Research Article Characterizations of ( 2,2,0)...

Research ArticleCharacterizations of119873(2 2 0) Algebras

Fang-an Deng Lu Chen ShouHeng Tuo and ShengZhang Ren

School of Mathematics and Computer Science Shaanxi University of Technology Hanzhong 723001 China

Correspondence should be addressed to Fang-an Deng dengfangans126com

Received 11 July 2015 Revised 20 November 2015 Accepted 8 December 2015

Academic Editor Antonio M Cegarra

Copyright copy 2016 Fang-an Deng et al 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

The so-called ideal and subalgebra and some additional concepts of119873(2 2 0) algebras are discussed A partial order and congruencerelations on119873(2 2 0) algebras are also proposed and some properties are investigated

1 Introduction

In the past years fuzzy algebras and their axiomatizationhave become important topics in theoretical research andin the applications of fuzzy logic The implication con-nective plays a crucial role in fuzzy logic and reasoning[1 2] Recently some authors studied fuzzy implicationsfrom different perspectives [3] Naturally it is meaningful toinvestigate the common properties of some important fuzzyimplications used in fuzzy logic Consequentially ProfessorWu [4] introduced a class of fuzzy implication algebras FI-algebras for short in 1990

In the past two decades some authors focused on FI-algebras Various interesting properties of FI-algebras [56] regular FI-algebras [4 7 8] commutative FI-algebras[9] 119882119889-FI-algebras [10] and other kinds of FI-algebras[11] were reported and some concepts of filter ideal andfuzzy filter of FI-algebras were proposed [4 12ndash18] Rela-tionships between FI-algebra and BCK-algebra [19 20] MV-algebras [21] and Rough set algebras [22 23] were partlyinvestigated and FI-algebras were axiomatized [24] In therecent work the relationships between these FI-algebras andseveral famous fuzzy algebras were systematically discussed[25ndash30]

In 1900Hilbert proposed the famous problem11986710 whichis the tenth problem of Hilbert11986710 argues whether there is auseful approach to determine that the general Diophantineequation is solvable Until 1970 this problem had beensolved by Y Matiyasevich It is clear that there is not amethod to judge whether the Diophantine equation can besolved In 1987 Siekmann and Szabo studied the unification

problem related to 11986710 It was concluded that unificationproblem of 119863119860-rewriting systems cannot be predicated Andit was stated that for every axiomatic system 120595 satisfying119863119860 sube 120595 sube 11986710 it is impossible to determine the problemof 120595ndash unification problem ⟨119878 = 119905⟩120595 119904 119905 isin 1198791 where1198791 is a set of term combined by the symbols oplus and otimes

[31]Our work constructed a new algebra system It was

called the 119873(2 2 0) algebra which is more general than the119863119860-rewriting systems We studied the basic properties of119873(2 2 0) algebra (119878 lowast Δ 0) In addition if the operation lowast

is idempotent then (119878 lowast Δ 0) is a rewriting system In thiscase it is impossible to determine the problem of 119873(2 2 0)algebra unification problem Analogously if the operation Δis nilpotent then (119878 Δ 0) is associated BCI-algebra

The present paper is organized as follows In Section 2we review some basic concepts and the main propertiesof 119873(2 2 0) algebras and show relationship between the119873(2 2 0) algebras and the related fuzzy logic algebras ThenSections 3ndash6 contain main results of inverse semigroup of119873(2 2 0) algebra partial order on semigroup of 119873(2 2 0)algebra congruence on semigroup of 119873(2 2 0) algebra andideal and subalgebra of 119873(2 2 0) algebra which will beconsidered respectively

2 Basis Concepts and Properties of119873(2 20) Algebras

In this part we firstly review some relevant concepts anddefinitions

Hindawi Publishing CorporationAlgebraVolume 2016 Article ID 2752681 7 pageshttpdxdoiorg10115520162752681

2 Algebra

Definition 1 (see [4]) Let 119883 be a universe set and 0 isin 119883and let rarr be a binary operation on 119883 A (2 0)-type algebra(119883 rarr 0) is called a fuzzy implication algebra shortly FI-algebra if the following five conditions hold for all 119909 119910 119911 isin

119883

(I1) 119909 rarr (119910 rarr 119911) = 119910 rarr (119909 rarr 119911)(I2) (119909 rarr 119910) rarr ((119910 rarr 119911) rarr (119909 rarr 119911)) = 1(I3) 119909 rarr 119909 = 1(I4) if 119909 rarr 119910 = 119910 rarr 119909 = 1 then 119909 = 119910(I5) 0 rarr 119909 = 1where 1 = 0 rarr 0

Then119873(2 2 0) algebra is an algebra of type (2 2 0) Thenotion was first formulated in 1996 by Deng and Xu andsome properties were obtained (see [32]) This notion wasoriginated from the motivation based on fuzzy implicationalgebra introduced byWu (see [4]) He proved that in a fuzzyimplication algebra (119883 rarr 0) the order relation le satisfying119909 le 119910 if and only if119909 rarr 119910 = 1 is a partial order In [32] Dengand Xu introduced a binary operation lowast defined on fuzzyimplication algebra (119883 rarr 0) such that for all 119886 119887 119906 isin 119883

119906 le 119886 997888rarr 119887 lArrrArr 119886 lowast 119906 le 119887 (1)

where (lowast rarr ) is an adjoint pair on119883

In the corresponding fuzzy logic the operation lowast isrecognized as logic connective ldquoconjunctionrdquo and rarr isconsidered as ldquoimplicationrdquo If the above expression holds fora product lowast then rarr is the residunm of lowast For a product lowastthe corresponding residunm rarr is uniquely defined by

119886 997888rarr 119887 = or 119909 | 119886 lowast 119909 le 119887 (2)

Let us note that 119886 rarr 119887 is the greatest element of the set119906 | 119886 lowast 119906 le 119887

We proved that if forall119886 119887 119906 isin 119883 the following hold

119906 997888rarr (119886 lowast 119887) = 119887 997888rarr (119906 997888rarr 119886)

(119886 lowast 119906) 997888rarr 119887 = 119906 997888rarr (119886 997888rarr 119887)

(3)

then (119883 lowast) is a semigroupIn fact the multiplication defined as above is associativeIt was shown in [4] that for a fuzzy implication algebra

(119883 rarr 0) considering any 119886 isin 119883 there is 1 rarr 119886 = 119886

119886 lowast (119887 lowast 119888) = 1 997888rarr

(119886 lowast (119887 lowast 119888)) = (119887 lowast 119888) 997888rarr

(1 997888rarr 119886) = (119887 lowast 119888) 997888rarr

119886 = 119888 997888rarr (119887 997888rarr 119886)

(119886 lowast 119887) lowast 119888 = 1 997888rarr

((119886 lowast 119887) lowast 119888) = 119888 997888rarr

(1 997888rarr (119886 lowast 119887)) = 119888 997888rarr

(119886 lowast 119887) = 119887 997888rarr (119888 997888rarr 119886)

(4)

Since 119887 rarr (119888 rarr 119886) = 119888 rarr (119887 rarr 119886) we have (119886 lowast119887) lowast 119888 = 119886 lowast (119887 lowast 119888) So it can be concluded that (119883 lowast) is asemigroup

By generalizing the expressions (3) we obtain the basicequations of119873(2 2 0) algebra

Definition 2 (see [32]) 119873(2 2 0) algebra is a nonempty set119878 with a constant 0 and two binary operations lowast Δ for all119886 119887 119888 isin 119878 satisfying the axioms

(1198731) 119886 lowast (119887 Δ 119888) = 119888 lowast (119886 lowast 119887)(1198732) (119886 Δ 119887) lowast 119888 = 119887 lowast (119886 lowast 119888)(1198733) 0 lowast 119886 = 119886

By substitutinglowast andΔ in expressions (1198731) and (1198732)withrarr and lowast respectively we arrive at the expressions (3)

Theorem 3 (see [32]) Let (119878 lowast Δ 0) be 119873(2 2 0) algebraThen for all 119886 119887 119888 isin 119878 the following holds

(1) 119886 lowast 119887 = 119887 Δ 119886(2) (119886 lowast 119887) lowast 119888 = 119886 lowast (119887 lowast 119888) (119886 Δ 119887) Δ 119888 = 119886Δ (119887 Δ 119888)(3) 119886 lowast (119887 lowast 119888) = 119887 lowast (119886 lowast 119888) (119886 Δ 119887) Δ 119888 = (119886 Δ 119888) Δ 119887

Corollary 4 (see [32 33]) If (119878 lowast Δ 0) is 119873(2 2 0) algebrathen (119878 lowast) and (119878 Δ) are semigroups

Therefore the119873(2 2 0) algebra is an algebra system witha pair of dual semigroups Several interesting properties of119873(2 2 0) algebra have been discussed earlier (see [32 33])Throughout this paper we will denote the semigroup (119878 lowast)of119873(2 2 0) algebra (119878 lowast Δ 0) by 119878

Theorem 5 Let (119878 lowast Δ 0) be 119873(2 2 0) algebra and let thefollowing hold for every 119909 in 119878 119909 lowast 0 = 119909 then (119878 lowast) is acommutative semigroup

Proof Since 119909 119910 isin 119878 this shows we have 119909lowast119910 = 119909lowast(119910lowast0) =

119910lowast(119909lowast0) = 119910lowast119909 This implies that 119909lowast119910 = 119910lowast119909 This yields(119878 lowast) is a commutative semigroup

So in 119873(2 2 0) algebra (119878 lowast Δ 0) for any 119909 in 119878 if 119909 lowast0 = 119909 holds then (119878 lowast) and (119878 Δ) are the same and are acommutative monoid as well

Definition 6 (see [34]) A residuated poset is a structure(119860 le rarr sdot 0 1) such that

(1198771) (119860 le 0 1) is a bounded poset(1198772) (119860 sdot 1) is a commutative monoid(1198773) it satisfies the adjointness property that is

119909 sdot 119910 le 119911 lArrrArr 119909 le 119910 997888rarr 119911 (5)

Applying Definition 6 and Theorem 5 to 119873(2 2 0) alge-bra we immediately obtain the following

Remark 7 Let (119878 lowast Δ 0) be 119873(2 2 0) algebra If (119878 lowast 0) issemigroup with the induced order le that is 119909 le 119910 hArr 119909 rarr

119910 = 1 for all 119909 119910 isin 119878 where 1 = 0 rarr 0 and 119909 lowast 0 = 119909 forall 119909 isin 119878 then semigroup (119878 lowast 0) is a residuated poset

Algebra 3

Remark 8 If a fuzzy implication algebra (119883 rarr 0) is with apartial order ldquolerdquo such that any 119886 119887 119906 isin 119883 119886 le 119887 hArr 119886 rarr

119887 = 1 119906 le 119886 rarr 119887 hArr 119886 lowast 119906 le 119887 and for all 119886 119887 119906 isin 119883 thefollowing conditions hold



(6)

then (119883 rarr lowast 0) is119873(2 2 0) algebra

Remark 9 Let (119878 lowast Δ 0) be 119873(2 2 0) algebra then semi-groups (119878 lowast) and (119878 Δ) are a pair of dual semigroup A pairof dual operations (lowast Δ) form an adjoint pair (rarr lowast) that is119906 le 119886 rarr 119887 hArr 119886 lowast 119906 le 119887 for every 119886 119887 119906 isin 119878

Theorem 10 Let (119878 lowast Δ 0) be119873(2 2 0) algebra and for every119909 119910 119911 in 119878 then the following hold

119909Δ (119910 lowast 119911) = 119910 lowast (119909 Δ 119911)

119909 lowast (119910 Δ 119911) = 119910Δ (119909 lowast 119911)

(7)

Proof With Theorem 3 we have 119909 Δ 119910 = 119910 lowast 119909 in 119878 Hence119909 Δ(119910 lowast 119911) = (119910 lowast 119911) lowast 119909 = 119910lowast (119911 lowast 119909) = 119910lowast (119909Δ119911) Similarly119909 lowast (119910 Δ 119911) = 119910 Δ(119909 lowast 119911)

It is well known that partial ordering and congruencesplay an important role in the theory of semigroups In thispaper we have given a partial order relation and a congruenceon semigroup of119873(2 2 0) algebra

Definition 11 (see [35]) A semigroup 119878 is called

(i) medial if it satisfies the equation 119909119910119911119906 = 119909119911119910119906

(ii) trimedial (dimedial resp) if every subsemigroup of119878 generated by at most three (two resp) elements ismedial

(iii) left (right or middle resp) semimedial if it satisfiesthe identity 1199092119910119911 = 119909119910119909119911(119911119910119909

2= 119911119909119910119909 119909119910119911119909 =

119909119911119910119909 resp)

(iv) semimedial if it is both left and right semimedial

(v) strongly semimedial if it is semimedial and middlesemimedial

(vi) exponential (119909119910)119899 = 119909119899119910119899 for every positive integer 119899

(vii) left (right resp) distributive if it satisfies 119909119910119911 =

119909119910119909119911(119911119910119909 = 119911119909119910119909)

Theorem 12 (1) Every semigroup (119878 lowast) of119873(2 2 0) algebra ismedial and semimedial

(2) Every semigroup (119878 lowast) of 119873(2 2 0) algebra is stronglysemimedial

(3) Every medial semigroup (119878 lowast) of 119873(2 2 0) algebra isexponential

Proof It is easy

3 On Inverse Semigroup of 119873(2 20) Algebra

Wewill assume the reader is familiar with elementary inversesemigroup theory as described in [36] Some properties ofinverse semigroup have been discovered in literature (see [3637]) We present here some related definitions and examplesof inverse semigroup of119873(2 2 0) algebra as follows

Definition 13 A semigroup (119878 lowast) of (119873 2 2 0) algebra(119878 lowast Δ 0) is called an inverse semigroup if for every 119886 isin 119878

there exists 119909 isin 119878 such that (119886lowast119909)lowast119886 = 119886 and (119909lowast119886)lowast119909 = 119909Henceforth we define 119909 as the inverse of 119886

Theorem 14 The inverses of semigroup (119878 lowast) of (119873 2 2 0)algebra are not unique

Proof If 1198861015840 is an inverse of 119886 and 1198871015840 is an inverse of 119887 then

((119886lowast119887)lowast(1198861015840lowast1198871015840))lowast(119886lowast119887) = ((119886lowast119886

1015840)lowast(119887lowast119887

1015840))lowast(119886lowast119887) = ((119886lowast

1198861015840)lowast119886)lowast((119887lowast119887

1015840)lowast119887) = 119886lowast119887 and ((1198861015840lowast1198871015840)lowast(119886lowast119887))lowast(1198861015840lowast1198871015840) =

((1198861015840lowast119886)lowast(119887

1015840lowast119887))lowast(119886

1015840lowast1198871015840) = ((119886

1015840lowast119886)lowast119886

1015840)lowast119886lowast((119887

1015840lowast119887)lowast119887

1015840) =

1198861015840lowast 1198871015840 imply that (119886 lowast 119887)1015840 = 119886

1015840lowast 1198871015840

If 119909 and 119910 are both inverse of 119886 then we can show that

119909 lowast 119886 = 119909 lowast ((119886 lowast 119910) lowast 119886) = (119886 lowast 119910) lowast (119909 lowast 119886)

= 119886 lowast (119910 lowast (119909 lowast 119886)) = 119910 lowast (119886 lowast (119909 lowast 119886))

= 119910 lowast ((119886 lowast 119909) lowast 119886) = 119910 lowast 119886

(8)

If 119886 lowast 119909 = 119886 lowast 119910 then

119909 = (119909 lowast 119886) lowast 119909 = (119910 lowast 119886) lowast 119909 = 119910 lowast (119886 lowast 119909)

= 119910 lowast (119886 lowast 119910) = (119910 lowast 119886) lowast 119910 = 119910

(9)

In general the expression 119909lowast 119886 = 119910lowast 119886 does not hold In thiscase the inverse is not unique

We will give some examples in the next section

Example 15 Let 119878 = 0 119886 119887 119888 119889 119890 be equipped with theoperation lowast defined by the following Caylayrsquos table

lowast 0 119886 119887 119888 119889 119890

0 0 119886 119887 119888 119889 119890

119886 0 119886 119887 119888 119889 119890

119887 119887 119887 119887 119887 119887 119887

119888 119888 119888 119887 119888 119887 119890

119889 119887 119887 119887 119887 119887 119887

119890 119890 119890 119887 119890 119887 119890

Δ 0 119886 119887 119888 119889 119890

0 0 0 119887 119888 119887 119890

119886 119886 119886 119887 119888 119887 119890

119887 119887 119887 119887 119887 119887 119887

119888 119888 119888 119887 119888 119887 119890

119889 119889 119889 119887 119887 119887 119887

119890 119890 119890 119887 119890 119887 119890

(10)

4 Algebra

Then (119878 lowast Δ 0) is 119873(2 2 0) algebra but (119878 lowast) is not aninverse semigroup because 119886 119887 are inverse of 119886 or 119887 and 119888

is inverse of 119888 and 119889 is inverse of 119889 but 119890 has no inverse

Example 16 Let 119878 = 0 119886 119887 119888 119889 be equipped with theoperation lowast defined by the following Caylayrsquos table

lowast 0 119886 119887 119888 119889

0 0 119886 119887 119888 119889

119886 119886 119886 119886 119886 119886

119887 0 119886 119887 119888 119889

119888 0 119886 119887 119888 119889

119889 0 119886 119887 119888 119889

Δ 0 119886 119887 119888 119889

0 0 119886 0 0 0

119886 119886 119886 119886 119886 119886

119887 119887 119886 119887 119887 119887

119888 119888 119886 119888 119888 119888

119889 119889 119886 119889 119889 119889

(11)

It is clear that the above system (119878 lowast) forms an inversesemigroup Obviously 119886 119888 119889 are inverse elements of 119886 or 119888or 119889 and 0 is inverse of 0 and 119887 is inverse of 119887 in (119878 lowast)

Clearly (119878 lowast Δ 0) is 119873(2 2 0) algebra and (119878 lowast) is aninverse semigroup

4 Partial Order on Semigroup of119873(2 20) Algebra

We will now investigate the partial order on semigroup of119873(2 2 0) algebra

If 119886 119887 are elements of a semigroup (119878 lowast) in 119873(2 2 0)

algebra (119878 lowast Δ 0) then by 119886 le 119887 we mean 119886 = 119890 lowast 119887 where 119890is an idempotent in 119878

If 119890 and 119891 are idempotents in 119878 we have

119890 lowast 119891 = (119890 lowast 119890) lowast (119891 lowast 119891) = 119890 lowast ((119890 lowast 119891) lowast 119891)

= (119890 lowast 119891) lowast (119890 lowast 119891) = (119890 lowast 119891)2

(12)

by repeatedly using 119886 lowast (119887 lowast 119888) = 119887 lowast (119886 lowast 119888) That means theproduct of idempotents is an idempotent

Theorem 17 The relation le is partial order relation on aninverse semigroup (119878 lowast) of119873(2 2 0) algebra (119878 lowast Δ 0)

Proof Since (119878 lowast) is an inverse semigroup of 119873(2 2 0)

algebra (119878 lowast Δ 0) for each 119886 isin 119878 there exists 1198861015840 isin 119878 such that119886 = (119886 lowast 119886

1015840) lowast 119886 Now (119886 lowast 119886

1015840) lowast (119886 lowast 119886

1015840) = 119886 lowast ((119886 lowast 119886

1015840) lowast 1198861015840) =

119886 lowast ((1198861015840lowast 119886) lowast 119886

1015840) = 119886 lowast 119886

1015840 This implies that 119886 lowast 1198861015840 is an

idempotent Hence 119886 = 119890 lowast 119886 for 119890 = 119886 lowast 1198861015840 So 119886 le 119886

Let 119886 le 119887 This implies that 119886 = 119890 lowast 119887 where 119890 is anidempotent Let 119887 le 119886 then 119887 = 119891 lowast 119886 If 119891 is an idempotentwe have 119890 lowast 119886 = 119890 lowast (119890 lowast 119887) = (119890 lowast 119890) lowast 119887 = 119890 lowast 119887 = 119886

and 119886 = 119890 lowast 119887 = 119890 lowast (119891 lowast 119886) = 119891 lowast (119890 lowast 119886) = 119891 lowast 119886 = 119887 Thisshows that le is antisymmetric

Let 119886 le 119887 and 119887 le 119888 then there exist idempotent 119890 and 119891such that 119886 = 119890 lowast 119887 and 119887 = 119891lowast 119888 So 119886 = 119890 lowast 119887 = 119890 lowast (119891 lowast 119888) =

(119890lowast119891)lowast119888This implies that 119886 le 119888 as 119890lowast119891 is an idempotent

Theorem 18 If (119878 lowast) is an inverse semigroup of 119873(2 2 0)algebra (119878 lowast Δ 0) 119886 119887 isin 119878 then

(1) 119886 le 119887 implies 119888 lowast 119886 le 119888 lowast 119887 and 119886 lowast 119888 le 119887 lowast 119888 for all119888 isin 119878

(2) 119886 le 119887 implies 1198861015840 le 1198871015840

(3) 119886 le 119887 and 119888 le 119889 then 119886 lowast 119888 le 119887 lowast 119889

Proof (1) If 119886 le 119887 then for an idempotent 119890 119886 = 119890 lowast 119887 Now119886 = 119890 lowast 119887 implies that 119888 lowast 119886 = 119888 lowast (119890 lowast 119887) = 119890 lowast (119888 lowast 119887) Hence119888lowast119886 le 119888lowast119887 Also since 119890 is an idempotent 119886lowast119888 = (119890lowast119887)lowast119888 =

119890 lowast (119887 lowast 119888) Hence 119886 lowast 119888 le 119887 lowast 119888(2) Now 119886 le 119887 implies that for an idempotent 119890 119886 = 119890lowast119887

So 1198861015840 = (119890lowast 119887)1015840= 1198901015840lowast1198871015840 Since 1198901015840 = 119890 is an idempotent hence

1198861015840le 1198871015840

(3) Let 119886 le 119887 and 119888 le 119889Thenwehave idempotents 119890 and119891such that 119886 = 119890lowast119887 and 119888 = 119891lowast119889 Now 119886lowast119888 = (119890lowast119887)lowast(119891lowast119889) =

(119890 lowast119891) lowast (119887 lowast 119889) implies that 119886 lowast 119888 le 119887 lowast 119889 This completes theproof

Theorem 19 Let (119878 lowast) and (119878 Δ) be two associative bands of119873(2 2 0) algebra and 119890 119886 119887 isin 119878Then 119890lowast119886 = 119890lowast119887 implies that119886 lowast 119890 = 119887 lowast 119890 and 119886 Δ 119890 = 119887 Δ 119890 implies that 119890 Δ 119886 = 119890 Δ 119887

Proof Suppose that 119890 lowast 119886 = 119890 lowast 119887 then 119886 lowast 119890 = 119886 lowast 119886 lowast 119890 lowast 119890 =

119886 lowast 119890 lowast 119886 lowast 119890 = 119890 lowast 119886 lowast 119886 lowast 119890 = 119890 lowast 119887 lowast 119886 lowast 119890 = 119887 lowast 119890 lowast 119886 lowast 119890 =

119887 lowast 119890 lowast 119887 lowast 119890 = 119887 lowast 119887 lowast 119890 lowast 119890 = 119887 lowast 119890And in an associative band (119878 Δ) suppose that 119886Δ 119890 =

119887Δ 119890 then 119890 Δ 119886 = 119890Δ 119890Δ 119886Δ 119886 = 119890Δ 119886Δ 119890Δ 119886 =

119890Δ 119886Δ 119887Δ 119890 = 119890Δ 119887Δ 119886Δ119890 = 119890Δ 119887Δ 119887Δ 119890 = 119890Δ 119890Δ 119887Δ 119887 =

119890Δ 119887

5 Congruence on Semigroup of119873(2 20) Algebra

Analogous characterization of the maximum idempotent-separating congruence on an eventually orthodox semigroupis given (see [38]) As important consequences some suffi-cient conditions for an eventually regular subsemigroup 119879

of 119878 to satisfy (119878) | 119879 = (119879) are obtained whence if 119878 isfundamental then so is 119879 Similarly to the treatises byMitsch[39] and Luo and Li (see [38]) we consider a semigroup 119878 of119873(2 2 0) algebra Let 119886 119887 119888 isin 119878 If 119886120588119887 implies that 119888119886120588119888119887then the relation 120588 is called left compatible If 119886120588119887 impliesthat 119886119888120588119887119888 then the relation 120588 is called right compatible If119886120588119887 and 119888120588119889 imply that 119886119888120588119887119889 then the relation 120588 is calledcompatible A compatible equivalence relation is called acongruence In this paper we have defined a congruencerelation on an inverse semigroup 119878 of119873(2 2 0) algebra

If 120588 is a congruence relation on an inverse semigroup 119878 of119873(2 2 0) algebra then we can define a binary operation in119878120588 in a natural way as (119886120588) lowast (119887120588) = (119886 lowast 119887)120588

Algebra 5

If (119886120588) = (1198861120588) and (119887120588) = (1198871120588) for 119886 119887 1198861 1198871 isin 119878 thenwe have 1198861205881198861 and 1198871205881198871 and hence 11988611988712058811988611198871 Thus the binaryoperation is well-defined 119878120588 consists of elements of the form119886120588 where 119886 isin 119878

Let (119886120588) and (119887120588) be in 119878120588 then (119886120588) lowast (119887120588) = (119886 lowast 119887)120588 isin 119878120588 Now if 119886 = (119886 lowast 119886

1015840) lowast 119886 then (119886120588) = ((119886 lowast 119886

1015840) lowast 119886)120588 =

((119886120588) lowast (1198861015840120588)) lowast (119886120588) If 1198861015840 = (119886

1015840lowast 119886) lowast 119886

1015840 then (1198861015840120588) = ((1198861015840lowast

119886) lowast 1198861015840)120588 = ((119886

1015840120588) lowast 119886120588) lowast (119886

1015840120588) This shows that 119886120588 and 1198861015840120588

are inverse of each other So 119878120588 is an inverse semigroup oncondition that 119878 is an inverse semigroup

Let 119890 be an idempotent in an inverse semigroup 119878 then 119890120588belongs to 119878120588 and (119890120588) lowast (119890120588) = 119890

2120588 = 119890120588 implies that 119890120588 is

idempotent in 119878120588

Theorem 20 Let 120588 be a congruence relation on an inversesemigroup 119878 of 119873(2 2 0) algebra If (119886120588) is an idempotent in119878120588 then there exists an idempotent 119890 isin 119878 such that (119886120588) =(119890120588)

Proof If (119886120588) is an idempotent in 119878120588 then (119886120588) lowast (119886120588) =

(1198862120588) = (119886120588) This implies that 1198861205881198862 and 1198862120588119886 Let 119909 be an

inverse of 1198862 in 119878Then (1198862lowast119909)lowast1198862 = 1198862 and (119909lowast1198862)lowast119909 = 119909

If 119890 = (119886 lowast 119909) lowast 119886 then 1198902 = ((119886 lowast 119909) lowast 119886) lowast ((119886 lowast 119909) lowast 119886) =

119909 lowast (1198862lowast 119909 lowast 119886

2) = 119909 lowast 119886

2= 119886 lowast 119909 lowast 119886 = 119890 So 119890 = 119886 lowast 119909 lowast 119886

is an idempotent Now 1198861205881198862 implies (119886 lowast 119909)120588(1198862 lowast 119909) Again

1198861205881198862 and (119886 lowast 119909)1205881198862 imply that ((119886 lowast 119909) lowast 119886)120588(1198862 lowast 119909) lowast 1198862 or

119890120588(1198862lowast 119909) lowast 119886

2 or 1198901205881198862 So 1198901205881198862 and 1198862120588119886 imply that (119886120588) =(119890120588)

6 Ideal and Subalgebra on Semigroup of119873(2 20) Algebra

Recently Jun and Song (see [40]) considered int-soft (gen-eralized) bi-ideals of semigroups further properties andcharacterizations of int-soft left (right) ideals are studied andthe notion of int-soft (generalized) bi-ideals is introducedRelations between int-soft generalized bi-ideals and int-softsemigroups are discussed and characterizations of (int-soft)generalized bi-ideals and int-soft bi-ideals are consideredRefer to the literature (see [37 40ndash42]) in this paper we focusour attention on the ideal and subalgebra on semigroup of119873(2 2 0) algebra

The semigroup (119878 lowast) of 119873(2 2 0) algebra determines anorder relation on 119878 119886 le 119887 hArr 119886 lowast 119887 = 0 where 0 = 119886 lowast 119886 is anelement that does not depend on the choice of 119886 isin 119878

Definition 21 A nonempty subset 119860 of119873(2 2 0) algebra 119878 iscalled an ideal of 119878 if it satisfies (1) 0 isin 119860 (2) (forall119909 isin 119878) (forall119910 isin

119860) (119909 lowast 119910 isin 119860 rArr 119909 isin 119860)

Lemma22 An ideal119860 of119873(2 2 0) algebra 119878 has the followingproperty (forall119909 isin 119878) (forall119910 isin 119860) (119909 le 119910 rArr 119909 isin 119860)

Proof Ifforall119909 isin 119878 forall119910 isin 119860 119909 le 119910 then119910 = 0lowast119910 = (119909lowast119910)lowast119910 =

119909 lowast (119910 lowast 119910) = 119909 lowast 0 by Definition 21 we get 119909 isin 119860

Definition 23 Let 119878 be119873(2 2 0) algebra For any 119886 119887 isin 119878 let119866(119886 119887) = 119909 isin 119878 | 119909lowast119886 le 119887 Then 119878 is said to be complicatedif for any 119886 119887 isin 119878 the set 119866(119886 119887) has a greatest element

Theorem 24 Let 119878 be 119873(2 2 0) algebra and let 119868 be a subsetof 119878 Then 119868 is an ideal of 119878 if and only if 119866(119909 119910) sube 119868 for all119909 119910 isin 119868

Proof (rArr) 119868 is an ideal and 119909 119910 are elements of 119868 For any119911 isin 119866(119909 119910) we have 119911 lowast 119909 le 119910 Hence we have 119911 lowast 119909 isin 119868 withLemma 22 Then we obtain 119911 isin 119868 since 119868 is an ideal

(lArr) If 119866(119909 119910) sube 119868 for all 119909 119910 isin 119868 we have 0 isin 119868 since0 isin 119866(119909 119910) For any 119887 isin 119868 and 119886 isin 119878 let 119886 lowast 119887 isin 119868 Then119866(119886 lowast 119887 119887) sube 119868 Hence since 119886 lowast (119886 lowast 119887) le 119887 we obtain 119886 isin

119866(119886 lowast 119887 119887) sube 119868 Hence 119886 isin 119868

Theorem 25 Let 119878 be119873(2 2 0) algebra for any 119886 119887 119888 119889 isin 119878defining119866(119886 119887) = 119909 isin 119878 | 119909lowast119886 le 119887 the following statementsare true

(1) If 0 isin 119866(119886 119887) then 119886 le 119887

(2) If 119886 isin 119866(119886 119887) then 119887 = 0

(3) If 119887 isin 119866(119886 119887) then 119886 lowast 0 = 0

(4) If 119909 isin 119866(119886 119887) and 119910 isin 119866(119888 119889) then 119909 lowast 119910 isin 119866(119886 lowast

119888 119887 lowast 119889)

Proof (1) If 0 isin 119866(119886 119887) then 119886 = 0lowast119886 le 119887 119886lowast119887 = (0lowast119886)lowast119887 =

0 119886 le 119887(2) If 119886 isin 119866(119886 119887) then 119886 lowast 119886 le 119887 so 0 = (119886 lowast 119886) lowast 119887 =

0 lowast 119887 = 119887 hence 119887 = 0(3) Assume that 119887 isin 119866(119886 119887) implies 0 = (119887 lowast 119886) lowast 119887 =

119887 lowast (119886 lowast 119887) = 119886 lowast (119887 lowast 119887) = 119886 lowast 0 Thus 119886 lowast 0 = 0(4) If 119909 isin 119866(119886 119887) then 119909 lowast 119886 le 119887 we have (119909 lowast 119886) lowast 119887 = 0

119910 isin 119866(119888 119889) then 119910 lowast 119888 le 119889 and we have (119910 lowast 119888) lowast 119889 = 0 weget 119909lowast119910lowast119886lowast 119888lowast 119887lowast119889 = (119909lowast119886lowast 119887) lowast (119910lowast 119888lowast119889) = 0lowast 0 = 0Hence 119909 lowast 119910 isin 119866(119886 lowast 119888 119887 lowast 119889)

Definition 26 Let 119878 be 119873(2 2 0) algebra and 119880 a nonemptysubset of 119878 Then 119880 is called a subalgebra of 119878 if 119909 lowast 119910 isin 119880

and 119910 lowast 119909 isin 119880 whenever 119909 119910 isin 119880

Theorem 27 Let 119878 be119873(2 2 0) algebra for any 119909 isin 119878 if 119909 lowast0 = 0 then 119866(119886 119887) is a subalgebra of 119878

Proof Let forall119909 119910 isin 119866(119886 119887) then 119909 lowast 119886 le 119887 119910 lowast 119886 le 119887 that is(119909 lowast 119886) lowast 119887 = 0 and (119910 lowast 119886) lowast 119887 = 0 imply ((119909 lowast 119910) lowast 119886) lowast 119887 =119909 lowast ((119910 lowast 119886) lowast 119887) = 119909 lowast 0 = 0 so (119909 lowast 119910) lowast 119886 le 119887 that is119909 lowast 119910 isin 119866(119886 119887) 119910 lowast 119909 isin 119866(119886 119887) can be proved in a similarway Therefore if 119909 lowast 0 = 0 then 119866(119886 119887) is a subalgebra of119878

Theorem 28 Let 119878 be 119873(2 2 0) algebra 119879 is a subalgebra of119878 if the condition 119909 lowast 0 = 0 forall119909 isin 119879 is satisfied and then anyideal 119868 of 119879 is119873(2 2 0) algebra

Proof For any 119909 119910 isin 119868 we have (119909 lowast 119910) lowast 119909 = 119910 lowast (119909 lowast

119909) = 119910 lowast 0 = 0 if 119868 is ideal of 119879 we obtain 119909 lowast 119910 isin 119868 with(119909 lowast 119910) lowast 119909 = 119910 lowast (119909 lowast 119909) = (119910 lowast 119909) lowast 119909 = 0 we get 119910 lowast 119909 isin 119868Hence 119909 lowast 119910 119910 lowast 119909 isin 119868 So 119868 is 119873(2 2 0) algebra where 119868 isan ideal of 119879

6 Algebra

Theorem 29 Let 119878 be 119873(2 2 0) algebra and 119886 isin 119878 Then theset119867(119886) = 119909 isin 119878 | 119909 lowast 119886 = 119886 is ideal of 119878 and is a subalgebraof 119878 as well

Proof (1) For any 119886 isin 119878 we have 0 lowast 119886 = 119886 so 0 isin 119867(119886)forall119909 isin 119878 forall119910 isin 119867(119886) there is 119909 lowast 119910 isin 119867(119886) and 119910 lowast 119886 = 119886 weget 119886 = 119909 lowast 119910 lowast 119886 = 119909 lowast 119886 so we have 119909 isin 119867(119886) Hence119867(119886)is ideal of 119878

(2) If 119909 isin 119867(119886) 119910 isin 119867(119886) then 119909 lowast 119886 = 119886 119910 lowast 119886 = 119886 then(119909 lowast 119910) lowast 119886 = 119909 lowast (119910 lowast 119886) = 119909 lowast 119886 = 119886 and 119909 lowast 119910 isin 119867(119886)Similarly it can be proved that 119910 lowast 119909 isin 119867(119886) Hence119867(119886) isa subalgebra of 119878

Conflict of Interests

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

Acknowledgment

This work is supported by NSF of China (11305097)

References

[1] M Baczynski and B Jayaram Fuzzy Implications Springer2008

[2] M Mas M Monserrat J Torrens and E Trillas ldquoA surveyon fuzzy implication functionsrdquo IEEE Transactions on FuzzySystems vol 15 no 6 pp 1107ndash1121 2007

[3] S Massanet G Mayor R Mesiar and J Torrens ldquoOn fuzzyimplications an axiomatic approachrdquo International Journal ofApproximate Reasoning vol 54 no 9 pp 1471ndash1482 2013

[4] W M Wu ldquoFuzzy implication algebrardquo Fuzzy Systems andMathematics vol 4 no 1 pp 56ndash64 1990

[5] Z W Li and G H Li ldquoSome properties of fuzzy implicationalgebrasrdquo Fuzzy Systems and Mathematics vol 14 supplement1 pp 19ndash21 2000 (Chinese)

[6] Z W Li and G H Li ldquoCharacterization of fuzzy implicationalgebrasrdquo Journal of Mathematics vol 28 no 6 pp 701ndash7052008 (Chinese)

[7] W Chen ldquoSome chararacters of regular fuzzy implicationalgebrasrdquo Fuzzy System and Mathematics vol 15 no 4 pp 24ndash27 2001 (Chinese)

[8] ZW Li L M Sun and C Y Zheng ldquoRegular fuzzy implicationalgebrasrdquo Fuzzy System and Mathematics vol 16 no 2 pp 22ndash26 2002 (Chinese)

[9] D Wu ldquoCommutative fuzzy implication algebrasrdquo Fuzzy Sys-tems and Mathematics vol 13 no 1 pp 27ndash30 1999 (Chinese)

[10] F A Deng and J L Li ldquo119882119889-fuzzy implication algebrasrdquo Journalof Harbin Normal University (Natural Science Edition) vol 12no 2 pp 18ndash21 1996 (Chinese)

[11] H R Zhang and L C Wang ldquoProperties of LFI-algebras andresiduated latticerdquo Fuzzy System andMathematics vol 18 no 2pp 13ndash17 2004 (Chinese)

[12] C H Liu and L X Xu ldquoMP-filters of fuzzy implicationalgebrasrdquo Fuzzy Systems andMathematics vol 23 no 2 pp 1ndash62009 (Chinese)

[13] C H Liu and L X Xu ldquoVarious concepts of fuzzy filters in FI-algebrasrdquo Fuzzy Systems and Mathematics vol 24 no 2 pp 21ndash27 2010

[14] Y Zhu and Y Xu ldquoOn filter theory of residuated latticesrdquo Infor-mation Sciences vol 180 no 19 pp 3614ndash3632 2010

[15] M Kondo andW A Dudek ldquoFilter theory of BL-algebrasrdquo SoftComputing vol 12 no 5 pp 419ndash423 2008

[16] M Haveshki and E Eslami ldquon-fold filters in BL-algebrasrdquoMathematical Logic Quarterly vol 54 no 2 pp 176ndash186 2008

[17] R A Borzooei S K Shoar and R Ameri ldquoSome types of filtersinMTL-algebrasrdquo Fuzzy Sets and Systems vol 187 no 1 pp 92ndash102 2012

[18] M Haveshki A Saeid and E Eslami ldquoSome types of filters inBL algebrasrdquo Soft Computing vol 10 no 8 pp 657ndash664 2006

[19] B Q Hu ldquoEquivalence of fuzzy implication algebras andbounded BCK-algebrasrdquo Applied Mathematics vol 6 no 2 pp233ndash234 1993 (Chinese)

[20] H Jiang and F A Deng ldquoRelations between fuzzy mplicationalgebras and bouned BCK-algebrasrdquo Fuzzy system and Mathe-matics vol 5 no 1 pp 88ndash89 1991 (Chinese)

[21] Z W Li and C Y Zheng ldquoRelations between fuzzy implicationalgebra and MV algebrardquo Journal of Fuzzy Mathematics vol 9no 1 pp 201ndash205 2001 (Chinese)

[22] S L Chen ldquoRough sets and fuzzy implication algebrasrdquo Com-puting Science and Engineering vol 31 no 4 pp 134ndash136 2009(Chinese)

[23] H X Wei and X Q Li ldquoRough set algebras and fuzzyimplication algebrasrdquo Computer Engineering and Applicationsvol 45 no 18 pp 38ndash39 2009 (Chinese)

[24] Y Q Zhu ldquoA logic system based on FI-algebras and its com-pletenessrdquo Fuzzy Systems and Mathematics vol 19 no 2 pp25ndash29 2005 (Chinese)

[25] D Pei ldquoA survey of fuzzy implication algebras and their axiom-atizationrdquo International Journal of Approximate Reasoning vol55 no 8 pp 1643ndash1658 2014

[26] Y B Jun ldquoFuzzy positive implicative and fuzzy associative filtersof lattice implication algebrasrdquo Fuzzy Sets and Systems vol 121no 2 pp 353ndash357 2001

[27] Y B Jun and S Z Song ldquoOn fuzzy implicative filters of latticeimplication algebrasrdquo Journal of Fuzzy Mathematics vol 10 no4 pp 893ndash900 2002

[28] L Liu and K Li ldquoFuzzy implicative and Boolean filters of 1198770-algebrasrdquo Information Sciences vol 171 pp 61ndash71 2005

[29] L Liu and K Li ldquoFuzzy filters of BL-algebrasrdquo InformationSciences vol 173 no 1ndash3 pp 141ndash154 2005

[30] L Liu and K Li ldquoFuzzy Boolean and positive implicative filtersof BL-algebrasrdquo Fuzzy Sets and Systems vol 152 no 2 pp 333ndash348 2005

[31] J Siekmann and P Z Szabo ldquoThe undecidability of the D119860-unification problemrdquoThe Journal of Symbolic Logic vol 54 no2 pp 402ndash414 1989

[32] F-A Deng and Y Xu ldquoOn N(220) algebrardquo Journal of South-west Jiaotong University vol 131 pp 457ndash463 1996

[33] F-A Deng and L-Q Yong ldquoRegular smeigroup of N(2 2 0)algebrardquo Advances in Mathematics vol 6 pp 655ndash662 2012(Chinese)

[34] I Chajda and J Paseka ldquoTense operators in fuzzy logicrdquo FuzzySets and Systems vol 276 no 1 pp 100ndash113 2015

[35] F Abdullahu and A Zejnullahu ldquoNotes on semimedial semi-groupsrdquoCommentationesMathematicaeUniversitatis Carolinaevol 50 no 3 pp 321ndash324 2009

[36] J M Howie An Introduction to Semigroup Theory AcademicPress San Diego Calif USA 1976

Algebra 7

[37] M Brion ldquoOn algebraic semigroups and monoids IIrdquo Semi-group Forum vol 88 no 1 pp 250ndash272 2014

[38] Y Luo and X Li ldquoThe maximum idempotent-separating con-gruence on eventually regular semigroupsrdquo Semigroup Forumvol 74 no 2 pp 306ndash318 2007

[39] H Mitsch ldquoSemigroups and their lattice of congruencesrdquoSemigroup Forum vol 26 no 1-2 pp 1ndash63 1983

[40] Y B Jun and S Song ldquoInt-soft (generalized) Bi-ideals ofsemigroupsrdquo The Scientific World Journal vol 2015 Article ID198672 6 pages 2015

[41] N Kehayopulu and M Tsingelis ldquoArchimedean ordered semi-groups as ideal extensionsrdquo Semigroup Forum vol 78 no 2 pp343ndash348 2009

[42] J Fountain and G M S Gomes ldquoFinite abundant semigroupsin which the idempotents form a subsemigrouprdquo Journal ofAlgebra vol 295 no 2 pp 303ndash313 2006

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of


Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of


Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of


Mathematical PhysicsAdvances in

Complex AnalysisJournal of


OptimizationJournal of


CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of


Operations ResearchAdvances in

Journal of


Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences


The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014


Algebra

Discrete Dynamics in Nature and Society



Decision SciencesAdvances in

Discrete MathematicsJournal of


Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

2 Algebra

Definition 1 (see [4]) Let 119883 be a universe set and 0 isin 119883and let rarr be a binary operation on 119883 A (2 0)-type algebra(119883 rarr 0) is called a fuzzy implication algebra shortly FI-algebra if the following five conditions hold for all 119909 119910 119911 isin

119883

(I1) 119909 rarr (119910 rarr 119911) = 119910 rarr (119909 rarr 119911)(I2) (119909 rarr 119910) rarr ((119910 rarr 119911) rarr (119909 rarr 119911)) = 1(I3) 119909 rarr 119909 = 1(I4) if 119909 rarr 119910 = 119910 rarr 119909 = 1 then 119909 = 119910(I5) 0 rarr 119909 = 1where 1 = 0 rarr 0

Then119873(2 2 0) algebra is an algebra of type (2 2 0) Thenotion was first formulated in 1996 by Deng and Xu andsome properties were obtained (see [32]) This notion wasoriginated from the motivation based on fuzzy implicationalgebra introduced byWu (see [4]) He proved that in a fuzzyimplication algebra (119883 rarr 0) the order relation le satisfying119909 le 119910 if and only if119909 rarr 119910 = 1 is a partial order In [32] Dengand Xu introduced a binary operation lowast defined on fuzzyimplication algebra (119883 rarr 0) such that for all 119886 119887 119906 isin 119883

119906 le 119886 997888rarr 119887 lArrrArr 119886 lowast 119906 le 119887 (1)

where (lowast rarr ) is an adjoint pair on119883

In the corresponding fuzzy logic the operation lowast isrecognized as logic connective ldquoconjunctionrdquo and rarr isconsidered as ldquoimplicationrdquo If the above expression holds fora product lowast then rarr is the residunm of lowast For a product lowastthe corresponding residunm rarr is uniquely defined by

119886 997888rarr 119887 = or 119909 | 119886 lowast 119909 le 119887 (2)

Let us note that 119886 rarr 119887 is the greatest element of the set119906 | 119886 lowast 119906 le 119887

We proved that if forall119886 119887 119906 isin 119883 the following hold



(3)

then (119883 lowast) is a semigroupIn fact the multiplication defined as above is associativeIt was shown in [4] that for a fuzzy implication algebra

(119883 rarr 0) considering any 119886 isin 119883 there is 1 rarr 119886 = 119886

119886 lowast (119887 lowast 119888) = 1 997888rarr

(119886 lowast (119887 lowast 119888)) = (119887 lowast 119888) 997888rarr

(1 997888rarr 119886) = (119887 lowast 119888) 997888rarr

119886 = 119888 997888rarr (119887 997888rarr 119886)

(119886 lowast 119887) lowast 119888 = 1 997888rarr

((119886 lowast 119887) lowast 119888) = 119888 997888rarr

(1 997888rarr (119886 lowast 119887)) = 119888 997888rarr

(119886 lowast 119887) = 119887 997888rarr (119888 997888rarr 119886)

(4)

Since 119887 rarr (119888 rarr 119886) = 119888 rarr (119887 rarr 119886) we have (119886 lowast119887) lowast 119888 = 119886 lowast (119887 lowast 119888) So it can be concluded that (119883 lowast) is asemigroup

By generalizing the expressions (3) we obtain the basicequations of119873(2 2 0) algebra

Definition 2 (see [32]) 119873(2 2 0) algebra is a nonempty set119878 with a constant 0 and two binary operations lowast Δ for all119886 119887 119888 isin 119878 satisfying the axioms

(1198731) 119886 lowast (119887 Δ 119888) = 119888 lowast (119886 lowast 119887)(1198732) (119886 Δ 119887) lowast 119888 = 119887 lowast (119886 lowast 119888)(1198733) 0 lowast 119886 = 119886

By substitutinglowast andΔ in expressions (1198731) and (1198732)withrarr and lowast respectively we arrive at the expressions (3)

Theorem 3 (see [32]) Let (119878 lowast Δ 0) be 119873(2 2 0) algebraThen for all 119886 119887 119888 isin 119878 the following holds

(1) 119886 lowast 119887 = 119887 Δ 119886(2) (119886 lowast 119887) lowast 119888 = 119886 lowast (119887 lowast 119888) (119886 Δ 119887) Δ 119888 = 119886Δ (119887 Δ 119888)(3) 119886 lowast (119887 lowast 119888) = 119887 lowast (119886 lowast 119888) (119886 Δ 119887) Δ 119888 = (119886 Δ 119888) Δ 119887

Corollary 4 (see [32 33]) If (119878 lowast Δ 0) is 119873(2 2 0) algebrathen (119878 lowast) and (119878 Δ) are semigroups

Therefore the119873(2 2 0) algebra is an algebra system witha pair of dual semigroups Several interesting properties of119873(2 2 0) algebra have been discussed earlier (see [32 33])Throughout this paper we will denote the semigroup (119878 lowast)of119873(2 2 0) algebra (119878 lowast Δ 0) by 119878

Theorem 5 Let (119878 lowast Δ 0) be 119873(2 2 0) algebra and let thefollowing hold for every 119909 in 119878 119909 lowast 0 = 119909 then (119878 lowast) is acommutative semigroup

Proof Since 119909 119910 isin 119878 this shows we have 119909lowast119910 = 119909lowast(119910lowast0) =

119910lowast(119909lowast0) = 119910lowast119909 This implies that 119909lowast119910 = 119910lowast119909 This yields(119878 lowast) is a commutative semigroup

So in 119873(2 2 0) algebra (119878 lowast Δ 0) for any 119909 in 119878 if 119909 lowast0 = 119909 holds then (119878 lowast) and (119878 Δ) are the same and are acommutative monoid as well

Definition 6 (see [34]) A residuated poset is a structure(119860 le rarr sdot 0 1) such that

(1198771) (119860 le 0 1) is a bounded poset(1198772) (119860 sdot 1) is a commutative monoid(1198773) it satisfies the adjointness property that is

119909 sdot 119910 le 119911 lArrrArr 119909 le 119910 997888rarr 119911 (5)

Applying Definition 6 and Theorem 5 to 119873(2 2 0) alge-bra we immediately obtain the following

Remark 7 Let (119878 lowast Δ 0) be 119873(2 2 0) algebra If (119878 lowast 0) issemigroup with the induced order le that is 119909 le 119910 hArr 119909 rarr

119910 = 1 for all 119909 119910 isin 119878 where 1 = 0 rarr 0 and 119909 lowast 0 = 119909 forall 119909 isin 119878 then semigroup (119878 lowast 0) is a residuated poset

Algebra 3





(6)




119909Δ (119910 lowast 119911) = 119910 lowast (119909 Δ 119911)

119909 lowast (119910 Δ 119911) = 119910Δ (119909 lowast 119911)

(7)







2= 119911119909119910119909 119909119910119911119909 =

119909119911119910119909 resp)





119909119910119909119911(119911119910119909 = 119911119909119910119909)




Proof It is easy








1015840)lowast(119887lowast119887




((1198861015840lowast119886)lowast(119887

1015840lowast119887))lowast(119886

1015840lowast1198871015840) = ((119886

1015840lowast119886)lowast119886

1015840)lowast119886lowast((119887

1015840lowast119887)lowast119887

1015840) =


1015840lowast 1198871015840





(8)




(9)




lowast 0 119886 119887 119888 119889 119890

0 0 119886 119887 119888 119889 119890

119886 0 119886 119887 119888 119889 119890

119887 119887 119887 119887 119887 119887 119887

119888 119888 119888 119887 119888 119887 119890

119889 119887 119887 119887 119887 119887 119887

119890 119890 119890 119887 119890 119887 119890

Δ 0 119886 119887 119888 119889 119890

0 0 0 119887 119888 119887 119890

119886 119886 119886 119887 119888 119887 119890

119887 119887 119887 119887 119887 119887 119887

119888 119888 119888 119887 119888 119887 119890

119889 119889 119889 119887 119887 119887 119887

119890 119890 119890 119887 119890 119887 119890

(10)

4 Algebra




lowast 0 119886 119887 119888 119889

0 0 119886 119887 119888 119889

119886 119886 119886 119886 119886 119886

119887 0 119886 119887 119888 119889

119888 0 119886 119887 119888 119889

119889 0 119886 119887 119888 119889

Δ 0 119886 119887 119888 119889

0 0 119886 0 0 0

119886 119886 119886 119886 119886 119886

119887 119887 119886 119887 119887 119887

119888 119888 119886 119888 119888 119888

119889 119889 119886 119889 119889 119889

(11)










(12)






1015840) lowast (119886 lowast 119886

1015840) = 119886 lowast ((119886 lowast 119886

1015840) lowast 1198861015840) =


1015840) = 119886 lowast 119886









(2) 119886 le 119887 implies 1198861015840 le 1198871015840





1198861015840le 1198871015840







119887Δ 119890 then 119890 Δ 119886 = 119890Δ 119890Δ 119886Δ 119886 = 119890Δ 119886Δ 119890Δ 119886 =

119890Δ 119886Δ 119887Δ 119890 = 119890Δ 119887Δ 119886Δ119890 = 119890Δ 119887Δ 119887Δ 119890 = 119890Δ 119890Δ 119887Δ 119887 =

119890Δ 119887





Algebra 5




1015840) lowast 119886)120588 =

((119886120588) lowast (1198861015840120588)) lowast (119886120588) If 1198861015840 = (119886

1015840lowast 119886) lowast 119886

1015840 then (1198861015840120588) = ((1198861015840lowast

119886) lowast 1198861015840)120588 = ((119886

1015840120588) lowast 119886120588) lowast (119886




2120588 = 119890120588 implies that 119890120588 is








2) = 119909 lowast 119886




119890120588(1198862lowast 119909) lowast 119886
















(1) If 0 isin 119866(119886 119887) then 119886 le 119887

(2) If 119886 isin 119866(119886 119887) then 119887 = 0



119888 119887 lowast 119889)













6 Algebra






Acknowledgment


References





































Algebra 7














Volume 2014




Journal of











Journal of


Function Spaces






Algebra









Algebra 3





(6)




119909Δ (119910 lowast 119911) = 119910 lowast (119909 Δ 119911)

119909 lowast (119910 Δ 119911) = 119910Δ (119909 lowast 119911)

(7)







2= 119911119909119910119909 119909119910119911119909 =

119909119911119910119909 resp)





119909119910119909119911(119911119910119909 = 119911119909119910119909)




Proof It is easy








1015840)lowast(119887lowast119887




((1198861015840lowast119886)lowast(119887

1015840lowast119887))lowast(119886

1015840lowast1198871015840) = ((119886

1015840lowast119886)lowast119886

1015840)lowast119886lowast((119887

1015840lowast119887)lowast119887

1015840) =


1015840lowast 1198871015840





(8)




(9)




lowast 0 119886 119887 119888 119889 119890

0 0 119886 119887 119888 119889 119890

119886 0 119886 119887 119888 119889 119890

119887 119887 119887 119887 119887 119887 119887

119888 119888 119888 119887 119888 119887 119890

119889 119887 119887 119887 119887 119887 119887

119890 119890 119890 119887 119890 119887 119890

Δ 0 119886 119887 119888 119889 119890

0 0 0 119887 119888 119887 119890

119886 119886 119886 119887 119888 119887 119890

119887 119887 119887 119887 119887 119887 119887

119888 119888 119888 119887 119888 119887 119890

119889 119889 119889 119887 119887 119887 119887

119890 119890 119890 119887 119890 119887 119890

(10)

4 Algebra




lowast 0 119886 119887 119888 119889

0 0 119886 119887 119888 119889

119886 119886 119886 119886 119886 119886

119887 0 119886 119887 119888 119889

119888 0 119886 119887 119888 119889

119889 0 119886 119887 119888 119889

Δ 0 119886 119887 119888 119889

0 0 119886 0 0 0

119886 119886 119886 119886 119886 119886

119887 119887 119886 119887 119887 119887

119888 119888 119886 119888 119888 119888

119889 119889 119886 119889 119889 119889

(11)










(12)






1015840) lowast (119886 lowast 119886

1015840) = 119886 lowast ((119886 lowast 119886

1015840) lowast 1198861015840) =


1015840) = 119886 lowast 119886









(2) 119886 le 119887 implies 1198861015840 le 1198871015840





1198861015840le 1198871015840







119887Δ 119890 then 119890 Δ 119886 = 119890Δ 119890Δ 119886Δ 119886 = 119890Δ 119886Δ 119890Δ 119886 =

119890Δ 119886Δ 119887Δ 119890 = 119890Δ 119887Δ 119886Δ119890 = 119890Δ 119887Δ 119887Δ 119890 = 119890Δ 119890Δ 119887Δ 119887 =

119890Δ 119887





Algebra 5




1015840) lowast 119886)120588 =

((119886120588) lowast (1198861015840120588)) lowast (119886120588) If 1198861015840 = (119886

1015840lowast 119886) lowast 119886

1015840 then (1198861015840120588) = ((1198861015840lowast

119886) lowast 1198861015840)120588 = ((119886

1015840120588) lowast 119886120588) lowast (119886




2120588 = 119890120588 implies that 119890120588 is








2) = 119909 lowast 119886




119890120588(1198862lowast 119909) lowast 119886
















(1) If 0 isin 119866(119886 119887) then 119886 le 119887

(2) If 119886 isin 119866(119886 119887) then 119887 = 0



119888 119887 lowast 119889)













6 Algebra






Acknowledgment


References





































Algebra 7














Volume 2014




Journal of











Journal of


Function Spaces






Algebra









4 Algebra




lowast 0 119886 119887 119888 119889

0 0 119886 119887 119888 119889

119886 119886 119886 119886 119886 119886

119887 0 119886 119887 119888 119889

119888 0 119886 119887 119888 119889

119889 0 119886 119887 119888 119889

Δ 0 119886 119887 119888 119889

0 0 119886 0 0 0

119886 119886 119886 119886 119886 119886

119887 119887 119886 119887 119887 119887

119888 119888 119886 119888 119888 119888

119889 119889 119886 119889 119889 119889

(11)










(12)






1015840) lowast (119886 lowast 119886

1015840) = 119886 lowast ((119886 lowast 119886

1015840) lowast 1198861015840) =


1015840) = 119886 lowast 119886









(2) 119886 le 119887 implies 1198861015840 le 1198871015840





1198861015840le 1198871015840







119887Δ 119890 then 119890 Δ 119886 = 119890Δ 119890Δ 119886Δ 119886 = 119890Δ 119886Δ 119890Δ 119886 =

119890Δ 119886Δ 119887Δ 119890 = 119890Δ 119887Δ 119886Δ119890 = 119890Δ 119887Δ 119887Δ 119890 = 119890Δ 119890Δ 119887Δ 119887 =

119890Δ 119887





Algebra 5




1015840) lowast 119886)120588 =

((119886120588) lowast (1198861015840120588)) lowast (119886120588) If 1198861015840 = (119886

1015840lowast 119886) lowast 119886

1015840 then (1198861015840120588) = ((1198861015840lowast

119886) lowast 1198861015840)120588 = ((119886

1015840120588) lowast 119886120588) lowast (119886




2120588 = 119890120588 implies that 119890120588 is








2) = 119909 lowast 119886




119890120588(1198862lowast 119909) lowast 119886
















(1) If 0 isin 119866(119886 119887) then 119886 le 119887

(2) If 119886 isin 119866(119886 119887) then 119887 = 0



119888 119887 lowast 119889)













6 Algebra






Acknowledgment


References





































Algebra 7














Volume 2014




Journal of











Journal of


Function Spaces






Algebra









Algebra 5




1015840) lowast 119886)120588 =

((119886120588) lowast (1198861015840120588)) lowast (119886120588) If 1198861015840 = (119886

1015840lowast 119886) lowast 119886

1015840 then (1198861015840120588) = ((1198861015840lowast

119886) lowast 1198861015840)120588 = ((119886

1015840120588) lowast 119886120588) lowast (119886




2120588 = 119890120588 implies that 119890120588 is








2) = 119909 lowast 119886




119890120588(1198862lowast 119909) lowast 119886
















(1) If 0 isin 119866(119886 119887) then 119886 le 119887

(2) If 119886 isin 119866(119886 119887) then 119887 = 0



119888 119887 lowast 119889)













6 Algebra






Acknowledgment


References





































Algebra 7














Volume 2014




Journal of











Journal of


Function Spaces






Algebra









6 Algebra






Acknowledgment


References





































Algebra 7














Volume 2014




Journal of











Journal of


Function Spaces






Algebra









Algebra 7














Volume 2014




Journal of











Journal of


Function Spaces






Algebra
















Volume 2014




Journal of











Journal of


Function Spaces






Algebra









Research Article Characterizations of ( 2,2,0)...

Documents

Transcript of Research Article Characterizations of ( 2,2,0)...