Research Article The Coarse Structure of the …downloads.hindawi.com/archive/2014/529804.pdfof the...

Research ArticleThe Coarse Structure of the Representation Algebra ofa Finite Monoid

Mary Schaps

Department of Mathematics Bar-Ilan University 52900 Ramat Gan Israel

Correspondence should be addressed to Mary Schaps mschapsmacsbiuacil

Received 30 June 2013 Accepted 12 November 2013 Published 30 January 2014

Academic Editor Nantel Bergeron

Copyright copy 2014 Mary SchapsThis is an open access article distributed under the Creative Commons Attribution License whichpermits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

Let M be a monoid and let L be a commutative idempotent submonoid We show that we can find a complete set of orthogonalidempotents 119871

0of the monoid algebra A ofM such that there is a basis of A adapted to this set of idempotents which is in one-to-

one correspondence with elements of the monoid The basis graph describing the Peirce decomposition with respect to

119871

0gives a

coarse structure of the algebra of which any complete set of primitive idempotents gives a refinement and we give some criterionfor this coarse structure to actually be a fine structure which means that the nonzero elements of the monoid are in one-to-onecorrespondence with the vertices and arrows of the basis graph with respect to a set of primitive idempotents with this basis graphbeing a canonical object

1 Introduction

When we speak of a coarse structure we mean the decompo-sition of the monoid algebra into Peirce components cor-responding to the elements of a commutative idempotentsubmonoid The fine structure is then the refinement whichoccurs when one breaks down each of these idempotents intoa sum of primitive idempotents The basic idea of this workis to try to understand the semigroup representation theoryinsofar as possible without delving into the group theory andto determine criteria for monoids for which there is a coarsestructure which corresponds to the fine structure

We assume that the field 119870 over which we are takingrepresentations of amonoid119872 is of characteristic which doesnot divide the order of any of the maximal subgroups of 119890119872119890as 119890 runs over the idempotents of119872ThenMaschkersquos theoremapplies and the group algebras of the maximal subgroups areall semisimple

The irreducible representations of the semigroup corre-spond to the disjoint union of the irreducible representationsof the various maximal subgroups Thus one condition wewill need for the coarse structure to coincide with the finestructure is that the monoid be aperiodic that is to say thatmaximal subgroups be trivial

The quiver of a finite dimensional algebra is a combina-torial object of central importance in studying its represen-tations theory There has been in recent years a great deal ofinterest in determining properties of the quiver and relationsof the monoid algebra as in the work of Saliola [1] and ofMargolis and Steinberg [2] The object used in this paperthe basis graph is closely related to the quiver For somepurposes particularly deformation theory the basis graph ispreferable and in this work we claim that for certain typesof monoids the basis graph gives a much clearer pictureof the relationship between the monoid and the monoidalgebra than the quiver doesThese are themonoids for whichthe coarse structure described above can be obtained for acomplete set of primitive idempotents and one of the mainresults of the paper is that this happens when the monoidis aperiodic and has commutative idempotents Examples ofthis phenomenon given below include thematrixmonoid andthe set of order-preserving and extensive maps from a finiteset into itself

Let us comment that there has also been an other recentwork on trying to unravel the algebra structure of themonoidalgebra from invariants of the monoid Thiery [3] is able toderive information about the Cartan matrix of the algebrafrom combinatorial Cartan matrix 119862 of the monoid and

Hindawi Publishing CorporationJournal of Discrete MathematicsVolume 2014 Article ID 529804 7 pageshttpdxdoiorg1011552014529804

2 Journal of Discrete Mathematics

similarly the third section in Dentonrsquos thesis [4] takes upthis themeThe anonymous referee pointed out the similarityof our concept to the third section of [5] and we have infact switched from our original notation to a version of theirnotation ldquolfixrdquo and ldquorfixrdquo to emphasize this similarity In thecase given in our Section 5 in which the coarse and finestructures coincide the entries in the matrix 119862 appear to bethe sizes of what we call the Peirce sets of themonoid definedin our Section 3

Let 119872 be a finite monoid and let 119864(119872) be its set ofidempotents We recall that the J-class of an element 119886 of119872 is the set of all 119887 isin 119872 such the 119872119886119872 = 119872119887119872 A J-class is regular if it contains an idempotent Two idempotents119890 and 119891 are called conjugate if they lie in the same J-classand this happens if and only if there are elements 119909 and 119910 inthe monoid such that 119890 = 119909119910 and 119891 = 119910119909

2 The Basis Graph of a FiniteDimensional Algebra

For any complete orthogonal setE of idempotents in a finitedimensional algebra119860with Jacobson radical 119869 we can definea basis graph [6] to be the directed graph with one vertexlabeled 119891

119894for each idempotent 119891

119894and the following loops and

arrows

(i) 1198990119894119894minus 1 loops of weight zero

(ii) 119899119896119894119895arrows or loops of weight 119896 for 119894 = 119895 or 119896 gt 0 where

119899

119896

119894119895= dim

119870(119891

119894(

119869

119896

119869

119896+1)119891

119895) (1)

Definition 1 A basis 119861 of 119860 which is a union of bases forthe different Peirce components 119891

119894119860119891

119895is said to respect the

idempotent set E

A refinement E1015840 of E is a complete set of orthogonalidempotents such that every element 119890 of E is a sum ofelements of E1015840 If we have such a refinement then we geta refinement of the basis graph in which some of the loopsare replaced by vertices and arrows For example the uppertriangular 2 times 2 matrices over a field 119896 considered withrespect to a set of idempotents containing only the identitymatrix would have a basis graph consisting of one vertexand two loops one of weight zero and one of weight 1 Wecan take a refinement of the idempotent set consisting ofthe idempotent diagonal matrices 119864

11and 119864

22 The refined

basis graph will have two vertices corresponding to the twoidempotents and one arrow of weight one

The basis graph for any complete orthogonal set ofprimitive idempotents is an invariant of the algebra sinceany two such sets are conjugate If the algebra is basic thenthe quiver is with respect to right modules just the basisgraph with all arrows of weight greater than one erasedsince the quiver is defined by considering the dimension of(119891

119894(119869119869

2)119891

119895) (if one uses leftmodules onemust take the dual)

If the algebra is not basic then all matrix blocks togetherwith the arrows of weight zero representing matrix units

must be shrunk to points in the quiver with a correspondingcoalescence of arrows

3 Finite Monoids with a Chosen CommutativeIdempotent Submonoid

Our aim in this section is to construct an alternative basis of119860 which still corresponds one-to-one to the set of elementsof the monoid but which behaves well with respect to thebasis graph defined above In particular we will want theidempotents to be orthogonal and the other basis elementsto respect the idempotent set as in Definition 1

We now consider a finite monoid (119872) with a chosencommutative idempotent submonoid 119871 If119872 has an absorb-ing zero element 119885 we assume that 119885 is also in 119871 A trivialchoice would be to take 119871 to be the identity element togetherwith 119885 if it exists Because of the commutativity the set 119871 ispartially ordered by the relation 119909 ⪯ 119910 hArr 119909 119910 = 119909 andis a semilattice under this ordering with the greatest lowerbound of two elements being their product Since 119871 containsa maximal element the identity then 119871 is also a lattice sincethe least upper bound of two elements 119909 and 119910 is the greatestlower bound of all the elements of 119871 which dominate both 119909

and 119910 and this set is nonempty because the identity alwaysdominates both 119909 and 119910

Definition 2 Let 119872 be a monoid with operation Let 119871 bea commutative idempotent submonoid For an element 119886 ofthe monoid 119872 the left 119871-idempotent lfix

119871(119886) is the product

of all the idempotents 119890 in the lattice 119871 such that 119890 119886 = 119886Similarly the right 119871-idempotent rfix

119871(119886) is the product of all

the idempotents 119890 in the lattice 119871 such that 119886 119890 = 119886 At leastone such idempotent always exists since the identity of themonoid satisfies the condition

Remark 3 If119872 has an absorbing zero119885 then119885 is not the leftor right idempotent of any element except itself

We now pass to the reduced monoid algebra 119860 of themonoid119872 over a field 119870 where 119870 is a field sufficiently largethat the quotient of 119860 by its radical 119869 splits completely as asum of matrix blocks over119870 One might for example take119870to be algebraically closed If119872 does not contain an absorbingzero element then we set 119860 = 119870119872 and 119861 = 119870119871 and define119872

0= 119872 and 119871

0= 119871 If119872 does contain a necessarily unique

zero element 119885 then we let 119860 be the quotient 119870119872119870119885

with subalgebra 119861 = 119870119871119870119885 This reduction is madebecause the full monoid algebra 119870119872 is isomorphic to adirect product of an algebra isomorphic to 119860 and a copyof the 119870 representing the ideal 119870119885 which gives amongother disadvantages an algebra which is decomposable anddisconnected quiver where algebra representation theoristsprefer to work with indecomposable algebras From the pointof view of representation theory the object of study is thealgebra 119860 We define sets 119872

0= 119872 minus 119885 and 119871

0= 119871 minus 119885

Note that in the second case 119872

0and 119871

0may no longer be

monoids since the product of two elements different from 119885

might be 119885

Journal of Discrete Mathematics 3

For each 119898 isin 119872 we let 119898 isin 119860 be equal 119898 in the casewithout the absorbing zero and 119898 + 119870119885 in the case with theabsorbing zeroThis will allow us to treat both cases togetherWe take as bases of 119860 and 119861 the sets119872

0= 119898 | 119898 isin 119872

0 and

119871

0= 119890119890 isin 119871

0 respectively

Definition 4 Amultiplicative basis B in a finite dimensionalalgebra 119860 is a basis such that the product of two elements ofthe basis is either zero or an element of the basis Thus B isa multiplicative basis of 119860 if and only if 119860 is the semigroupalgebra ofB

The basis 119872

0is a multiplicative basis of the reduced

monoid algebra 119860 defined above and 119871

0is a multiplicative

basis for 119861By the Munn-Ponizovskii theorem [7] the isomorphism

classes of simple modules are in one-to-one correspondencewith the simple modules of the various isomorphism classesof maximal subgroups one for each J-class and we alsoassume that the characteristic of119870 does not divide the ordersof any of the maximal subgroups

It is a standard fact about rings that if 119890 and 119891 areidempotents satisfying 119891 ⪯ 119890 then 119890 minus 119891 is an idempotentorthogonal to 119891 By a construction of Solomon [8] thereduced monoid algebra 119861 which is a subalgebra of 119860 hasa basis of orthogonal idempotents in one-to-one correspon-dence with the elements of 119871

0 obtained by a process of

Moebius inversion These idempotents are actually primitivein 119861 but they are not necessarily primitive in 119860 as theexample of taking 119871 to be the identity demonstrates TheMoebius function 120583(119891 119890) for elements of a partially orderedset is defined recursively to be 0 if 119891119890 to be 1 if 119891 = 119890 and tobe minussum

119891le119911lt119890120583(119891 119911) for 119891 lt 119890

Definition 5 For each 119890 isin 119871

0 let 119890 = sum

119891le119890120583(119891 119890)119891 be the

corresponding primitive idempotent of 119861 The set of all the 119890will be denoted by 119871

0

For any 119886 isin 119872

0 let

119886 =

lfix119871 (119886) sdot 119886 sdot

rfix119871 (119886) isin 119860

(2)

Note that each idempotent is the sum of an element 119890 for119890 isin 119871

0with a linear combination of lower idempotents119891The

set of all 119886 will be denoted by

119872

0 so that for each 119886 isin 119872

0 we

have an element 119886 of the multiplicative basis1198720of 119860 and an

element of the set 1198720 also of 119860 Our aim in this section is to

prove that 1198720is an alternative basis of 119860 which behaves well

with respect to the basis graph

Lemma 6 An idempotent 119890 isin 119871 is its own left and right 119871-idempotent

Proof Obviously 119890 119890 = 119890 However if119891 is any other elementof 1198710such that 119890 119891 = 119890 then by the definition of the partial

ordering on idempotents 119890 ⪯ 119891This shows that 119890 is aminimalright idempotent and the proof on the left is dual

We define a collection of subsets of119872 by

119878

0= 119890

1015840119872

0 119890

10158401015840| 119890

1015840 119890

10158401015840isin 119871

0 (3)

This is the full collection of 119871-Peirce sets of 119872 if there is nozero in themonoidWhen there is a zero119885 there would be anadditional Peirce set 119885 which we ignore because it vanishesunder the passage to the reduced monoid algebra In mildabuse of notation we refer to the sets in 119878

0as 1198710-Peirce sets

We impose a linear ordering on 119878

0which is subordinate

to the natural partial ordering on 119871

0times 119871

0and then a linear

ordering on 119872

0which is subordinate to the ordering on 119878

0

We impose the same ordering on 119872

0 We now make the

following claim

Lemma 7 The linear transformation 119891 119860 rarr 119860 whichmaps 119886 isin 119872

0to 119886 written in the ordered basis 119872

0 is upper

triangular with 1 on the diagonal and thus invertible

Proof By the construction of the primitive idempotents of119870119871

0 the idempotent 119890 is given by 119890 with coefficient 1 and a

linear combination of lower idempotents Thus 119886 is the sumof 119886with coefficient 1 and a linear combination of elements of119872

0from strictly smaller 119871

0-Peirce setsThis gives the desired

result

Proposition 8 The set

119872

0= 119886 | 119886 isin 119872

0 is a basis for

119860 whose elements lie in the components of the Peirce decom-position of 119860 by the idempotent set 119871

0 Thus the dimension of

each Pierce component 1198901015840119860

119890

10158401015840 can be obtained from themonoidby calculating the number of elements with left idempotent 1198901015840and right idempotent 11989010158401015840

Proof By Lemma 7 the matrix mapping1198720to

119872

0is invert-

ible Since1198720is a basis of 119860 so is 119872

0

Since every 119886 has left and right idempotents from

119871

0 the

set 1198720respects the set of idempotents 119871

0as in Definition 1

and each 119886 lies in an

119871

0-Peirce component Since they are

linearly independent the number of 119886 in any

119871

0-Peirce

component of

119872

0is less than or equal to the dimension

but since the total number of elements in

119872

0is equal to the

dimension of 119860 each inequality must in fact be equality

This proposition demonstrates the possibility of choosinga coarse set of idempotents which after conversion to appro-priate representation algebra idempotents using inclusion-exclusion methods will give a one-to-one correspondencebetween semigroup elements and basis elements for a basisof the monoid algebra respecting

119871

0 thus indicating that

the aspects of the monoid algebra visible at this level ofrefinement are natural to the semigroup To get themaximumuse out of the proposition one should choose the coarse set ofidempotents as fine as possible without losing commutativityThe best situation of all is that in which the set of idempotents

119871

0is actually a complete set of primitive idempotents and we

will address that case in the last section


4 Examples

In general

119872

0will not be a multiplicative basis and even

when it is as in Example 1 below the multiplication will notnecessarily coincide with the multiplication in the monoidthat is we may have 119886119887 =

119886119887Our first example is a very natural one in which

119871

0is a

complete set of primitive idempotents

Example 1 (monoid with multiplicative basis in which thesdot-operator does not respect multiplication) Let 119872 be themonoid of 119899 times 119899 matrix units 119864

119894119895

119899

119894119895=1 together with an

identity element 1 and a zero element 119885 The monoid 119871 istaken to be the matrix units of the diagonal together with1 and 119885 while 119871

0= 119871 minus 119885 The reduced monoid algebra

119860 is isomorphic to the semisimple algebra 119870 oplus 119872

119899(119870) In

this case since the poset of idempotents is given by 119885 le 119864

119894119894

for all 119894 = 1 119899 the inclusion and exclusion processes arevery simple giving 119864

119894119894minus 119885 for 119894 = 1 119899 119885 for 119885 and

1minussum

119899

119894=1119864

119894119894+(119899minus1)119885 Dividing119870119872 by the ideal119870119885 to get119860

we then have

119864

119894119894= 119864

119894119894= 119864

119894119894+ 119870119885 for 119894 = 1 119899 The basis

graph is a directed graph with an isolated vertex for 1 and119899 vertices 119890

119894119894 119894 = 1 119899 with one arrow in each direction

between any pair of the 119899 vertices In this case the basis 1198710is

multiplicative since

119864

119894119895

119864

119896ℓ=

119864

119894119895119864

119896ℓand

1 sdot

119864

119894119895=

119864

119894119895sdot

1 = 0Note that this last equation shows that we do not always have119886 sdot

119887 =

119886119887 even when the basis is multiplicative

This monoid has three J-classes one containing all thematrix units one containing the identity and one containingonly 119885 The quiver of 119860 consists of two points since theequations 119864

119894119894= 119864

119894119895119864

119895119894and 119864

119895119895= 119864

119895119894119864

119894119895show that all

of the idempotents in 119871

0are conjugate as defined in the

IntroductionExample 1 represents one extreme possibility in which

almost all idempotents of 1198710are in a single J-class The

opposite extreme in which each element of 1198710corresponds

to a different J-class is actually a common situation for theimportant class of linear algebraic monoids which have beenextensively studied by Putcha [9]

Definition 9 A cross-section lattice 119871 for119872 is a commutativeidempotent submonoid which contains exactly one elementfrom each regularJ-class

Linear algebraic monoids arise as the Zariski closures oflinear algebraic groups but there are more general examplesThe cross section lattices in the linear algebraicmonoids ariseas the set of idempotents inside a maximal torus Most linearalgebraic monoids are infinite but they have finite versionsdefined over finite fields

Example 2 (monoid with cross section lattice) If 119860 is anyfinite dimensional algebra then its monoid (119860 sdot) is a linearalgebraic monoid Over a finite field this will be a finitemonoid Let 119890

1 119890

119899 be a complete set of primitive

idempotents for 119860 as an algebra ordered so that 1198901 119890

119896

with 119896 le 119899 being representatives of conjugacy classes of

idempotents If as a concrete instance of this construction119860 was the119898times119898matrix monoid over a field of two elementsthen the matrix units 119864

119894119894 119894 = 1 119899 would be such a set of

primitive idempotents but they would all be conjugate so wewould have 119896 = 1

Returning to the general case every subset 119868 sube 1 119896let 119890119868= sum

119894isin119868119890

119894 Then 119890

119868| 119868 sube 1 119896 is a cross section

lattice of (119860 sdot) There are 2119896 regularJ-classes in (119860 sdot)

Most finite monoids do not have cross section lat-tices When a cross-section lattice exists it is a naturalbut not inevitable choice for the commutative idempotentsubmonoid 119871

Example 3 (another monoid with cross section lattice) Let(119875 ⪯) be a poset with 119899 elements 119909

1 119909

119899arranged so that

119909

119894⪯ 119909

119895implies that 119894 le 119895 Let 119872 be a submonoid of the

119899 times 119899 matrices over the field F2of two elements in which an

element 119886119894119895= 0 if 119909

119894997410 119909

119895 In particular all the matrices in

119872 are upper triangular Let 119871 be the set of all 2119899 diagonalmatrices which are all idempotents because 0 and 1 areidempotents of the field The submonoid 119871 contains the zeromatrix and is a cross section lattice as in Definition 9 above

The idempotents 119890 in

119871

0are primitive only when the

diagonal matrix has only one nonzero entry Otherwise thesubmonoid 119890119872119890 has a nontrivial maximal subgroup 119866 andin order to find a set of primitive idempotents for 119890119860119890 we willhave to decompose119870119866 into its irreducible representations

In an earlier paper [10] we proposed an even coarser setof idempotents based on intervals along the diagonal

There are many important examples of cross sectionlattices The one we will consider with particular care whichmotivated our search for a general coarse structure is themonoid of endofunctions on a finite set This monoid has nozero element

Example 4 (obtaining a complete decomposition of the iden-tity into orthogonal idempotents for a specific representationof 119879119899) Let 119879

119899be the monoid of functions from the set 119873 =

1 2 119899 into itself acting on the right with compositionas the operation The monoid 119879

119899has 119899

119899 elements Thismonoid has a natural filtration by the two-sided ideals

119871

119894= 119891 isin 119879

119899|

1003816

1003816

1003816

1003816

119873 sdot 119891

1003816

1003816

1003816

1003816

le 119894 (4)

with 119871119899= 119879

119899 We define a representation 120588 of 119879

119899as operating

from the right on a vector space 119881 = ⟨V1 V

119899⟩ by sending

119891 isin 119879

119899to 120588(119891) the matrix with entries 1 in positions (119894 119894 sdot 119891)

and 0 everywhere else Then 119891 isin 119871

119894if and only if 120588(119891) has

rank 119894 If we denote the constant map with unique image 119895 by119895 then 120588(119895) is the matrix with all ones in column 119895 These arethe only matrices in the image of 120588 of rank 1 so

119871

1= 1 119899 (5)


These constant maps will play a special role in what followsso we note for future reference that if 119904 is an arbitrary elementof 119879119899 then

119904 ∘ 119894 = 119894

119894 ∘ 119904 = 119894 ∘ 119904

(6)

We now fix a distinguished index 1 and define embed-dings 120572

119894of119879119894in119879119899by letting 120572

119894(119891) be the element of119879

119899which

acts like 119891 on 1 119894 and is constant equal to 1 on theremaining elements This embedding is a homomorphism ofsemigroups though not a homomorphism of monoids If welet id119894represent the identity element of 119879

119894 then we set

119890

119894= 120572

119894(id119894) (7)

Note that 1198901= 120572

1(id1) = 1 and 119890

119899= id119899

Since each 119890

119894is the image of an idempotent under a

homomorphism of semigroups it is itself an idempotentFurthermore

119890

119894∘ 119890

119895= 119890min(119894119895) (8)

so that under the ordering of idempotents in a semigroup bywhich

119891 ⪯ 119890 lArrrArr 119890 ∘ 119891 = 119891 = 119891 ∘ 119890 (9)

we have that the ordering of 119890119894is according to the ordering of

the indices and by the order of J-classes Furthermore theset 119871 = 119890

119894 is in fact a cross section lattice in 119879

119899 We define

119891

1= 119890

1= 1

119891

119894= 119890

119894minus 119890

119894minus1 119894 = 2 119899

(10)

which is the Moebius inversion for a chain Then the idem-potent 119891

119894is orthogonal to all the 119890

119895with 119895 lt 119894 and thus to all

the idempotents119891119895with 119895 lt 119894 A simple telescoping argument

shows that

119891

1+ sdot sdot sdot + 119891

119899= id119899 (11)

so 119891

1 119891

119899 is a complete orthogonal set of idempotents

for 119860

Definition 10 The right rank rr(119891) of a map f is the width ofits image that is the largest number in the image If rr(119891) =119894 then the last 119899 minus 119894 columns of 120588(119891) are zero and the 119894thcolumn is nonzero The left rank lr(119891) of a map is the indexof the largest number 119895 such that 119895 sdot119891 = 1 sdot119891 unless themap isa constant in such case the left rank is 1 In terms of matricesthis means that 119895 is the lowest row of 120588(119891) which is not equalto the first row

In fact the right rank of 119891 is the smallest 119894 such that119891 ∘ 119890

119894= 119891 since 119890

119894acts as the identity on the first 119894 columns

and the ldquotailrdquo of 119890119894does not act because all those columns are

zero The product is composition but because the action isfrom the right first 119891 acts and then 119890

119894 The left rank of 119891 is

the smallest 119894 such that 119890119894∘ 119891 = 119891 since the first 119894 elements go

as they go in 119891 and the remainder go to 1 sdot 119891 The constantmap 119895 has right rank 119895 and left rank 1The left and right ranksof 119890119894are 119894The basis graph of119879

1is the point119891

1The basis graph of119879

2

with respect to the idempotent set 1198911 119891

2 consists of the two

vertices an arrow from119891

1to1198912given by themap 2 and a loop

at1198912given by themap119891which transposes 1 and 2Thiswill be

the appropriate coarse structure If we refine this by splitting119891

2into two primitive idempotents then the loop vanishes

to produce the second idempotent and the arrow comes outof only one of the primitive idempotents corresponding tothe nontrivial representation of the corresponding maximalsubgroup 119862

2

It should be pointed out that where a choice is involvedin the selection of the lattice 119871 the operation of sending119886 to 119886 may behave strangely with regard to the J-classesIn the example of the mappings 119879

119899 all the constant maps

are idempotents lying in the lowest J-class By the choicewe made of the lattice 119871 1 =

1 remains an idempotentwhereas all the other 119895 = 119895 minus 1 become arrows in the radicalAll the various choices are conjugate obtained one from theother by permutations of the underlying set 1 2 119899 onwhich the mappings act From the point of view of algebrarepresentation theory this is quite natural since the differencebetween two conjugate idempotents in a basic algebra isa linear combination of nonlooped arrows However itdoes mean that not all members of a J-class have similarrepresentations in the basis graph

One solution to this problem then would be to restrictourselves to J-trivial monoids those for which there isa unique idempotent in each regular J-class HoweverExample 1 above shows that this would be too restrictive

Returning to the question of multiplicative bases it isnot hard to check by direct calculation in the case of 119879

2

of Example 4 above that the basis

119872

0is multiplicative We

turn now to1198793and the 14-dimensional component with right

and left idempotents 1198913 in order to show that 119872

0is not in

general a multiplicative basisThe elements not in the radicalcorrespond to the six permutations of 1 2 and 3 and theelements in the radical-squared correspond under the map-ping of 119886 to 119886 to all the arrangements of two distinct numberssuch that 3 is among them and the first and last numbers aredifferent These are 311 322 133 233 331 332 113 and223 Let us set 119908 = 231 and calculate

119908 119908 (12)

A slightly tedious calculation of the sixteen elements in theproduct shows that eight cancel each other out and theremaining eight correspond to

312 minus

322 =119908119908 minus

119908 119890

2 119908 notin

119872

0 (13)

Thus the basis is not multiplicative


5 The Fine Structure

In semigroup theory a pseudovariety is a collection of finitesemigroups closed under homomorphic images subsemi-groups and finite direct products Although we have beenstudying monoids the property of being a monoid is notstable under taking subsemigroups aswe see fromExample 1since the matrix units together with 119885 are a semigroupSince every semigroup can be converted into a monoid byadjoining an identity element we will content ourselves withtrying to find pseudovarieties for which there is a coarsestructure which coincides with the fine structure The pointat which this correspondence most readily breaks down isat the point where the local subgroups are broken downinto irreducible representations Thus for discussing the finestructure we will consider only semigroups for which themaximal subgroups are all trivial This is the pseudovarietyA of aperiodic semigroups

Example 5 (aJ-trivial monoid) Consider the monoid of allpartial 1-1 maps from 119873 = 1 2 119899 to itself for somenatural number 119899 This has a submonoid119872 consisting of allthe partial maps 119891 which are order preserving and extensivewhich means that if 119894119891 is defined for some 119894 isin 119873 then 119894119891 ge 119894These monoids are not just aperiodic they are actually J-trivial [5]

For this particular example the connected components ofthe basis graph are determined by a natural number 119889 whichis the commondimension of the domain and imageThe rightand left idempotents are the identitymaps on the domain andimageThe quiver is theHasse diagram of the partial order onsubsets 119878 and 119879 of size 119904 given by 119878 ge 119879 if there is an order-preserving extensive119891 119878 rarr 119879The basis graph is the entirediagram generated by the partial order

The representation theory of J-trivial monoids hasreceived considerable attention recently [5] There is ahomomorphism from a J-trivial monoid into its latticeof idempotents and when this is extended to the monoidalgebras the kernel corresponds to the radical in the rep-resentation algebra There is a one-to-one correspondencebetween the idempotents of the monoid and the irreduciblerepresentations in the representation algebra See [5] fordetails

The idempotents of aJ-trivial monoid do not in generalcommute and thus we cannot apply our theory to every J-trivial monoid However in Example 5 the idempotents arepartial identity maps which do commute with each otherThus the idempotents in that example form a lattice in whichthe meet is given by multiplication We conclude that theidempotents 119890 are all primitive and thus by Proposition 8 thecoarse and fine structures coincide

The simplest case in which we can find a semilattice ofidempotents is a case like Example 5 where all the idempo-tents commute the variety IC Ash [11] proved that if a semi-group lies in this pseudovariety then it is a homomorphicimage of a subsemigroup of an inverse semigroup one forwhich to each element 119904 is associated a unique element 119904minus1such that 119904119904minus1119904 = 119904 and 119904

minus1119904119904

minus1= 119904

minus1 By uniqueness since

for any idempotent the choice 119904 = 119904

minus1 satisfies this conditioneach idempotent is its own partial inverse

Example 6 (monoid where the coarse and fine structurescoincide) Let 119873 = 1 2 119899 for some natural number 119899and let 119872 be the set of one-to-one order-preserving partialmaps 119891 119873 rarr 119873 The idempotents in this monoidare the partial identity maps and thus commute with eachother It is aperiodic because for any idempotent 119890 the onlyelement in 119890119872119890 is 119890 itself for if 119878 is the common domain andrange of 119890 the only order-preserving one-to-one map from119878 to itself is the identity Although aperiodic the monoid isnot J-trivial The J-classes are determined by the number119894 of elements in 119878 so that the J-class of 119890 contains theidempotents corresponding to all the 119899119894(119899 minus 119894) subset of119873with 119894 elements as well as all order-preserving partial mapswhose domain and range have 119894 elements Since for eachpair of sets 119878 and 119879 with 119894 elements there is a unique one-to-one order-preserving map between them the principalfactor corresponding to each J-class has a reduced algebraisomorphic to the matrix algebra of the size of the numberof idempotents in J Since these matrix algebras generatethe matrix blocks of the Munn-Ponizovskii theorem theidempotents 119890 are in fact primitive since they correspond tothe diagonal idempotents of the matrix blocks

For any semigroup 119878 an ideal 119868 is a subset closedunder multiplication by elements of 119878 and the Rees quotientsemigroup 119878119868 is the semigroup obtained by replacing theelements of 119868 by a single zero element 119885 The principal factorof a semigroup corresponding to aJ-class 119869 is 119878119869119878119873 where119873 = 119878119869119878 minus 119869 and if 119873 = 0 then we understand the quotientto be 119869The semigroups which can appear as principal factorsare of a limited number of typesThey can be semigroupswithzero multiplication 0-simple semigroups or ideal simplesemigroups

Proposition 11 Let 119872 be a monoid with zero 119885 If themonoid is aperiodic and has commuting idempotents thecoarse structure of 119872 with respect to the set of all non-zeroidempotents will coincide with its fine structureThere is a one-to-one correspondence between the non-zero elements of themonoid and the set of vertices and arrows of the basis graphs

Remark 12 The monoid thus belongs to A cap IC the inter-section of the pseudovariety A of aperiodic semigroupsand the pseudovariety IC of semigroups with commutingidempotents We mention this notation because some of thetheorems we rely on are phrased in the language of varieties

Proof Because we are in the pseudovariety ICwe can choose119871 to be the set of all idempotents and it will be a submonoidWe let 119869

1 119869

119898 be the set of regular J-classes in 119872

where 119869

1is the class of 1 and 119869

119898is the class of 119885 By

the Munn-Ponizovskii theorem the radical quotient of therepresentation algebra 119860 of119872 is of the form

119860

1015840=

119898minus1

⨁

119894=1

119872

119886119894(119870119866

119894) (14)


where the 119866

119894are the maximal subgroups of the regular J-

class 119869119894and 119886

119894is a number depending on the structure of the

class 119869119894 We have assumed that the maximal subgroups are

trivial so each regular J-class gives a single matrix blockFurthermore by the standard reference [12] we find that 119886

119894

for a monoid in IC is exactly the number of idempotents intheJ-class

By general properties of the semigroups in the pseudova-riety IC the set Reg(119872) of elements in regular J-classesis a submonoid of 119872 Since each of these regular J-classescontains an idempotent the regular principal factors cannothave zero multiplication and since we have assumed that 119872has a zero it cannot be ideal simple so it must be 0-simpleand in fact completely 0-simple since it contains a regularregularJ-class A regular semigroup with commuting idem-potents is an inverse semigroup that is every element 119904 hasa unique inverse 119904minus1 belonging to the sameJ-class as 119904 Eachregular factor is thus also an inverse subsemigroup and is infact a Brandt semigroup a completely 0-simple semigroupwhich is an inverse semigroup It is also aperiodic and a finiteaperiodic Brandt semigroup is a semigroup of matrix unitsThus for each regular J-class 119869 for a monoid in IC cap A theprincipal factor is a matrix unit semigroup If 119890 and 119891 areequivalent idempotents there must be elements 119909 isin 119890119872119891

and 119910 isin 119891119872119890 such that 119890 = 119909119910 and 119891 = 119910119909 In thiscase 119909 and 119910 are unique and are precisely the matrix unitswhich transfer from 119890 to 119891 and back The degree is equal tothe number of idempotents equivalent to 119890 The basis graphfor the corresponding matrix block is the complete directedgraph with a number of vertices equal to the degree of thematrix block where for a directed graph completeness givesone arrow in each direction between any pair of vertices

Thus for each regular J-class the orthogonal idempo-tents obtained by inclusion-exclusion from the idempotentsin the J-class are equal in number to the dimension of thematrix block If so they must be primitive and the coarsestructure determined by this choice of idempotents doesindeed coincide with the fine structure Thus by applyingProposition 8 we have a one-to-one correspondence betweenthe elements of themonoid and the vertices and arrows of thebasis graph

If we were to require the semigroup idempotents tobe primitive then the regular part would have to be anannihilating sum of Brandt semigroups as mentioned inthe remark after Theorem 3 of [13] However since we onlyrequire the algebra idempotents after inclusion-exclusion tobe primitive we have access to a much richer collection ofsemigroups

Conflict of Interests

The author declares that there is no conflict of interestsregarding the publication of this paper

References

[1] F V Saliola ldquoThe quiver of the semigroup algebra of a left reg-ular bandrdquo International Journal of Algebra and Computationvol 17 no 8 pp 1593ndash1610 2007

[2] S Margolis and B Steinberg ldquoQuivers of monoids with basicalgebrasrdquo Compositio Mathematica vol 148 no 5 pp 1516ndash1560 2012

[3] N M Thiery ldquoCartan invariant matrices for finite monoidsexpression and computation using charactersrdquo DMTCS Pro-ceedings pp 887ndash898 2012

[4] T Denton Excursions into Algebra and Combinatorics at q = 02012

[5] T Denton F Hivert A Schilling and N M Thiery ldquoOn therepresentation theory of finite 119869-trivial monoidsrdquo SeminaireLotharingien de Combinatoire vol 64 article B64d 2011

[6] M Schaps ldquoDeformations of finite-dimensional algebras andtheir idempotentsrdquo Transactions of the American MathematicalSociety vol 307 no 2 pp 843ndash856 1988

[7] W D Munn ldquoMatrix representations of semigroupsrdquo Proceed-ings of the Cambridge Philosophical Society vol 53 pp 5ndash121957

[8] L Solomon ldquoThe Burnside algebra of a finite grouprdquo Journal ofCombinatorial Theory vol 2 pp 603ndash615 1967

[9] M S Putcha Linear Algebraic Monoids vol 133 of LondonMathematical Society Lecture Note Series 1988

[10] M Gerstenhaber and M Schaps ldquoFinite posets and theirrepresentation algebrasrdquo International Journal of Algebra andComputation vol 20 no 1 pp 27ndash38 2010

[11] C J Ash ldquoFinite semigroups with commuting idempotentsrdquoAustralianMathematical Society A vol 43 no 1 pp 81ndash90 1987

[12] A H Clifford and G B PrestonThe Algebraic Theory of Semi-groups AMS 1961

[13] G Lallement and M Petrich ldquoSome results concerning com-pletely 0-simple semigroupsrdquo Bulletin of the American Mathe-matical Society vol 70 pp 777ndash778 1964

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Algebra

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Stochastic AnalysisInternational Journal of


similarly the third section in Dentonrsquos thesis [4] takes upthis themeThe anonymous referee pointed out the similarityof our concept to the third section of [5] and we have infact switched from our original notation to a version of theirnotation ldquolfixrdquo and ldquorfixrdquo to emphasize this similarity In thecase given in our Section 5 in which the coarse and finestructures coincide the entries in the matrix 119862 appear to bethe sizes of what we call the Peirce sets of themonoid definedin our Section 3

Let 119872 be a finite monoid and let 119864(119872) be its set ofidempotents We recall that the J-class of an element 119886 of119872 is the set of all 119887 isin 119872 such the 119872119886119872 = 119872119887119872 A J-class is regular if it contains an idempotent Two idempotents119890 and 119891 are called conjugate if they lie in the same J-classand this happens if and only if there are elements 119909 and 119910 inthe monoid such that 119890 = 119909119910 and 119891 = 119910119909

2 The Basis Graph of a FiniteDimensional Algebra

For any complete orthogonal setE of idempotents in a finitedimensional algebra119860with Jacobson radical 119869 we can definea basis graph [6] to be the directed graph with one vertexlabeled 119891

119894for each idempotent 119891

119894and the following loops and

arrows

(i) 1198990119894119894minus 1 loops of weight zero

(ii) 119899119896119894119895arrows or loops of weight 119896 for 119894 = 119895 or 119896 gt 0 where

119899

119896

119894119895= dim

119870(119891

119894(

119869

119896

119869

119896+1)119891

119895) (1)

Definition 1 A basis 119861 of 119860 which is a union of bases forthe different Peirce components 119891

119894119860119891

119895is said to respect the

idempotent set E

A refinement E1015840 of E is a complete set of orthogonalidempotents such that every element 119890 of E is a sum ofelements of E1015840 If we have such a refinement then we geta refinement of the basis graph in which some of the loopsare replaced by vertices and arrows For example the uppertriangular 2 times 2 matrices over a field 119896 considered withrespect to a set of idempotents containing only the identitymatrix would have a basis graph consisting of one vertexand two loops one of weight zero and one of weight 1 Wecan take a refinement of the idempotent set consisting ofthe idempotent diagonal matrices 119864

11and 119864

22 The refined

basis graph will have two vertices corresponding to the twoidempotents and one arrow of weight one

The basis graph for any complete orthogonal set ofprimitive idempotents is an invariant of the algebra sinceany two such sets are conjugate If the algebra is basic thenthe quiver is with respect to right modules just the basisgraph with all arrows of weight greater than one erasedsince the quiver is defined by considering the dimension of(119891

119894(119869119869

2)119891

119895) (if one uses leftmodules onemust take the dual)

If the algebra is not basic then all matrix blocks togetherwith the arrows of weight zero representing matrix units

must be shrunk to points in the quiver with a correspondingcoalescence of arrows

3 Finite Monoids with a Chosen CommutativeIdempotent Submonoid

Our aim in this section is to construct an alternative basis of119860 which still corresponds one-to-one to the set of elementsof the monoid but which behaves well with respect to thebasis graph defined above In particular we will want theidempotents to be orthogonal and the other basis elementsto respect the idempotent set as in Definition 1

We now consider a finite monoid (119872) with a chosencommutative idempotent submonoid 119871 If119872 has an absorb-ing zero element 119885 we assume that 119885 is also in 119871 A trivialchoice would be to take 119871 to be the identity element togetherwith 119885 if it exists Because of the commutativity the set 119871 ispartially ordered by the relation 119909 ⪯ 119910 hArr 119909 119910 = 119909 andis a semilattice under this ordering with the greatest lowerbound of two elements being their product Since 119871 containsa maximal element the identity then 119871 is also a lattice sincethe least upper bound of two elements 119909 and 119910 is the greatestlower bound of all the elements of 119871 which dominate both 119909

and 119910 and this set is nonempty because the identity alwaysdominates both 119909 and 119910

Definition 2 Let 119872 be a monoid with operation Let 119871 bea commutative idempotent submonoid For an element 119886 ofthe monoid 119872 the left 119871-idempotent lfix

119871(119886) is the product

of all the idempotents 119890 in the lattice 119871 such that 119890 119886 = 119886Similarly the right 119871-idempotent rfix

119871(119886) is the product of all

the idempotents 119890 in the lattice 119871 such that 119886 119890 = 119886 At leastone such idempotent always exists since the identity of themonoid satisfies the condition

Remark 3 If119872 has an absorbing zero119885 then119885 is not the leftor right idempotent of any element except itself

We now pass to the reduced monoid algebra 119860 of themonoid119872 over a field 119870 where 119870 is a field sufficiently largethat the quotient of 119860 by its radical 119869 splits completely as asum of matrix blocks over119870 One might for example take119870to be algebraically closed If119872 does not contain an absorbingzero element then we set 119860 = 119870119872 and 119861 = 119870119871 and define119872

0= 119872 and 119871

0= 119871 If119872 does contain a necessarily unique

zero element 119885 then we let 119860 be the quotient 119870119872119870119885

with subalgebra 119861 = 119870119871119870119885 This reduction is madebecause the full monoid algebra 119870119872 is isomorphic to adirect product of an algebra isomorphic to 119860 and a copyof the 119870 representing the ideal 119870119885 which gives amongother disadvantages an algebra which is decomposable anddisconnected quiver where algebra representation theoristsprefer to work with indecomposable algebras From the pointof view of representation theory the object of study is thealgebra 119860 We define sets 119872

0= 119872 minus 119885 and 119871

0= 119871 minus 119885

Note that in the second case 119872

0and 119871

0may no longer be

monoids since the product of two elements different from 119885

might be 119885



0= 119898 | 119898 isin 119872

0 and

119871

0= 119890119890 isin 119871

0 respectively


The basis 119872









119891le119911lt119890120583(119891 119911) for 119891 lt 119890


0 let 119890 = sum

119891le119890120583(119891 119890)119891 be the


0

For any 119886 isin 119872

0 let

119886 =

lfix119871 (119886) sdot 119886 sdot

rfix119871 (119886) isin 119860

(2)




119872


0 we









119878

0= 119890

1015840119872

0 119890

10158401015840| 119890

1015840 119890

10158401015840isin 119871

0 (3)






0times 119871

0and then a linear

ordering on 119872


0


0 We now make the

following claim



0 is upper







result


119872

0= 119886 | 119886 isin 119872

0 is a basis for




119890



119872

0is invert-


0


119871

0 the


0as in Definition 1


119871



119871

0-Peirce

component of

119872



119872

0is equal to the



119871



119871




4 Examples

In general

119872




119871

0is a



119894119895

119899





119899(119870) In


119894119894


119894119894minus 119885 for 119894 = 1 119899 119885 for 119885 and

1minussum

119899

119894=1119864


we then have

119864

119894119894= 119864

119894119894= 119864

119894119894+ 119870119885 for 119894 = 1 119899 The basis





119864

119894119895

119864

119896ℓ=

119864

119894119895119864

119896ℓand

1 sdot

119864

119894119895=

119864

119894119895sdot


119887 =



119894119894= 119864

119894119895119864

119895119894and 119864

119895119895= 119864

119895119894119864











1 119890



119896






119894isin119868119890

119894 Then 119890





1 119909


119909

119894⪯ 119909



element 119886119894119895= 0 if 119909

119894997410 119909




119871








119899has 119899


119871

119894= 119891 isin 119879

119899|

1003816

1003816

1003816

1003816

119873 sdot 119891

1003816

1003816

1003816

1003816

le 119894 (4)

with 119871119899= 119879


119899as operating



119891 isin 119879



119894if and only if 120588(119891) has


119871

1= 1 119899 (5)



119904 ∘ 119894 = 119894

119894 ∘ 119904 = 119894 ∘ 119904

(6)


119894of119879119894in119879119899by letting 120572

119894(119891) be the element of119879

119899which


119894 then we set

119890

119894= 120572

119894(id119894) (7)

Note that 1198901= 120572

1(id1) = 1 and 119890

119899= id119899

Since each 119890



119890

119894∘ 119890

119895= 119890min(119894119895) (8)


119891 ⪯ 119890 lArrrArr 119890 ∘ 119891 = 119891 = 119891 ∘ 119890 (9)




119899 We define

119891

1= 119890

1= 1

119891

119894= 119890

119894minus 119890

119894minus1 119894 = 2 119899

(10)





shows that

119891


119899= id119899 (11)

so 119891

1 119891


for 119860



119894= 119891 since 119890







1is the point119891


2









2







2


119872




0is not in


119908 119908 (12)


312 minus

322 =119908119908 minus

119908 119890

2 119908 notin

119872

0 (13)










minus1119904119904

minus1= 119904









1 119869


where 119869




119860

1015840=

119898minus1

⨁

119894=1

119872

119886119894(119870119866

119894) (14)


where the 119866


class 119869119894and 119886




119894








References





















Volume 2014




Journal of











Journal of


Function Spaces






Algebra











0= 119898 | 119898 isin 119872

0 and

119871

0= 119890119890 isin 119871

0 respectively


The basis 119872









119891le119911lt119890120583(119891 119911) for 119891 lt 119890


0 let 119890 = sum

119891le119890120583(119891 119890)119891 be the


0

For any 119886 isin 119872

0 let

119886 =

lfix119871 (119886) sdot 119886 sdot

rfix119871 (119886) isin 119860

(2)




119872


0 we









119878

0= 119890

1015840119872

0 119890

10158401015840| 119890

1015840 119890

10158401015840isin 119871

0 (3)






0times 119871

0and then a linear

ordering on 119872


0


0 We now make the

following claim



0 is upper







result


119872

0= 119886 | 119886 isin 119872

0 is a basis for




119890



119872

0is invert-


0


119871

0 the


0as in Definition 1


119871



119871

0-Peirce

component of

119872



119872

0is equal to the



119871



119871




4 Examples

In general

119872




119871

0is a



119894119895

119899





119899(119870) In


119894119894


119894119894minus 119885 for 119894 = 1 119899 119885 for 119885 and

1minussum

119899

119894=1119864


we then have

119864

119894119894= 119864

119894119894= 119864

119894119894+ 119870119885 for 119894 = 1 119899 The basis





119864

119894119895

119864

119896ℓ=

119864

119894119895119864

119896ℓand

1 sdot

119864

119894119895=

119864

119894119895sdot


119887 =



119894119894= 119864

119894119895119864

119895119894and 119864

119895119895= 119864

119895119894119864











1 119890



119896






119894isin119868119890

119894 Then 119890





1 119909


119909

119894⪯ 119909



element 119886119894119895= 0 if 119909

119894997410 119909




119871








119899has 119899


119871

119894= 119891 isin 119879

119899|

1003816

1003816

1003816

1003816

119873 sdot 119891

1003816

1003816

1003816

1003816

le 119894 (4)

with 119871119899= 119879


119899as operating



119891 isin 119879



119894if and only if 120588(119891) has


119871

1= 1 119899 (5)



119904 ∘ 119894 = 119894

119894 ∘ 119904 = 119894 ∘ 119904

(6)


119894of119879119894in119879119899by letting 120572

119894(119891) be the element of119879

119899which


119894 then we set

119890

119894= 120572

119894(id119894) (7)

Note that 1198901= 120572

1(id1) = 1 and 119890

119899= id119899

Since each 119890



119890

119894∘ 119890

119895= 119890min(119894119895) (8)


119891 ⪯ 119890 lArrrArr 119890 ∘ 119891 = 119891 = 119891 ∘ 119890 (9)




119899 We define

119891

1= 119890

1= 1

119891

119894= 119890

119894minus 119890

119894minus1 119894 = 2 119899

(10)





shows that

119891


119899= id119899 (11)

so 119891

1 119891


for 119860



119894= 119891 since 119890







1is the point119891


2









2







2


119872




0is not in


119908 119908 (12)


312 minus

322 =119908119908 minus

119908 119890

2 119908 notin

119872

0 (13)










minus1119904119904

minus1= 119904









1 119869


where 119869




119860

1015840=

119898minus1

⨁

119894=1

119872

119886119894(119870119866

119894) (14)


where the 119866


class 119869119894and 119886




119894








References





















Volume 2014




Journal of











Journal of


Function Spaces






Algebra










4 Examples

In general

119872




119871

0is a



119894119895

119899





119899(119870) In


119894119894


119894119894minus 119885 for 119894 = 1 119899 119885 for 119885 and

1minussum

119899

119894=1119864


we then have

119864

119894119894= 119864

119894119894= 119864

119894119894+ 119870119885 for 119894 = 1 119899 The basis





119864

119894119895

119864

119896ℓ=

119864

119894119895119864

119896ℓand

1 sdot

119864

119894119895=

119864

119894119895sdot


119887 =



119894119894= 119864

119894119895119864

119895119894and 119864

119895119895= 119864

119895119894119864











1 119890



119896






119894isin119868119890

119894 Then 119890





1 119909


119909

119894⪯ 119909



element 119886119894119895= 0 if 119909

119894997410 119909




119871








119899has 119899


119871

119894= 119891 isin 119879

119899|

1003816

1003816

1003816

1003816

119873 sdot 119891

1003816

1003816

1003816

1003816

le 119894 (4)

with 119871119899= 119879


119899as operating



119891 isin 119879



119894if and only if 120588(119891) has


119871

1= 1 119899 (5)



119904 ∘ 119894 = 119894

119894 ∘ 119904 = 119894 ∘ 119904

(6)


119894of119879119894in119879119899by letting 120572

119894(119891) be the element of119879

119899which


119894 then we set

119890

119894= 120572

119894(id119894) (7)

Note that 1198901= 120572

1(id1) = 1 and 119890

119899= id119899

Since each 119890



119890

119894∘ 119890

119895= 119890min(119894119895) (8)


119891 ⪯ 119890 lArrrArr 119890 ∘ 119891 = 119891 = 119891 ∘ 119890 (9)




119899 We define

119891

1= 119890

1= 1

119891

119894= 119890

119894minus 119890

119894minus1 119894 = 2 119899

(10)





shows that

119891


119899= id119899 (11)

so 119891

1 119891


for 119860



119894= 119891 since 119890







1is the point119891


2









2







2


119872




0is not in


119908 119908 (12)


312 minus

322 =119908119908 minus

119908 119890

2 119908 notin

119872

0 (13)










minus1119904119904

minus1= 119904









1 119869


where 119869




119860

1015840=

119898minus1

⨁

119894=1

119872

119886119894(119870119866

119894) (14)


where the 119866


class 119869119894and 119886




119894








References





















Volume 2014




Journal of











Journal of


Function Spaces






Algebra











119904 ∘ 119894 = 119894

119894 ∘ 119904 = 119894 ∘ 119904

(6)


119894of119879119894in119879119899by letting 120572

119894(119891) be the element of119879

119899which


119894 then we set

119890

119894= 120572

119894(id119894) (7)

Note that 1198901= 120572

1(id1) = 1 and 119890

119899= id119899

Since each 119890



119890

119894∘ 119890

119895= 119890min(119894119895) (8)


119891 ⪯ 119890 lArrrArr 119890 ∘ 119891 = 119891 = 119891 ∘ 119890 (9)




119899 We define

119891

1= 119890

1= 1

119891

119894= 119890

119894minus 119890

119894minus1 119894 = 2 119899

(10)





shows that

119891


119899= id119899 (11)

so 119891

1 119891


for 119860



119894= 119891 since 119890







1is the point119891


2









2







2


119872




0is not in


119908 119908 (12)


312 minus

322 =119908119908 minus

119908 119890

2 119908 notin

119872

0 (13)










minus1119904119904

minus1= 119904









1 119869


where 119869




119860

1015840=

119898minus1

⨁

119894=1

119872

119886119894(119870119866

119894) (14)


where the 119866


class 119869119894and 119886




119894








References





















Volume 2014




Journal of











Journal of


Function Spaces






Algebra

















minus1119904119904

minus1= 119904









1 119869


where 119869




119860

1015840=

119898minus1

⨁

119894=1

119872

119886119894(119870119866

119894) (14)


where the 119866


class 119869119894and 119886




119894








References





















Volume 2014




Journal of











Journal of


Function Spaces






Algebra










where the 119866


class 119869119894and 119886




119894








References





















Volume 2014




Journal of











Journal of


Function Spaces






Algebra
















Volume 2014




Journal of











Journal of


Function Spaces






Algebra









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