119900

119871119900= 119901 119871

119901= 119895

119880119896= 119890 119894 and 119890

1015840

cap 1198941015840

minus 1198961015840

= 0 119896 isin 119868119891

119880119901= 119889 ℎ 119900 and 119889

1015840

cap ℎ1015840

cap 1199001015840

minus 1199011015840

= 2 119901 notin 119868119891


119871119887= ℎ 119900 and 119887

1015840

minus ℎ1015840

cup 1199001015840

= 5 119887 notin 119868119888

119871119888= 119894 119895 and 119888

1015840

minus 1198941015840

cup 1198951015840

= 1 4 5 6 15 119888 notin 119868119888

119871119889= 119892 119897 119901 and 119889

1015840

minus 1198951015840

cup 1198971015840

cup 1199011015840

= 2 16 119889 notin 119868119888

119871119891= 119894 119895 and 119891

1015840

minus 1198941015840

cup 1198951015840

= 12 119891 notin 119868119888

119871ℎ= 119894 119901 and ℎ

1015840

minus 1198941015840

cup 1199011015840

= 2 ℎ notin 119868119888

119871119895= 119898 119899 and 119895

1015840

minus 1198981015840

cup 1198991015840

= 0 119895 isin 119868119888



119863119888= 119863

119900= 119863

119901= 119886 119887 119888 119889 119890 119891 119892 ℎ 119894 119896 119897 119898 119899 119900 119901

5 Conclusion




Acknowledgment


References






































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










(123 f)

(13 c)

(3 a)

(234 e)

(24 d)

Figure 6 (119867119862

119898 le) of (119866119872 119868

119888

) in Table 2


)

119866 119886 119888 119889 119890 119891

1 times times

2 times times times


4 times times





(1) If (1198981015840


1015840

hArr 119892 119898 in(119866119872 119868

119888

)(2) If (1198981015840


1015840

minus ⋃119904isin119878

1198981015840

119904hArr 119892

119898 in (119866119872 119868119888

) where119898119904isin 119871

119898(119904 isin 119878)


119862

119898=






119868119888hArr (119898

1015840

119898) notin Min(119867119898) and 119898

1015840

minus ⋃119904isin119878

1198981015840

119904= 0 where 119898

119904isin

119871119898(119904 isin 119878)


119898 notin 119868119888



119872 119868119888




(1) 119898 isin 119868119901hArr 119898 isin 119868

119900hArr 119898 isin 119868

119888

(2) 119898 isin 119870119901hArr 119898 isin 119870

119900hArr 119898 isin 119870

119888

(3) 119898 isin 119862119901hArr 119898 isin 119862

119900hArr 119898 isin 119862

119888

(4) 119863119888= 119863

119901= 119863

119900



119898 le)

(3) Find 119871119898 If |119871

119898| ge 2 and 119898

1015840

= ⋃119898119904isin119871119898

1198981015840

119904 then 119898 isin

119868119888


119901= 119863

119900= 119872 119868

119888


119886| = 2 |119871

119888| = 1 |119871

119889| = 1 |119871

119890| = 1


119871119886= 119888 119890 and 119886

1015840

= 1198881015840

cup 1198901015840 we have 119886 isin 119868

119888by Theorem 17

Thereby 119863119888

= 119863119901

= 119863119900





follows

1198861015840

= 3 6 12 13 141198871015840

= 1 2 3 4 5 6 7 8 9 10 11 171198881015840

= 1 4 5 6 7 8 9 10 11 15 171198891015840

= 1 2 3 7 8 9 10 11 12 13 14 15 161198901015840

= 2 3 15 16 171198911015840

= 7 8 9 10 11 12 171198921015840

= 7 8 9 10 12 14 15ℎ1015840

= 1 2 3 7 8 9 10 11 171198941015840

= 7 8 9 11 17



119866 119886 119887 119888 119889 119890 119891 119892 ℎ 119894 119895 119896 119897 119898 119899 119900 119901




4 times times times

5 times times













a

l

m

g

d b

o

p

j

h

n

c

i

k

f e


1198951015840

= 7 8 9 10 111198961015840

= 171198971015840

= 12 13 14 15119898

1015840

= 8 9 10 111198991015840

= 7 9 10 111199001015840

= 1 2 3 4 6 7 8 9 10 111199011015840

= 1 3 7 8 9 10 11



119898(Table 6)


119880119894= 119888 ℎ 119891 and 119888

1015840

cap ℎ1015840

cap 1198911015840

minus 1198941015840

= 10 119894 notin 119868119891

119880119895= 119888 119891 119901 and 119888

1015840

cap 1198911015840

cap 1199011015840

minus 1198951015840

= 0 119895 isin 119868119891

Table 6

119880119886= 0 119880

119887= 0

119871119886= 0 119871

119887= 119900 ℎ

119880119888= 0 119880

119889= 0

119871119888= 119894 119895 119871

119889= 119892 119897 119901

119880119890= 0 119880

119891= 0

119871119890= 119896 119871

119891= 119894 119895

119880119892= 119889 119880

ℎ= 119887

119871119892= 0 119871

ℎ= 119894 119901

119880119894= 119888 ℎ 119891 119880

119895= 119888 119891 119901

119871119894= 119896 119871

119895= 119898 119899

119880119896= 119890 119894 119880

119897= 119889

119871119896= 0 119871

119897= 0

119880119898= 119895 119880

119899= 119895

119871119898= 0 119871

119899= 0

119880119900= 119887 119880

119901= 119889 ℎ 119900

119871119900= 119901 119871

119901= 119895

119880119896= 119890 119894 and 119890

1015840

cap 1198941015840

minus 1198961015840

= 0 119896 isin 119868119891

119880119901= 119889 ℎ 119900 and 119889

1015840

cap ℎ1015840

cap 1199001015840

minus 1199011015840

= 2 119901 notin 119868119891


119871119887= ℎ 119900 and 119887

1015840

minus ℎ1015840

cup 1199001015840

= 5 119887 notin 119868119888

119871119888= 119894 119895 and 119888

1015840

minus 1198941015840

cup 1198951015840

= 1 4 5 6 15 119888 notin 119868119888

119871119889= 119892 119897 119901 and 119889

1015840

minus 1198951015840

cup 1198971015840

cup 1199011015840

= 2 16 119889 notin 119868119888

119871119891= 119894 119895 and 119891

1015840

minus 1198941015840

cup 1198951015840

= 12 119891 notin 119868119888

119871ℎ= 119894 119901 and ℎ

1015840

minus 1198941015840

cup 1199011015840

= 2 ℎ notin 119868119888

119871119895= 119898 119899 and 119895

1015840

minus 1198981015840

cup 1198991015840

= 0 119895 isin 119868119888



119863119888= 119863

119900= 119863

119901= 119886 119887 119888 119889 119890 119891 119892 ℎ 119894 119896 119897 119898 119899 119900 119901

5 Conclusion




Acknowledgment


References






































Volume 2014




Journal of











Journal of


Function Spaces






Algebra











119866 119886 119887 119888 119889 119890 119891 119892 ℎ 119894 119895 119896 119897 119898 119899 119900 119901




4 times times times

5 times times













a

l

m

g

d b

o

p

j

h

n

c

i

k

f e


1198951015840

= 7 8 9 10 111198961015840

= 171198971015840

= 12 13 14 15119898

1015840

= 8 9 10 111198991015840

= 7 9 10 111199001015840

= 1 2 3 4 6 7 8 9 10 111199011015840

= 1 3 7 8 9 10 11



119898(Table 6)


119880119894= 119888 ℎ 119891 and 119888

1015840

cap ℎ1015840

cap 1198911015840

minus 1198941015840

= 10 119894 notin 119868119891

119880119895= 119888 119891 119901 and 119888

1015840

cap 1198911015840

cap 1199011015840

minus 1198951015840

= 0 119895 isin 119868119891

Table 6

119880119886= 0 119880

119887= 0

119871119886= 0 119871

119887= 119900 ℎ

119880119888= 0 119880

119889= 0

119871119888= 119894 119895 119871

119889= 119892 119897 119901

119880119890= 0 119880

119891= 0

119871119890= 119896 119871

119891= 119894 119895

119880119892= 119889 119880

ℎ= 119887

119871119892= 0 119871

ℎ= 119894 119901

119880119894= 119888 ℎ 119891 119880

119895= 119888 119891 119901

119871119894= 119896 119871

119895= 119898 119899

119880119896= 119890 119894 119880

119897= 119889

119871119896= 0 119871

119897= 0

119880119898= 119895 119880

119899= 119895

119871119898= 0 119871

119899= 0

119880119900= 119887 119880

119901= 119889 ℎ 119900

119871119900= 119901 119871

119901= 119895

119880119896= 119890 119894 and 119890

1015840

cap 1198941015840

minus 1198961015840

= 0 119896 isin 119868119891

119880119901= 119889 ℎ 119900 and 119889

1015840

cap ℎ1015840

cap 1199001015840

minus 1199011015840

= 2 119901 notin 119868119891


119871119887= ℎ 119900 and 119887

1015840

minus ℎ1015840

cup 1199001015840

= 5 119887 notin 119868119888

119871119888= 119894 119895 and 119888

1015840

minus 1198941015840

cup 1198951015840

= 1 4 5 6 15 119888 notin 119868119888

119871119889= 119892 119897 119901 and 119889

1015840

minus 1198951015840

cup 1198971015840

cup 1199011015840

= 2 16 119889 notin 119868119888

119871119891= 119894 119895 and 119891

1015840

minus 1198941015840

cup 1198951015840

= 12 119891 notin 119868119888

119871ℎ= 119894 119901 and ℎ

1015840

minus 1198941015840

cup 1199011015840

= 2 ℎ notin 119868119888

119871119895= 119898 119899 and 119895

1015840

minus 1198981015840

cup 1198991015840

= 0 119895 isin 119868119888



119863119888= 119863

119900= 119863

119901= 119886 119887 119888 119889 119890 119891 119892 ℎ 119894 119896 119897 119898 119899 119900 119901

5 Conclusion




Acknowledgment


References






































Volume 2014




Journal of











Journal of


Function Spaces






Algebra











119863119888= 119863

119900= 119863

119901= 119886 119887 119888 119889 119890 119891 119892 ℎ 119894 119896 119897 119898 119899 119900 119901

5 Conclusion




Acknowledgment


References






































Volume 2014




Journal of











Journal of


Function Spaces






Algebra



















Volume 2014




Journal of











Journal of


Function Spaces






Algebra









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