Yingdan Ji and Yanfeng Luo

De Gruyter | Published online: February 9, 2016

# Abstract

We build up a multiplicative basis for a locally adequate concordant semigroup algebra by constructing Rukolaĭne idempotents. This allows us to decompose the locally adequate concordant semigroup algebra into a direct product of primitive abundant 0 - J * -simple semigroup algebras. We also deduce a direct sum decomposition of this semigroup algebra in terms of the * -classes of the semigroup obtained from the above multiplicative basis. Finally, for some special cases, we provide a description of the projective indecomposable modules and determine the representation type.

MSC 2010: 16G30; 16G60

## 1 Introduction

Munn [1, 2] gave a direct product decomposition of finite inverse semigroup algebras into matrix algebras over group algebras using principal ideal series. In [3], this result was independently obtained by Rukolaĭne. His approach was to construct a multiplicative basis by defining the so-called Rukolaĭne idempotents. Munn later showed that the technique developed by Rukolaĭne worked for inverse semigroups with idempotents sets locally finite, see [4].

Recent interest in Möbius functions arose due to the work of Solomon on decomposing the semigroup algebra of a finite semilattice into a direct product of fields [5], and the work of Brown on studying random walks on bands by using representation theory of their semigroup algebras [6]. By using the Möbius functions on the natural partial orders on inverse semigroups, Steinberg extended the results of Solomon and Munn on direct product decomposition of finite inverse semigroups to inverse semigroups with idempotents sets finite, and he explicitly computed the corresponding orthogonal central idempotents [7]. Guo generalized the results described above to finite locally inverse semigroups and finite ample semigroups, again using Möbius functions, see [8, 9].

Decomposing an algebra with an identity into direct sum of projective indecomposable modules is an important problem in representation theory because it provides a complete set of primitive orthogonal idempotents. It also allows for an explicit computation of the Gabriel quiver, the Auslander-Reiten quiver and the representation type of an algebra. It has shown that the semigroup algebras of finite -trivial monoids are basic; furthermore the projective indecomposable modules have been described, see [1012]. However, there is no much description of the projective indecomposable modules of the semigroup algebras which are not basic.

The first part of this paper is primarily concerned with carrying over certain results of inverse semigroup algebras to locally adequate concordant semigroup algebras. In general, the contracted semigroup algebras of locally adequate concordant semigroups are not basic. The remainder of the paper is devoted to exploring a description of the projective indecomposable modules and to determining whether or not these semigroup algebras are representation- finite.

The paper is organized as follows. In Section 2, we provide some background on semigroups and algebras. If R0 [S] is the contracted semigroup algebra of a locally adequate concordant semigroup S with idempotents set E(S) pseudofinite, in Section 3, we generalize the concepts and results of Rukolaĭne idempotents of inverse semigroup algebras to R0 [S]. Section 4 involves constructing a multiplicative basis ¯ of R0 [S], see Theorem 4.8, and developing basic properties of the semigroup S ¯ = B ¯ { 0 } . In Section 5, R0 [S], is decomposed into a direct product of primitive abundant 0 - J * -simple semigroup algebras, see Theorem 5.1. In Section 6, if R0[S] contains an identity, the multiplicative basis ¯ also allows for a direct sum decomposition of R0 [S]. Theorem 6.5 translates the problem involving the projective indecomposable modules of R0 [S] into cancellative monoids theory terms. Furthermore, we determine the representation type of these semigroup algebras.

## 2 Preliminaries

In this section, we recall some basic definitions and results on semigroups and representation theory of algebras. Throughout this paper, let R denote a commutative ring with identity, and denote the zero element of a R-algebra by the symbol 0.

We first recall some definitions and results on semigroups which can be found in [13, 14].

Without loss of generality, we always assume a semigroup S is with a zero element (denoted by θ). Denote the set of all nonzero elements of S and E(S) (the idempotents set of S) by S* and E(S)*, respectively.

Let S be a semigroup and K be an equivalence relation on S. The K -class containing an element a of the semigroup S will be denoted by Ka or Ka(S) in case of ambiguity. We denote the set of nonzero K -classes of S by ( S / K ) * .

Denote by S1 the semigroup obtained from a semigroup S by adding an identity if S has no identity, otherwise, let S1 = S. It is well known that Green’s relations play an important role in the theory of semigroups. They were introduced by Green in 1951: for a; bS

a L b S 1 a = S 1 b , a R b a S 1 = b S 1 , a J b S 1 a S 1 = S 1 b S 1 , H = L R , D = L R .

It is clear that (resp., ) is a right (resp., left) congruence on S and D J . A semigroup S is called regular if every -class and every -class contain idempotents. The regularity of a semigroup S can be characterized by the property that the set V(a) = {a′ ∈ S | aaa = a, aaa′ = a′} is nonempty for each aS.

Pastijn first extended the Green’s relations to the so called “Green’s *-relations” on a semigroup S [15]: for a; bS,

a L * b ( x , y S 1 ) ( a x = a y b x = b y ) , a R * b ( x , y S 1 ) ( x a = y a x b = y b , a J * b J * ( a ) = J * ( b ) , H * = L * R * and D * = L * R * ,
where J *( a) is the smallest ideal containing a which is saturated by * and * .

Clearly, * (resp., * ) is a right (resp., left) congruence on S. It is easy to see that * (resp., * ), and for a, b ∈ Reg (S), a b (resp., a b ) if and only if a * b (resp., a * b ). So = * , = * and J = J * on regular semigroups.

We say a semigroup is abundant if each * -class and each * -class of it contains an idempotent. It is clear that regular semigroups are abundant semigroups.

Let S be an abundant semigroup and aS*. We use a (resp., a*) to denote a typical idempotent related to a by * (resp., * ).

Define two partial orders ≤r and ≤l on S [16] by

a r b R a * R b * and a = a b for some a , a l b L a * L b * and a = b a * for some a * ,

The natural partial order ≤ on S is defined to be ≤=≤r ∩ ≤l. We have an alternative characterisation of ≤: for x, yS, xy if and only if there exist e, fE(S) such that x = ey = yf.

Let S be an abundant semigroup and eE(S)*. Define ω(e) = {fE(S) | fe}. Clearly, ω(e) = E(eSe). For convenience, denote the subsemigroup of S generated by w(e) by 〈e〉.

An abundant semigroup S is called idempotent connected (IC) [17], if for all aS* a R a * ( S ) E ( S ) and a * L a * ( S ) E ( S ) , there is an isomorphism

α a : a a * , with x a = a α a ( x ) ,
for each x ∈ 〈 a 〉. It is known that an abundant semigroup S is IC if and only if ≤ r=≤ l on S [ 16, Theorem 2.6].

A semigroup S is said to satisfy the regularity condition [16] if for all idempotents e and f of S the element ef is regular. If this is the case, the sandwich set S (e, f) = {gV (ef) ∩ E(S) | ge = fg = g} of idempotents e and f is non-empty, and takes the form

S ( e , f ) = { g E ( S ) | g e = f g = g , e g f = e f } .

A semigroup S is said to be concordant if S is IC abundant and satisfies the regularity condition, see [18]. It is known that regular semigroup is concordant, and in this case ≤ coincide with the natural partial order defined by Nambooripad [19].

An abundant semigroup with commutative idempotents is called an adequate semigroup. If each local submonoid eSe (eE(S)*) of a semigroup S is adequate (resp., inverse), then the semigroup S is said to be locally adequate(resp., locally inverse). We say a semigroup locally adequate concordant if it is both concordant and locally adequate.

By [20, Corollary 5.6], an IC abundant semigroup is locally adequate if and only if ≤ is compatible with multiplication. It is well known that inverse (resp., locally inverse) semigroups are regular adequate (resp., locally adequate) semigroups and conversely, so that locally adequate concordant semigroups generalize locally inverse semigroups, and hence generalize inverse semigroups.

Refer to [13, Chapter 8] for the definitions of a left (resp., right) S-system and an (S,T)-bisystem for monoids S, T. Let M be a (S, T)-bisystem. Then the mapping smsm (resp., mtmt) is an (S, T)-isomorphism from SSM (resp., MTT) onto M, and we call it a canonical isomorphism.

We recall the definition of blocked Rees matrix semigroups [14]. Let J and Λ be non-empty sets and be a non-empty set indexing partitions P(J) = {Jλ: λ ∈ Γ}, P(Λ) = {Λλ: λ ∈ Γ} of J and Λ, respectively. We make a convention that i, j, k, l will denote members of J; s, t, m, n will denote members of Λ and λ, μ, ν, κ will denote members of Γ.

By the (λ, μ)-block of a J × Λ matrix we mean those (j,s)-positions with jJλ and s ∈ Λμ. The (λ, λ)-blocks are called the diagonal blocks of the matrix.

For each pair (λ, μ) ∈ Γ × Γ, let Mλμ be a set such that for each λ, Mλλ = Tλ is a monoid and for λλ, either M λ μ = 0 or Mλμ is a (Tλ, Tμ)-bisystem. Moreover, we impose the following condition on {Mλμ :λ, μ ∈ Γ}.

(M) For all λ, μ, ν ∈ Γ, if Mλμ, Mμλ are both non-empty, then Mλμ is non-empty and there is a (Tλ, Tν)-homomorphism φλμν: MλμMμν such that if λ = μ or μ = ν, then φλμν is the canonical isomorphism and such that the square

is commutative.

Here, for aMλμ, bMμν, we denote (ab) φλνμ by ab. On the other hand, let 0 (zero) be a symbol not in any Mλμ with the convention that 0x = x0 = 0 for every element x of { 0 } { M λ μ : λ , μ Γ } .

Denote by (a)js the J × Λ-matrix with entry a in the (j,s)-position and zeros elsewhere. Let M be the set consisting all J × Λ-matrix (a)js, where (j, s) is in some (λ, μ)-block and aMλ, μ, and the zero matrix (denoted by θ). Define a Λ × J sandwich matrix P = (psi) where a nonzero entry in the (λ, μ)-block of P is a member of Mλμ.

Let A = (a)is, B = (b)jtM, by condition (M), the product AB = APB = (apsjb)it makes M be a semigroup, which we denote by 𝓜 0 ( M λ μ ; J , Λ , Γ ; P ) and call a blocked Rees matrix semigroup.

In addition, we call M a PA blocked Rees matrix semigroup if it satisfies the following conditions (C), (U) and(R):

(C) If a, a1, a2Mλμ, b, b1, b2Mμκ, then ab1 = ab2 implies b1 = b2; a1b = a2b implies a1 = a2;

(U) For each λ ∈ Γ and each s ∈ Λλ (resp., jJλ), there is a member j of Jλ (resp., s ∈ Λλ) such that psj is a unit in Mλλ;

(R) If Mλμ, Mare both non-empty where λμ, then abaa for all aMλμ. bMλμ.

We record some elementary properties of PA blocked Rees matrix semigroups in the following lemma.

Lemma 2.1

([14, Proposition 2.4]). Let M = 𝓜 0 ( M λ μ ; J , Λ , T ; P ) be a PA blocked Rees matrix semigroup. Then

(i) a non-zero element (a)isof M is an idempotent if and only if there is an element λ ∈ Γ such that (i, s) ∈ Jλ × Λλand a is a unit in Tλwith inverse psi;

(ii) all nonzero idempotents of M are primitive;

(iii) the non-zero elements (a)isand (b)jtof M are * -related if and only if i = j;

(iv) the non-zero elements (a)isand (b)jtof M are * -related if and only if s = t;

(v) M is abundant;

(vi) the non-zero idempotents e = (a)isand f = (b)jtof M with (i, s) ∈ Jλ Ø Λλand )j, t) ∈ (i, s) ∈ Jμ Ø Λμare D -related if and only if λ = μ;

(vii) the non-zero element (a)isof M is regular if and only if there is an element λ ∈ Γ such that (i, s) ∈ Jλ Ø Λλand a is a unit in Tλ.

Let M = 𝓜 0 ( M λ μ ; J , Λ , T ; P ) be a PA blocked Rees matrix semigroup. Then we can always assume that there exists 1λJλ 1λJλ ∩ Λλ such that H 1 λ 1 λ * = T λ is a cancellative monoid with an identity eλ(λ ∈ Γ).

Recall that a Munn algebra is an algebra ( A ; I , Λ ; p ) of matrix type over an algebra A [21] such that each row and each column of the sandwich matrix P contains a unit of A. Let M = 𝓜 0 ( G ; J , Λ ; P ) be a completely 0-simple semigroup. It is known that R 0 [ M ] 𝓜 ( R [ G ] ; J , Λ ; P ) , see [22, Lemma 5.17].

Let M = 𝓜 0 ( M λ μ ; J , Λ , Γ ; P ) be a PA blocked Rees matrix semigroup. Define the generalized Munn algebra ( R [ M μ λ ] ; J , Λ , Γ ; P ) of M to be the vector space consisting of all the J Ø Λ-matrices (ais) with only finitely many nonzero entries, where aisR[Mλμ] if (i, s) ∈ Jλ Ø Λλ, with multiplication defined by the formula (ais) ∘ (bjt) = (aisP (bjt).

In particular, if | Γ | = 1, the generalized Munn algebra is a Munn algebra.

The proof of the following result is similar to that of [22, Lemma 5.17].

Lemma 2.2.

R 0 [ M ] 𝓜 ( R [ M μ λ ] ; J , Λ , Γ ; P ) .

If ( a i s ) ( R [ M μ λ ] ; J , Λ , Γ ; P ) has only one nonzero entry ajt, we will write (j, a, t) or (a)jt instead of (ais).

Now we recall the definition of primitive abundant semigroups. Let S be an abundant semigroup. If eE(S)* is minimal under the natural order ≤ defined on S, e is said to be primitive. It is known that an idempotent eS is primitive if and only if e has the property that for each idempotent fE(S), fe = ef = fθ ⇒) f = e. The semigroup S is said to be primitive abundant if all its nonzero idempotents are primitive.

By Lemma 2.1(ii) and (v), PA blocked Rees matrix semigroups are primitive abundant. Conversely, if S is a primitive abundant, then S is isomorphic to a PA blocked Rees matrix semigroup 𝓜 0 ( M μ λ ; J , Λ , Γ ; P ) ; furthermore, there is variability in the sandwich matrix P on the choice of data in constructing the isomorphism.

We can simply take r 1 λ = q 1 λ = e λ , and thus for all λ Γ , p 1 λ 1 λ = q 1 λ r 1 λ = e λ , see [14, Theorem 3.8]. The sandwich matrix attaching to a PA blocked Rees matrix will be always assumed to be of such form.

A semigroup S is called 0 - J * -simple if S2 ≠ {θ} and S, {θ} are the only J * -classes of S. It is known that a primitive abundant semigroup is a 0-direct union of primitive abundant 0 - J * -simple semigroups. Recall that a semigroup S is said to be primitive adequate if S is adequate and all its nonzero idempotents are primitive.

We say that a semigroup S is a weak Brandt semigroup if the following conditions are satisfied:

(B1) if a, b, c are elements of S such that ac = bc ≠ 0 or ca = cb ≠ 0, then a = b;

(B2) if a, b, c are elements of S such that ab ≠ 0 and bc ≠ 0, then abc ≠ 0;

(B3) for each element a of S there is an element e of S such that ea = a and an element f of S such that af = a;

(B4) if e and f are nonzero idempotents of S, then there are nonzero idempotents e1 …, en of S with e1 = e, en = f such that for each i = 1, …, n1, one of eiSei + 1, ei + 1Sei is nonzero.

Obviously, a Brandt semigroup is a weak Brandt semigroup.

By [14, Corollary 5.6], a weak Brandt semigroup is just a 0 - J * -simple primitive adequate semigroup, or just a 0 - J * -simple PA blocked Rees matrix semigroup 𝓜 0 ( M λ μ ; J , J , Γ ; P ) with the properties that the sandwich matrix P is diagonal and pjj is equal to the identity eλ of the monoid Mλλ for each λ ∈ Γ and each jJλ.

Finally we list some basic definitions concerning semigroup algebras and the module theory of algebras which can be found in [21, 23].

Let S be a semigroup and let R[S] denote the semigroup algebra of S over R. If T is a subset of the semigroup S, then denote the set of all finite R-linear combinations of elements of T by R[T].

By the contracted semigroup algebra of S over R, denoted by R0 [S], we mean the factor algebra R[S]/R[θ].

If a = ∑ risiR0 [S], then the set supp a = {siS\{θ}| ri ≠ 0} is called the support of a.

Obviously, S\{θ} is a multiplicative basis of the contracted semigroup algebra R0 [S], because it is a R-basis of R0 and 0-closed (S2S ∪ {0}).

Let A be a R-algebra. A right A-module M is said to be indecomposable if M ≠ 0 and M has no direct sum decomposition M = NL, where N and L are nonzero right A-modules.

An idempotent eA is called primitive if eA is an indecomposable A-module. By [24, Corollary 6.4a], e is primitive in the algebra A if and only if e is primitive in the multiplicative semigroup Mult(A).

Suppose that A is a R-algebra with an identity. If the right A-module AA is a direct sum I1 ⊕ … ⊕ In of indecomposable right A-modules, then we call such a decomposition an indecomposable decomposition of A. It is known that this is the case if and only if there exists a complete set {e1, …, en} of primitive orthogonal idempotents of A such that Ii = eiA (i = 1, …, n).

Assume that A is a R-algebra with an identity and {e1,…, en} is a complete set of primitive orthogonal idempotents of A. The algebra A is called basic if eiAejA, for all ij.

The basic algebra associated toA is the algebra Ab = eAAeA, where e A = e j 1 + + e j m , and e j 1 , , e j m are chosen such that e j t A , 1 t m , are all the non-isomorphic projective indecomposable right A-modules.

It is known that Ab is basic and mod Ab ≇ mod A as categories (see, for example [23, Corollary 6.10]).

A right artinian algebra A is defined to be representation-finite if there are finitely many isomorphism classes of finitely generated, indecomposable right A-modules.

## 3 Rukolaĭne idempotents

In this section, we first recall the concept of Rukolaĭne idempotents of inverse semigroup algebras which was first introduced by Rukolaĭne [3]. Then we extend the Rukolaĭne idempotents to certain locally adequate concordant semigroup algebras.

Let E be a semilattice and e, fE. Then f is said to be maximal under e [25] or e covers f [4] if e > f and there is no gE such that e > g > f. Denote by ê the set {fE: e covers f}. Es is said to be pseudofinite if

1. (i)

for e, fE with e > f, there exists an element g such that e covers g and e > gf;

2. (ii)

ê is a finite set for each eE.

It is clear that finite semilattices are pseudofinite.

Let S be a finite inverse semigroup and eE(S)* Rukolaĭne [3] defined an element σ(e) ∈ R0[S] by

(1) σ ( e ) = e + { e i ..... e i j } e ^ ( 1 ) j e i 1 ... e i j ,
where { e i1, ... , eij} takes over all non-empty subset of ê. He proved that the set { σ( e) | eE( S)*} collects a family of orthogonal idempotents of R 0 [ S].

Let S be an inverse semigroup with E(S) pseudofinite. Then ê is a finite set whose elements are commutative, and hence σ(e) ∈ R0[S] is well defined for each eE(S)*. In this case, Munn [4] gave an obvious alternative definition of σ(e) as

σ ( e ) = Π g e ^ ( e g ) .

It is shown that {σ(e) |eE(S)*} is a set of orthogonal idempotents of R0[S], and σ(e), eE(S), are called the Rukolaĭne idempotents of R0[S].

Remark 3.1.

Let S be an inverse semigroup with E(S) pseudofinite. If eS is a minimal nonzero idempotent, that is e covers θ, then ê = 0 . In this case, we make the convention that σ(e) = e.

The idempotents set E(S) of a semigroup S is said to be locally pseudofinite (resp., locally finite) if E(eSe) is a pseudofinite (resp., a finite) semilattice for each eE(S).

Let S be a locally adequate IC abundant semigroup with E(S) locally pseudofinite. Then E(eSe) is a pseudofinite semilattice for each eE(S)* and so ê is a finite set with elements commutative since êE(eSe). As in [4], for each eE(S), let

σ ( e ) = Π g e ^ ( e g ) R 0 [ S ] .
We shall show that σ( e) is an idempotent of R 0[ S] for each eE( S). To this aim, we need the following results.

Lemma 3.2.

Let S be a locally adequate IC abundant semigroup with E(S) locally pseudofinite. Then for each aS* we have

(i) αa(a) = a*, where αa is the isomorphism from w(a) to w(a*);

(ii) if we still denote by αa the extension of αa to R0[w(a) by R-linearly, then αa(σ(a)) = σ(a*).

Proof. (i) By the hypothesis that S is IC abundant, there exists a semigroup isomorphism α a : a a * . Since S is locally adequate, a = ω ( a ) is a subsemilattice with identity a and a * = ω ( a * ) is a subsemilattice with identity a*. It follows that αa (a) = a* and (i) holds.

(ii) Note that αa is a semilattice isomorphism. It follows from the definition of a ^ and a ^ * that α a | a ^ is a bijection from a ^ onto a ^ * . Which together with the fact that αa(a) = a* implies that

α a ( σ ( a ) ) = Π g a ^ ( α a ( a ) α a ( g ) ) = Π f a ^ * ( a * f ) = σ ( a * ) .

Proposition 3.3.

Let S be a locally adequate IC abundant semigroup with E(S) locally pseudofinite. Then

(i) for each eE(S)*, σ(e) is an idempotent and eσ(e) = σ(e)e = σ(e);

(ii) σ(a)a = (a*) for each aS*;

(iii) for aS*, h R a * E ( S ) , f L a * E ( S ) , σ(h)a = (f).

Proof. (i) Let g, hê. Note that êE(eSe). Since S is locally adequate, E(eSe) is a semilattice, hence (eg)2 = eg and (eg) commutes with (eh). It follows that

σ ( e ) 2 = g e ^ ( e g ) 2 = g e ^ ( e g ) = σ ( e ) .
This shows that σ( e) is an idempotent. The rest is obvious.

(ii) By Lemma 3.2 (ii), we have αa(σ(a)) = σ(a). It follows that

a σ ( a * ) = a α a ( σ ( a ) ) = a α a ( g e ^ ( a g ) ) = a g e ^ ( α a ( a ) α a ( g ) ) = ( a α a ( a ) a α a ( t ) ) g e ^ \ { t } ( α a ( a ) α a ( g ) ) ( choose  t a ^ ) = ( a a t a ) g e ^ \ { t } ( α a ( a ) α a ( g ) ) ( since  S is IC ) = ( a t ) a g e ^ \ { t } ( α a ( a ) α a ( g ) ) = = g e ^ ( a g ) a = σ ( a ) a ,
as required.

(iii) It follows directly from (ii).

Remark 3.4.

(i) If S is an adequate semigroup, then {σ(e) | eE(S)*} ⊆ R0[S] is a set of pairwise orthogonal idempotents. Indeed, let e, fE(S)*, and there is no loss of generality in assuming ef. By Proposition 3.3(i), σ(e) σ(f) = σ(e)e f σ(f), thus it suffices to show σ(e)ef = 0. By hypothesis, e > efE(S). If ef = θ (in S), this is trivial. If efθ, there exists an idempotent gê such that g > ef. Then

σ ( e ) e f = ( h e ^ \ { g } ( e h ) ) ( e g ) e f = 0.
In either case, σ( e) ef = 0. Therefore σ( e) σ( f) = 0.

(ii) There exists a locally adequate IC abundant semigroup S with the property that the idempotents σ(e) (eE(S)*) are not pairwise orthogonal. To see this, let S = 𝓜 0 ( G ; I , I ; P ) be a completely 0-simple semigroup, where G is a group with identity e, I = {1, 2} and P is a I × I-matrix with p21 = 0 and pij = e otherwise. Obviously, S is a locally adequate IC abundant semigroup. Since g = (1, e, 1) and f = (2, e, 1) are primitive idempotents of S, we have σ(g) = g and σ(f) = f. Then

σ ( g ) σ ( f ) = g f = ( 1 , e , 1 ) ( 2 , e , 1 ) = ( 1 , e , 1 ) 0.
Consequently, { σ( e) | eE( S)*} is not a set containing pairwise orthogonal idempotents.

## 4 Multiplicative basis ℬ ¯ and semigroup S ¯

Let S be a locally adequate concordant semigroup with E(S) locally finite. In this section, first we construct a multiplicative basis ¯ of R0[S] by means of the Rukolaĭne idempotents defined in Section 3. Then we provide some properties of the semigroup ¯ { 0 } .

For each aS*, in view of Lemma 3.3 (ii) and (iii),

σ ( a ) a σ ( a * ) = ( a σ ( a * ) ) σ ( a * ) = a σ ( a * ) = σ ( a ) a
and σ( a ) ( a*) does not depend on the choice of the elements a* and a . Denote
a ¯ = σ ( a ) a σ ( a * ) .
Then by (1) we have
(2) a ¯ = a σ ( a * ) = a + { e i 1 , ... , e i j } a ^ * ( 1 ) j a e i 1 e i j .
Note that e i1e ije it < a* for t = 1, ... , j. Then ae i1e ijaa* = a since ≤ is compatible with the multiplication of S. Moreover, ae i1eij < a. Otherwise, suppose that ae i1eij = a. Since the elements of a * ^ commute, f = e i1eij is an idempotent and f < a*. Now a * a * implies a* f = a, hence f = a* f = a*. This is a contradiction. Therefore
(3) a ¯ a + b < a R b , a S *
In particular, we have ā ≠ 0 for each aS*. Now let
¯ = { a ¯ | a S * } .
We will show that ¯ is a multiplicative basis of R 0[ S].

Lemma 4.1.

Let S be a locally adequate concordant semigroup with E(S) locally pseudofinite. Then for a, bS*

a ¯ b ¯ = { a b ¯ , i f E ( S ) L a * R b * 0 , 0 , o t h e r w i s e .
In particular, ¯ is0 -closed.

Proof. Suppose that E ( S ) L a * R b * 0 . Let g E ( S ) L a * R b * . Then a * g * b . Since * (resp., * ) is a right (resp., left) congruence on S, we have a b * g b = b and a b * a g = a . Hence a b L b * R a * and so ab ≠ 0. On the other hand, ā = (g) and b ¯ = σ ( g ) b . It follows from Proposition 3.3 that

a ¯ b ¯ = a σ ( g ) σ ( g ) b = a σ ( g ) b (since  σ (g) is an idempotent) = a b σ ( b * ) (by Proposition 3 .3 (ii)) = a b σ ( ( a b ) * ) (since  a b * b ) = a b ¯
Suppose that E ( S ) L a * R b * 0 . Take e E ( S ) L a * and f E ( S ) R b * . Then ā = ( e) and b ¯ = σ ( f ) b . Note that σ( e) e = σ( e) and ( f) = σ( f).

If ef = θ, then ef = 0 in R0[S], and hence

a ¯ b ¯ = ( a σ ( e ) ) ( a ( f ) b ) = ( a σ ( e ) e ) ( f σ ( f ) b ) = a σ ( e ) ( e f ) σ ( f ) = 0.

If efθ, then θS(e, f) = {gE(S) | ge = fg = g, egf = ef}. Since S satisfies the regularity condition, S ( e , f ) 0 . Thus there exists a nonzero idempotent gS(e, f) and eg, gfE(S). Moreover, ege and gff. We claim that either gf < f or eg < e. Otherwise, suppose that gf = f and eg = e. Then g * f and g * e . So g L e * R f * E ( S ) = 0 , which is a contradiction. Without loss of generality, assume that eg < e. Then there exists hgê such that eghg since E(S) is pseudofinite. It follows that

σ ( e ) e f = σ ( e ) e g f = ( h e ^ ( e h ) ) e g f = ( h e ^ \ { h g } ( e h ) ) ( ( e h g ) e g ) f = ( h e ^ \ { h g } ( e h ) ) ( e e g h g e g ) f = ( h e ^ \ { h g } ( e h ) ) ( e g e g ) f = 0.
Therefore
a ¯ b ¯ = ( a σ ( e ) e ) ( f σ ( f ) b ) = a ( σ ( e ) e f ) σ ( f ) b = 0.

Remark 4.2.

For e, fE(S*), either E ( S ) L e * R f * = 0 or E ( S ) L e * R f * = S ( e , f ) .

In fact, if E ( S ) L e * R f * 0 , then there is a unique idempotent g E ( S ) L e * R f * such that ge = g, fg = g and egf = e. Hence gS(e, f) and so E ( S ) L e * R f * S ( e , f ) .

To prove the reverse inclusion, suppose that hS(e, f), we shall show that h * g . Note that hV(ef). Then

e 1 = e f h R e f * L h * E ( S ) , e 2 = g 1 e f R g 1 * L e f * E ( S ) .
It follows from he = h that ee 1 = e 1 = e 1 e. Since g L e * R f * , we have e f R e * L f * . Thus e 1 * e f * e and e 1 = e 1 e = e. Hence h * e 1 = e * g . Similarly, we may show that h * g . Therefore h * g and h = g E ( S ) L e * R f * . We have shown that S ( e , f ) E ( S ) L e * R f * . Consequently, E ( S ) L e * R f * S ( e , f ) .

Lemma 4.3.

Let S be a locally adequate concordant semigroup with E(S) locally pseudofinite. Then ¯ is linearly independent in R0[S].

Proof. Suppose to the contrary that ¯ is linearly dependent in R0[S]. Then there exist an nonzero integer n, distinct x ¯ 1 , . . . , x ¯ n , and r1, . . . , rnR\{0} such that

r 1 x ¯ 1 + + r n x ¯ n = 0
Let xl be a maximal element of { x 1, x 2, ... , xn} under the natural partial order ≤ on S. By (3) suppose that x ¯ 1 = x t + i = 1 k t r i l b i l with ril ≠ 0 and bil < xl for i = 1, ... , kl, l = 1, 2, ..., n. Then
r 1 ( x 1 + i = 1 k 1 r i 1 b i l ) + + r n ( x n + i = 1 k n r i n b i n ) = 0
Since S \ { θ} is a basis of R 0[ S] and rl ≠ 0, there exists at least an element bij for some jl and some i such that bij = xi. Thus xl = bijxj, which is a contradiction. Therefore ¯ is linearly independent. □

The next result, which is due to Lawson [16], gives a characterization of the natural partial order on an abundant semigroup.

Lemma 4.4.

([16, Proposition 2.5]). Let S be an abundant semigroup and a, cS*. Then ca if and only if there exists an idempotent fω(a*) such that f L , c * a f = c .

Let S be an abundant semigroup. By Lemma 4.4, if bgE(S), then bE(S). For each eE(S), if ω(e) = E(eSe) is finite, then the element θ f e f ¯ R 0 [ S ] is well defined. Whenever this can be done without ambiguity we shall use the notation f e f ¯ instead of θ f e f ¯ .

Lemma 4.5.

Let S be a locally adequate concordant semigroup with E(S) locally finite and eE(S)*. Then

e = f e f ¯

Proof. Since S is a locally adequate semigroup with E(S) locally finite, we have E(eSe) is a finite semilattice. It is clear that an idempotent fe if and only if fE(eSe). We prove the lemma by induction. If e is a minimal idempotent of S under the natural partial order, the lemma is obvious by Remark 3.1. Suppose the lemma is true for all idempotent g < e. Let ê = {e1, e2, . . . , en} with n ≤ 1. Then fe if and only if fes for some 1 ≤ Sn. By (1) we have

f e f ¯ = e ¯ + f < e f ¯ = e + f < e f ¯ + { e i 1 , , e i j } e ^ ( 1 ) j e i 1 e i j ,
where { e i1, . . . , eij} takes over all the non-empty subset of ê. It follows from the hypothesis that
f e f ¯ = e + f < e f ¯ + { e i 1 , , e i j } e ^ ( 1 ) j f e i 1 e i j f ¯ .
Fix some f < e. Let e t1 ... etm be a smallest (under the natural partial order) product of e 1, e 2, ... , en such that fe t1 ··· etm. Then f ¯ appears in the sum
{ e i 1 , , e i j } e ^ ( 1 ) j f e i 1 e i j f ¯
with coefficient ( 1 ) m + C m 1 ( 1 ) m 1 + C m 2 ( 1 ) m 2 + + C m m 1 ( 1 ) = 1 . Thus
{ e i 1 , , e i j } e ^ ( 1 ) j f e i 1 e i j f ¯ = f < e f ¯
and e = f e f ¯ . □

Let S be a locally adequate IC abundant semigroup and a, cS* with ca. Then by Lemma 4.4 there exists an idempotent fω(a*) such that f L c * and af = c. We claim that such an idempotent f is unique. Suppose that g is another such an idempotent. Then g * f , and hence fg = f, gf = g. Since f, ga*, we have f, ga* Sa*. It follows that gf = fg, and so that g = f. Denote by ec such unique idempotent.

Lemma 4.6.

Let S be a locally adequate IC abundant semigroup and aS*. Denote e = a*. Then (i) the mapping defined by

φ : { b S * | b a } { e a f | θ f e } b e b
is a bijection;

(ii) {bS* | ba} = {aeaf | θfe}.

Proof. (i) To show (i) holds, define a mapping by

ψ : { e a f | θ f e } { b S * | b a } , e a f a e a f .
We shall show that ϕ and ψ are mutually inverse. Let ba. Then b = aeb and so eb = eaeb ∈ { eaf | fe}. Thus ψϕ( b) = ψ( eb) = aeb = b. On the other hand, let fe. Since aeaf = af, we have ϕψ( eaf) = ϕψ( eaf) = ϕaeaf = eaf. Consequently ϕ is a bijection.

(ii) It is obvious.

Let S be a locally adequate IC abundant semigroup with E(S) locally finite. Then the set {bS* | ba} is finite. Hence the element θ b a b ¯ R 0 [ S ] is well defined. In what follows, we write b a b ¯ instead of θ b a b ¯ .

Lemma 4.7.

Let S be a locally adequate concordant semigroup with E(S) locally finite and aS*. Then

a = b a b ¯ .

Proof. Let e = a*. By Lemma 4.5, we have

a = a e = a f e f ¯ = f e , f L a f * a f ¯ + f e , f L a f * a f ¯ .

Now

f e , f L a f * a f ¯ = f e , f L a f * ( a f ) σ ( f ) = f e , f L a f * a e a f σ ( f ) .

Let fE(S)* with fe and f L a f * . Since eaf ℒ* af and af · 1 = af · f, we have eaf · 1 = eaf · f, that is, eaf = eaf f. Note that E(eSe) is a semilattice. Then eaf f = feaf since eaf, fE(eSe). Thus eaff. But eaff because f L a f * , hence eaf < f. By the fact E(eSe) is finite that there exists h f ^ such that eafh. Hence

e a f σ ( f ) = e a f ( t f ^ ( f t ) ) = ( e a f ( f h ) ) ( t f ^ \ { h } ( f t ) ) = ( e a f f e a f h ) ( t f ^ \ { h } ( f t ) ) = ( e a f e a f ) ( t f ^ \ { h } ( e t ) ) = 0.

Therefore f e , f L a f * a f ¯ = 0 . It follows that

a = f e , f L a f * a f ¯ = f e , f L a f * a f f ¯ ( since f ¯ = f σ ( f ) ) = f e , f L a f * a f ( a f ) * ¯ ( since f * ( a f ) * , by Proposition 3.3 ( iii ) ) = f e , f L a f * a f ¯ = f e a e a f ¯ ( since a f = a e a f ) = b a b . ¯ ( by Lemma 4.6 ( ii ) )

Summing up, we have

Theorem 4.8.

Let S be a locally adequate concordant semigroup with E(S) locally finite. Then ¯ is a multiplicative basis ofR0[S] with multiplication given by

a ¯ b ¯ = { 0 , o t h e r w i s e . a b ¯ i f E ( S ) L a * R b * 0 ,

Proof.It follows from Lemmas 4.1, 4.3 and 4.7 directly. □

Let

S ¯ = ¯ { 0 } .

Then, by Theorem 4.8, S ¯ is a subsemigroup of the multiplicative semigroup of R0[S] such that R 0 [ S ] = R 0 [ S ¯ ] .

In order to study R0[S] better via S ¯ , we need to give more properties of S ¯ . In the remainder of this section, we always assume that S is a locally adequate concordant semigroup with E(S) locally pseudofinite.

Lemma 4.9.

The map ϕ : S S ¯ given by aā and θ ↦ 0, where aS*, is a bijection.

Proof. Obviously, ϕ is surjective. It suffices to show that ϕ is injective. Suppose to the contrary that there exist a, cS such that ac and a ¯ = c ¯ . By (3), there exist a1, · · ·, as, c1, · · ·, ctS* with a1, · · ·, as < a, c1, · · ·, ct < c such that a ¯ = a + i = 1 s r i a i and c ¯ = c + i = 1 t r i c i for some r 1 , , r s , r 1 , , r t R * . Thus

a + i = 1 s r i a i = c + i = 1 t r i c i .

Because S* is a basis of R0[S] and ac, a must cancel with some ci, hence a = ci < c. Similarly, c = aj < a for some aj. Now a < c < a, a contradiction. Therefore ϕ is injective.

Lemma 4.10.

E ( S ¯ ) \ { 0 } = { e ¯ | e E ( S ) * } . .

Proof. Let eE(S)*. Note e L e * R e * . Then by Theorem 4.1, ēē = ē ≠ 0 and hence { e ¯ | e E ( S ) * } E ( S ¯ ) \ { 0 } . To prove the reverse inclusion, assume that a ¯ E ( S ¯ ) * . Then āā = ā ≠ 0 and so

a 2 ¯ = a ¯ a ¯ = a ¯

by Theorem 4.1. By Lemma 4.9 we have a2 = a, that is aE(S)*. Hence E ( S ¯ ) \ { 0 } { e ¯ | e E ( S ) * } , as required.

Lemma 4.11.

Let aS*. Then a ¯ * * ( S ¯ ) a ¯ and a ¯ * ( S ¯ ) a ¯ .

Proof. Note that L a * R z * E ( S ) = L a * * R z * E ( S ) for any zS. Then by Theorem 4.1 a ¯ z ¯ = 0 if and only if a ¯ * z ¯ = 0 . Suppose that a ¯ x ¯ = a ¯ y ¯ for some x ¯ , y ¯ S ¯ 1 . If a ¯ x ¯ = a ¯ y ¯ = 0 , then a ¯ * x ¯ = 0 = a ¯ * y ¯ . On the other hand, if a ¯ x ¯ = a ¯ y ¯ 0 , then a x ¯ = a ¯ x ¯ = a ¯ y ¯ = a y ¯ . Hence ax = ay. Which together with a**a implies that a*x = a*y. Therefore a ¯ * x ¯ = a * x ¯ = a * y ¯ = a ¯ * y ¯ . Dually, if a ¯ * x ¯ = a ¯ * y ¯ for some x ¯ , y ¯ S ¯ 1 , we may show that a ¯ x ¯ = a ¯ y ¯ Consequently a ¯ * * a ¯ . The case for * is a dual. □

The following result describes the relationship between the Green *-relations of S and the Green *-relations of S ¯ .

Lemma 4.12.

Leta, bS. Then

(i) a * ( S ) b a ¯ * ( S ¯ ) b ¯ ; ;

(ii) a * ( S ) b a ¯ * ( S ¯ ) b ¯ ; ;

(iii) a D * ( S ) b a ¯ D * ( S ¯ ) b ¯ .

Proof. Note that * is the dual of * and D * = * * . It suffices to prove (i) is true.

Assume that a * ( S ) b . Then a * * ( S ) b * and so a*b* = a*, b*a* = b*. Clearly, a * L b * * R a * * and b * L a * * R b * * . It follows from Theorem 4.1 that a ¯ * b ¯ * = a * b * ¯ = a ¯ * and b ¯ * a ¯ * = b * a * ¯ = b ¯ * , that is, a ¯ * * ( S ¯ ) b ¯ * . It follows from Lemma 4.11 that a ¯ * ( S ¯ ) a ¯ * * ( S ¯ ) b ¯ * * ( S ¯ ) b ¯ . Conversely, suppose that a ¯ * ( S ¯ ) b ¯ . Then by Lemma 4.11 a ¯ * * ( S ¯ ) b ¯ * , that is, a ¯ * b ¯ * = a ¯ * and b ¯ * a ¯ * = b ¯ * . Hence a * b * ¯ = a ¯ * and b * a * ¯ = b ¯ * . By Lemma 4.9 we have a*b* = a* and b*a* = b*. Thus a* * (S)b*, hence a* *(S) b. □

Let K = { * , * , D * } . From Lemmas 4.9 and 4.12, for each K K , the mapping

φ K : ( S / K ) * ( S ¯ / K ) * K a * K a ¯ * ,
where a ∈* S*, is a bijection. Throughout this paper, we identify the set ( S / D * ) * (resp., ( S / * ) * , ( S / * ) * with the set ( S ¯ / D * ) * (resp., ( S ¯ / * ) * , ( S ¯ / * ) * ), and denote it by Y (resp., I, L). For each α ( S / K ) * , if K α * = K a * for some aS*, then we denote by K α * ¯ or K a ¯ * the nonzero K -class φ K ( K α * ) of S ¯ , and let K α * ¯ 0 = K α * ¯ { 0 } and K a ¯ * 0 = K a ¯ * { 0 } .

As a direct consequence of Lemma 4.12, we have

Corollary 4.13.

For each aS, we have

(i) R a ¯ * 0 . (resp., L a ¯ * 0 ) is a right (resp., left) ideal of S ¯ ;

(ii) D a ¯ * 0 is an ideal of S ¯ .

Proof. It follows from Lemma 4.12 and the proof of Lemma 4.1. □

Lemma 4.14.

Let eE(S)*. Then H e * ( S ) H e ¯ * ( S ¯ ) .

Proof. By Lemma 4.12, it is easy to see that the map ϕ defined in Lemma 4.9 sends H e * onto H e ¯ * . Let ϕ e : H e * H e ¯ * be the restriction of ϕ to H e * . Clearly ϕe is a bijection. Let a, b H e * . Then e H e * = L α * R b * . It follows from Theorem 4.1 that

ϕ e ( a b ) = a b ¯ = a ¯ b ¯ = φ e ( a ) φ e ( b ) .

Hence ϕe is an isomorphism, as required. □

Theorem 4.15.

S ¯ is primitive abundant.

Proof. That S ¯ is abundant follows directly from Lemmas 4.10 and 4.11. To show S ¯ is primitive, let e ¯ , f ¯ E ( S ¯ ) \ { 0 } with e ¯ f ¯ . Thus e ¯ f ¯ = f ¯ e ¯ = e ¯ 0 . By Theorem 4.1 there exists g ¯ L e ¯ * R f ¯ * E ( S ¯ ) . Hence e ¯ = e ¯ f ¯ L f ¯ * R e ¯ * . Similarly, we have e ¯ = e ¯ f ¯ R f ¯ * L e ¯ * . Therefore e ¯ * ( S ¯ ) = f ¯ and e ¯ = f ¯ . Consequently S ¯ is primitive. □

Recall a semigroup T with zero θ is called a 0-direct union of semigroups Tα (αX) if T = ∪αXTα and TαTβ = TαTβ = {θ} for all αβ.

Theorem 4.16.

(i) For each α ∈ Y, D α * ¯ 0 is 0- J * -simple primitive abundant;

(ii) S ¯ is a 0-direct union of D α * ¯ 0 ( α Y )

(iii) On S ¯ , D * = J * .

Proof. (i) Let a ¯ , b ¯ D α * ¯ . If E ( S ) L a * R b * = 0 , then a ¯ b ¯ = 0 D α * ¯ 0 . If E ( S ) L a * R b * = 0 , then by the proof of Lemma 4.1 we have

a ¯ b ¯ = a b ¯ R a ¯ * L b ¯ * D α * ¯ .

It follows that D α * ¯ 0 is a subsemigroup of S ¯ . It follows from the fact S ¯ is primitive abundant that D α * ¯ 0 is primitive abundant. In particular, ( D α * ¯ 0 ) 2 0 . That D α * ¯ 0 is 0- J * -simple is obvious.

(ii) Note that { D α * ¯ 0 | α Y } collects all the nonzero D * -classes of S ¯ . Thus S ¯ is a 0-disjoint union of the subsemigroups D α * ¯ 0 ( α Y ) . That D α * ¯ 0 D β * ¯ 0 = { 0 } whenever αβ follows from Theorem 4.1. Therefore S ¯ is a 0-direct union of D α * ¯ 0 ( α Y ) .

(iii) Let α S / D * and a ¯ D α * ¯ . Notice that a ¯ D α * ¯ is a D * -class of S ¯ . Since J*(ā) is an ideal of S ¯ containing ā which is saturated by * and * , we have D α * ¯ 0 J * ( a ¯ ) . On the other hand, by (ii), D α * 0 ¯ is an ideal of S ¯ , and hence it is an ideal saturated by * and * . Then the fact J*(ā) is the smallest ideal containing ā which is saturated by * and * implies that D α * ¯ 0 J * ( a ¯ ) . Note that for all b ¯ , c ¯ S ¯ , b ¯ J * c ¯ if and only if J * ( b ¯ ) = J * ( c ¯ ) . It follows that b ¯ J * c ¯ if and only if b ¯ , c ¯ D β * ¯ for some . This shows that for each β S / D * is a J * -class of S ¯ . (iii) follows.

Let T be an abundant semigroup. In [18], S. Armstrong defined the *-trace of T to be the partial groupoid tr*(T) = (T,·) with partial binary operation “·” defined by

a b = { ab , if E ( S ) L a * R b * 0 , undefined , otherwise .

It is clear that tr*(T) is a disjoint union of D * -classes of T, which is closed under ·. The multiplication “·” on tr*(T) can be extended to tr*(T)0 = tr*(T) ∪ {0} by setting undefined products equal to 0, where 0 is a symbol not in T and acts as zero element. Then tr*(T)0 is a semigroup under this multiplication. Armstrong [18] studied and characterized the *-trace of a concordant semigroup, in particular, he proved that tr*(T)0 is a primitive abundant semigroup.

Remark 4.17.

Let S be a locally adequate concordant semigroup with E(S) locally pseudofinite. Then S ¯ is a multiplicative subsemigroup of R0[S] and a good homomorphism image of tr*(S). Indeed, from Lemmas 4.9, 4.10 and 4.12, one can deduce that S ¯ is isomorphic to the semigroup obtained from tr*(S)0by equating θ (the zero element of S) with 0. And Lemma 4.14, Theorems 4.15 and 4.16 can also be obtained from the results of [18].

## 5 Direct product decomposition

Let S be a locally adequate concordant semigroup with E(S) locally finite. We have constructed a new basis for R0[S} in last section. As an application, we provide a direct product decomposition for R0[S] in this section.

Theorem 5.1.

Let S be a locally adequate concordant semigroup with E(S) locally finite and S ¯ = { a σ ( a * ) | a S * } { 0 } . Then

R 0 [ S ] α Y R [ D α * ¯ ]
is a direct product decomposition of R 0[ S], where Y = ( S / D * ) * and D α * ¯ , α Y , are all non-zero D * - classes of S ¯ .

Proof. Since S ¯ \ { 0 } is a basis of R0[S], we have R 0 [ S ] = R 0 [ S ] ¯ . It follows from Theorem 4.16 (ii) that R 0 [ S ] ¯ = a Y R 0 [ D α * ¯ 0 ] . Note that R [ D α * ¯ ] = R 0 [ D α * ¯ 0 ] . Then we have R 0 [ S ] a Y R [ D α * ¯ ] , as required. □

Next we consider the case R0[S] containning an identity. The following result is essential for us.

Lemma 5.2

([26, Theorem 1.4]). Let S be a semigroup. If the semigroup ring R0[S] contains an identity, then there exists a finite subset U of E(S) and for all sS, there exist e, fU such that s = es = sf.

Lemma 5.3.

Let S be a locally adequate concordant semigroup with E(S) locally finite. If R0[S] contains an identity, then S as well as S ¯ has finitely many * -classes (resp., * -classes, D * -classes). In particular, S as well as S ¯ has finitely many idempotents.

Proof. By Lemma 4.12, we only need to consider the case of S ¯ . Suppose that R0[S] contains an identity. Then by Lemma 5.2 there exists a finite subset U of E ( S ) ¯ such that for all s ¯ S ¯ , s ¯ = e ¯ s ¯ = s ¯ f ¯ for some e ¯ , f ¯ U . Thus, in order to show that S ¯ / * is finite, it suffices to show that U R s ¯ * 0 for each s ¯ S ¯ . Suppose to the contrary that U R a ¯ * 0 for some a ¯ S ¯ . Then there exists an idempotent e ¯ U such that e ¯ a ¯ a ¯ , but e ¯ R a ¯ * . Since S ¯ is abundant, there exists f ¯ E ( S ¯ ) R a ¯ * . Thus e ¯ f ¯ = f ¯ , and hence

( f ¯ e ¯ ) ( f ¯ e ¯ ) = f ¯ ( e ¯ f ¯ ) e ¯ = f ¯ e ¯
This shows that f ¯ e ¯ E ( S ¯ ) . We claim that f ¯ e ¯ 0 . Otherwise, suppose f ¯ e ¯ = 0 . Then
a ¯ = f ¯ a ¯ = f ¯ ( e ¯ a ¯ ) = ( f ¯ e ¯ ) a ¯ = 0
which is a contradiction. It follows from Lemma 3.3 [14] that f ¯ e ¯ R f ¯ * L e ¯ * . Thus f ¯ e ¯ = e ¯ f ¯ e ¯ = e ¯ , which together with e ¯ f ¯ = f ¯ implies that e ¯ * f ¯ . Hence e ¯ * f ¯ * a ¯ , a contradiction. Therefore S ¯ has finite many * -classes. Dually, S ¯ has finite many * -classes and so finite many D * -classes.

Since * = * * , S ¯ has finite many * -classes. Hence S ¯ has finite many idempotents since each * -class contains at most one idempotent. □

Corollary 5.4.

Let S be a locally adequate concordant semigroup with E(S) locally finite. If R0[S] contains an identity, then

R 0 [ S ] = α Y R [ D α * ¯ ]
where Y = S ¯ / D * and D α * ¯ , α Y , are all non-zero D * - classes of S ¯ .

Proof. If R0[S] contains an identity, then S as well as S ¯ has finitely many D * -classes. Now it follows from the proof of Theorem 5.1. □

To end this section, we consider two special cases: adequate and regular. As applications of Theorem 5.1, we give a direct product decomposition of IC adequate semigroup algebras and locally inverse semigroup algebras.

Recall that an IC adequate semigroup (sometimes called ample semigroup) is an adequate semigroup which is IC. Note that the set of idempotents of an adequate semigroup is a semilattice and adequate semigroups are locally adequate. Hence a locally adequate concordant semigroup is adequate if and only if it is IC adequate.

Corollary 5.5.

Let S be an IC adequate semigroup with E(S) locally finite. ThenR0[S] is a direct product of contracted weak Brandt semigroup algebras. Moreover,R0[S] contains an identity if and only if S / * and S / * are finite.

Proof. Let S be an IC adequate semigroup with E(S) locally finite and let S ¯ = { a ¯ | a S } { 0 } . Then

R 0 [ S ] α Y R [ D α * ¯ ] ,
where Y = ( S ¯ / D * ) * and D α * ¯ , α Y , are all non-zero D * -classes of S ¯ . Since S is adequate, it follows from Lemma 4.12 that S ¯ is also adequate. Then by Theorem 4.16, for each α Y = S ¯ / D * , D α * ¯ 0 is a 0 J * -simple primitive adequate semigroup. So it is a weak Brandt semigroup. Note that R [ D α * ¯ ] = R [ D α * ¯ 0 ] . Therefore R 0[ S] is a direct product of contracted weak Brandt semigroup algebras.

Suppose that R0[S] contains an identity. Then by Lemma 5.3 S ¯ / * and S ¯ / * are finite. Conversely, Suppose that S ¯ / * and S ¯ / * are finite. Then Y = S ¯ / D * is finite. It follows from the proof of Theorem 5.1 that

R 0 [ S ] = α Y R [ D α * ¯ ]
where Y = S ¯ / D * and D α * ¯ , α Y , are all non-zero D * -classes of S ¯ . Similar argument as above, D α * ¯ 0 is a weak Brandt semigroup for each αY. Let
D α * ¯ 0 = 0 ( M λ μ α ; I α , I α , Γ α ; P α )
for each αY, where Pα is a diagonal matrix with p i i α = e λ α for each i I λ α , and where e λ α is the identity of the monoid M λ λ α for each λ ∈ Γ α. Then the element
e = α Y λ Γ α , i I λ α ( e λ α ) i i R 0 [ S ]
is well defined, where ( e λ α ) i i is the | Iα| × | Iα| matrix with entry e λ α in the ( i, i) position and zeros elsewhere. Clearly, e is the identity of R 0[ S]. □

It is clear that a locally adequate concordant semigroup is regular if and only if it is locally inverse.

Corollary 5.6.

Let S be a locally inverse semigroup with E(S) locally finite. Then

(4) R 0 [ S ] α ( S / D ) * ( R [ G α ] ; I α , Λ α ; P α ) ,
where Gα is the maximal subgroup in Dα, Iα ( resp., Λ α) is the set of the -classes ( resp., - classes) contained in Dα, and P is a regular Λ α × Iα-matrix with entries in G α 0 for each α S / D .

Proof. It is clear that a regular 0 J * -simple primitive abundant is completely 0-simple. Note that Green’s *-relations coincide with Green’s relations in regular semigroups. Then D α * ¯ 0 is a completely 0-simple semigroup, say, D α * ¯ 0 = 0 ( G α ; I α , Λ α ; P α ) , for each α ( S ¯ / D * ) * = ( S / D ) * . Thus R 0 [ D α * ¯ 0 ] = ( R [ G α ] ; I α , Λ α ; P α ) . It follows from Theorem 5.1 that R 0 [ S ] α ( S / D ) * ( R [ G α ] ; I α , Λ α ; P α ) .

Corollaries 5.5 and 5.6 generalize the results on finite ample semigroups [9] and on finite locally inverse semigroups [8].

Corollary 5.7

([4, Theorem 6.5]). Let S be an inverse semigroup with E(S) locally finite. Then

R 0 [ S ] α ( S / D ) * M | I α | ( R [ G α ] ) ,
where Gα is the maximal subgroup in Dα and | Iα| denotes the number of the - classes of Dα for each α S / D .

Proof. By hypothesis, Lemmas 4.10 and 4.12, we deduce that S ¯ is an inverse semigroup. Then the fact S ¯ is a 0-direct union of D α ¯ 0 ( α S / D ) implies that each D α ¯ 0 is a Brandt semigroup. Say, D α ¯ 0 = 𝓜 0 ( G α ; I α ; I α ; P α ) where Gα is a maximal subgroup of S ¯ which is contained in D α ¯ , Iα is the set of -classes of D α ¯ , Pα is a diagonal Iα × Iα-matrix with (pα)ii is equal to the identity eα of Gα for each iIα. Furhtermore, by Lemma 4.14, Gα is isomorphic to any maximal subgroup of S contained in Dα; by Lemma 4.12, Iα is the set of -classes of Dα. Now it is easily verified that R [ D α ¯ ] M | I α | ( R [ G α ] ) . Consequently, by Theorem 5.1, we obatin the desired direct product decomposition. □

## 6 Projective indecomposable modules

Throughout this section, let S denote a locally adequate concordant semigroup with E(S) locally finite. Since projective indecomposable modules are discussed on algebras with identities, we always assume that the contracted semigroup algebra R0[S] contains an identity.

By Corollary 4.13, for i I = S ¯ / * , R i * ¯ 0 is a right ideal of S ¯ . Note that R 0 [ S ] = R 0 [ S ¯ ] . Then R [ R j * ¯ ] is a right ideal of R0[S] and can be considered as a right R0[S]-module for each ii.

We first give out a direct sum decomposition of R0[S].

Theorem 6.1.

If R0[S] has an identity, then

(5) R 0 [ S ] R 0 [ S ] = i I R [ R i * ¯ ]
is a finite direct sum decomposition of R 0[ S].

Proof. If R0[S] contains an identity, then S as well as S ¯ has finitely many *-classes and so I is finite. Since S ¯ * is a disjoint union of R i * ¯ ( i I ) , the right R0[S]-module R0[S]R0[S] is a direct sum of R [ R i * ¯ ] ( i I ) . Therefore (5) gives a finite direct sum decomposition of R0[S]. □

By Lemma 4.16, D α ¯ 0 ( α Y ) is a 0 J * -simple PA blocked Rees matrix semigroup, say, D α * ¯ 0 = 𝔻 0 ( M λ μ α ; J α ; Λ α ; Γ a ; P α ) .

Next we investigate conditions under which the projective R0[S]-modues R [ R i * ¯ ] are isomorphic.

Lemma 6.2.

Let α, β, ∈ Y, iJα; JJβ. If R [ R i * ¯ ] R [ R j * ¯ ] , then α = β.

Proof. Let ψ : R [ R i * ¯ ] R [ R j * ¯ ] be a right R0[S]-module isomorphism. Suppose to the contrary that αβ. Let x ¯ R i * ¯ . Since S ¯ is abundant, there exists an idempotent e ¯ L x ¯ * D α * ¯ . Then x ¯ e ¯ = x ¯ and ψ ( x ¯ e ¯ ) = ψ ( x ¯ ) 0 . On the other hand, ( ψ ( x ¯ ) , e ¯ ) D * , thus ψ ( x ¯ ) e ¯ = 0 by Lemma 4.16 (ii). Whence ψ ( x ¯ e ¯ ) ψ ( x ¯ ) e ¯ , a contradiction. Therefore α = β, as required. □

Let βY and R [ R i * ¯ ] R [ R β * ¯ ] . Then R [ R i * ¯ ] is a right R0[S]-module. By Theorem 5.1, R 0 [ S ] = α Y R [ D α * ¯ ] .

Thus we only need to consider R [ R i * ¯ ] as a right R [ D β * ¯ ] -module; M R [ R i * ¯ ] is an indecomposable right R0[S]-module if and only if M is an indecomposable right R [ D β * ¯ ] -module. Therefore, it suffices to find all the non-isomorphic projective indecomposable right R [ D α * ¯ ] -modules (αY).

Let M = 𝓜 0 ( M λ μ ; J , Λ , Γ ; P ) be a PA blocked Rees matrix semigroup and λ ∈ Γ, i; jJλ. For each μ ∈ Γ, define

R i μ * ¯ = S Λ μ H i s * ¯ , n i μ = | M λ μ | .

Here n = n. Since | H k s * ¯ | = | M λ μ | = | H l t * ¯ | for all (k, s), (l, t) in the (λ, μ)-block, we have | R i μ * ¯ | = n i μ | Λ μ | . We say the semigroup M satisfies the row-block condition if for all λν ∈ Γ, iJλ and jJν, there exists μ ∈ Γ such that nn.

Lemma 6.3.

Let D * ¯ be a D * -class of S ¯ and D * ¯ 0 = 𝔻 0 ( M λ μ ; J , Λ , Γ ; P ) , i, j, ∈ J.

(i) If i, jJλ for some λ ∈ Γ, then R [ R i * ¯ ] R [ R j * ¯ ] ;

(ii) If R [ R i * ¯ ] R [ R j * ¯ ] , then n = n for each μ ∈ Γ;

(iii) If D * ¯ 0 satisfies the row-block condition, then { R [ R 1 λ * ¯ ] | λ Γ } collects pairwise non-isomorphic projective right R [ D * ¯ ] -modules.

Proof. (i) Let i, jJλ for some λ ∈ Γ. Then for any μ ∈ Γ, n = n, and hence we can define a map ψ : R [ R i * ¯ ] R [ R j * ¯ ] by (i, ā, s) ↦ (j, ā, s), where s ∈ ⋀μ and āMλμ, and extend R-linearly. By definition, ψ restricts to a bijection R i * ¯ R j * ¯ . Hence ψ is a R-module isomorphism from R [ R i * ¯ ] to R [ R j * ¯ ] . We claim that ψ is a right R [ D * ¯ ] -module homomorphism. For this, let x ¯ = ( i , a ¯ , s ) R i * ¯ and y ¯ = ( k , b ¯ , t ) D * ¯ 0 , then

ψ ( x ¯ ) y ¯ = ( j , a ¯ p s k b ¯ , t ) = ψ ( i , a ¯ p s k b ¯ , t ) = ψ ( x ¯ , y ¯ ) .

Therefore ψ is a right R0[S]-module isomorphism, and (i) is proved.

(ii) Without loss of generality, suppose that ψ : R [ R i * ¯ ] R [ R j * ¯ ] is a R [ D * ¯ ] -module isomorphism with ψ ( R i * ¯ ) = R j * ¯ . Let μ ∈ Γ and x ¯ R i μ * ¯ . Since D * ¯ 0 is abundant and all its idempotents are in the diagonal blocks, there exist an element lJμ and an idempotent e ¯ D * ¯ such that e ¯ L x ¯ * R l * ¯ . But by the fact ψ ( x ¯ e ¯ ) = ψ ( x ¯ ) e ¯ and (4.8), we deduce that

E ( S ¯ ) L x ¯ * R l * ¯ 0 E ( S ¯ ) L ψ ( x ¯ ) * R l * ¯ 0 .

It follows from the fact 0 ψ ( x ¯ ) = ψ ( x ¯ e ¯ ) that L ψ ( x ¯ ) * R l * ¯ contains an idempotent, hence ψ ( x ¯ ) R j μ * ¯ . Therefore ψ ( R i μ * ¯ ) R j μ * ¯ . Notice that R k * ¯ = v Γ R k v * ¯ for each kJ. Because ψ is a bijection from R i * ¯ to R j * ¯ , we have n i μ | Δ μ | = | R i μ * ¯ | = | R j μ * ¯ | = n j μ | Λ μ | . This implies that n = n.

(iii) This follows from (i) and (ii). □

Let D * ¯ be a D * -clas of S ¯ and let D * ¯ 0 = 0 ( M λ μ ; J , Λ , Γ ; P ) satisfy the row-block condition. By (5), R [ D * ¯ ] is a direct sum of R [ R j * ¯ ] ( j J ) . For each pair i, jJ, according to Lemma 6.3, R [ R i * ¯ ] R [ R j * ¯ ] if and only if there exists λ ∈ Γ such that i, jJλ. Thus it suffices to find the non-isomorphic indecomposable direct summands of R [ R 1 λ * ¯ ] for each λ ∈ Γ.

Let λ ∈ Γ and let fλ, 1, ··· fλ,nλ, ··· fλ,nλ+mλ, be a complete set of primitive orthogonal idempotents of R[Tλ] such that fλ,1R[Tλ], ··· fμ,nλR[Tλ] are all the non-isomorphic projective indecomposable right R[Tλ]-modules. Notice that

R [ R 1 λ * ¯ ] = ( 1 λ , e λ , 1 λ ) R [ D * ¯ ] = 1 p n λ + m λ ( 1 λ , f λ , p , 1 λ ) R [ D * ¯ ] .

Lemma 6.4.

Let D * ¯ be a D * -class of S ¯ and D * ¯ 0 = 𝔻 0 ( M λ μ ; J , Λ , Γ ; P ) .

(i) For each pair u, υR[Tλ], the right R[Tλ]-modules uR[Tλ] ≅ υR[Tλ] if and only if the right R [ D * ¯ ] -modules ( 1 λ , u , 1 λ ) R [ D * ¯ ] ( 1 λ , υ , 1 λ ) R [ D * ¯ ] ;

(ii) Let fR[Tλ] be an idempotent. Then f R[Tλ] is an indecomposable right R[Tλ]-module if and only if ( 1 λ , f , 1 λ ) R [ D * ] ¯ is an indecomposable right R [ D * ¯ ] -module.

Proof. (i) Suppose that ϕ: uR[Tλ] → υR[Tλ] is a right R[Tλ]-module isomorphism. Let wR[Tλ] and ( i , y , s ) D * ¯ . Then (1λ, w, 1λ)(i, y, s) = (1λ, w(p1λ,iy), s). If i = 1λ, then p1λ,iy = eλy = y by our assumption on P. Therefore

( 1 λ , w , 1 λ ) R [ D * ¯ ] = μ Γ , s Λ μ , x M λ μ R ( 1 λ , w x , s ) .

By condition (C) in the definition of PA blocked Rees matrix semigroups, for all μ ∈ Γ and x, yMλμ, if wx = wy, then x = y in Mλμ. Thus the R-linear map

ϕ ˜ : ( 1 λ , u , 1 λ ) R [ D * ¯ ] ( 1 λ , υ , 1 λ ) R [ D * ¯ ] ( 1 λ , u x , s ) ( 1 λ , ϕ ( u ) x , s )
is well defined and is injective. We claim that φ ˜ is a right R [ D * ¯ ] -module isomorphism. Indeed, since ϕ is surjective, φ ˜ is also surjective, hence φ ˜ is a bijection. Let ( l , y , s ) D * ¯ . Then φ ˜ ( ( 1 λ , u x , s ) ( l , y , t ) ) = φ ˜ ( 1 λ , u x p s l y , t ) = φ ˜ ( ( 1 λ , u x , s ) ( l , y , t ) and consequently, φ ˜ is a R [ D * ¯ ] -homomorphism.

Conversely, suppose that φ ˜ : ( 1 λ , u , 1 λ ) R [ D * ¯ ] ( 1 λ , υ , 1 λ ) R [ D * ¯ ] is a right R [ D * ¯ ] -module isomorphism. For each wuR[Tλ],

φ ˜ ( 1 λ , w , 1 λ ) = φ ˜ ( 1 λ , w , 1 λ ) ) ( 1 λ , e λ , 1 λ ) ( 1 λ , υ R [ T λ ] , 1 λ ) .

Thus we can define a map ϕ : uR[Tλ] → μR[Tλ] by ( 1 λ , φ ( w ) , 1 λ ) = φ ˜ ( 1 λ , w , 1 λ ) . Obviously, ϕ is a bijection. It thus suffices to show ϕ(wx) = ϕ(w)x for all xR[Tλ]. Indeed,

( 1 λ , φ ( w x ) , 1 λ ) = φ ˜ ( ( 1 λ , w x , 1 λ ) ) = φ ˜ ( ( 1 λ , w , 1 λ ) ) ( 1 λ , x , 1 λ ) = ( 1 λ , φ ( w ) x , 1 λ ) ,
which implies ϕ( wx) = ϕ( w) x, and (i) follows.

(ii) Clearly, f′ = (1λ, f, 1λ) is an idempotent of R0[M]. We only need to show that f  Mult  R [ D * ¯ ] is primitive if and only if f ∈ Mult R[Tλ] is primitive. Indeed, let e  Mult  R [ D * ¯ ] be an idempotent. Then e′ < f′ if and only if there exists an idempotent eR[Tλ] such that e′ = (1λ, e, 1λ) and e < f, and hence (ii) follows. □

Notice that the results of Lemma 6.4 can be applied to general PA blocked Rees matrix semigroups.

Theorem 6.5.

Let S be a locally adequate concordant semigroup with E(S) locally finite and Y = S / D * . If

(i) for each αY, D α * ¯ 0 = 𝓜 0 ( M λ μ α ; J α , Λ α ; P α ) satisfies the row-block condition,

(ii) for each λ ∈ Γα, f λ , 1 α , , f λ , n λ α α , , f λ , n λ α + m λ α α is a complete set of primitive orthogonal idempotents of R [ T λ α ] such that f λ , 1 α R [ T λ α ] , , f λ , n λ α α R [ T λ α ] are all the non-isomorphic projective indecomposable right R [ T λ α ] -modules,

then the set a Y , λ Γ α { ( 1 λ α , f λ , q α , 1 λ α ) R 0 [ S ] | 1 q n λ α } collects all the non-isomorphic projective indecomposable right R0[S]-modules.

Proof. Let αY and λ ∈ Γα. By Lemma 6.4 and the hypotheses, the right R [ D α * ¯ ] -modules ( 1 λ α , f λ , q α , 1 λ α ) R [ D α * ¯ ] are indecomposable; furthermore, ( 1 λ α , f λ , q α , 1 λ α ) R [ D α * ¯ ] ( ( 1 λ α , f λ , p α , 1 λ α ) R [ D α * ¯ ] if and only if f λ , q α , R [ T λ α ] f λ , p α , R [ T λ α ] as right R [ T λ α ] -modules , where 1 ≤ q, 1 n λ α + m λ α . Therefore, ( p λ α f λ , q α , 1 λ α ) R [ D α * ¯ ] ( 1 q n λ α ) are all the non-isomorphic projective indecomposable right R [ D α * ¯ ] -modules.

As mentioned before, M is an indecomposable right R [ D α * ¯ ] -module if and only if M is an indecomposable right R0[S]-module. Consequently, ( 1 λ α f λ , q α , 1 λ α ) R 0 [ S ] are all the non-isomorphic projective indecomposable right R0[S]-modules, where the union takes over all αY, λ ∈ Γα and 1 q n λ α , □

For each D α * ¯ 0 = 𝓜 0 ( M λ μ ; J α , Λ α , Γ α ; P α ) , if |Γα| = 1 then the semigroup D α * ¯ 0 is isomorphic to a Rees matrix semigroup [27], say, D α * ¯ 0 = 𝓜 0 ( T α ; J α , Λ α ; P α ) . In the following proposition we specialize to this case.

Proposition 6.6.

Let S be a locally adequate concordant semigroup with E(S) locally finite and for each α Y = S / D * , D α * ¯ 0 = 𝓜 0 ( T α ; J α , Λ α ; P α ) be a Rees matrix semigroup over a cancellative monoid Tα.

(i) R 0 [ S ] b Π α Y R [ T α ] b ;

(ii)R0[S] is representation-finite if and only if for each αY, R[Tα] is representation-finite.

Proof. (i) It is clear that R0[S] satisfies the row-block condition. Let α Y = S / D * . Suppose that f 1 α , , f n α α , , f n α + m α α is a complete set of primitive orthogonal idempotents of R[Tα] such that f 1 α R [ T α ] , , f n α α R [ T α ] are all the non-isomorphic projective indecomposable right R[Tα]-modules. Then e R [ T α ] = f 1 α + + f n α α , and thus R[Tα]b = eR[Tα]. By Theorem 6.5, we have eR0[S] = ∑αY(1α, eR[Tα], 1α), where 1αJα denote the element 1 λ α for each αY. This, together with the fact R 0 [ S ] = α Y R [ D α * ] ¯ , implies that

R 0 [ S ] b = e R 0 [ S ] R 0 [ S ] e R 0 [ S ] = α Y ( 1 α , e R [ T α ] , 1 α ) R 0 [ S ¯ ] ( 1 α , e R [ T α ] , 1 α ) = α Y ( 1 α , e R [ T α ] , 1 α ) ( 1 α , R [ T α ] , 1 α ) ( 1 α , e R [ T α ] , 1 α ) = α Y ( 1 α , e R [ T α ] R [ T α ] e R [ T α ] , 1 α ) α Y R [ T α ] b .

(ii) This follows from (i) immediately. □

To end our paper, for regular case, we have the following results.

Corollary 6.7.

Let S be a locally inverse semigroup with idempotents set E(S) locally finite. Suppose that R0[S] contains an identity. Then R0[S] is representation-finite if and only if R[Gα]is representation-finite for each α α S / D .

Proof. let αY. Then

D α * ¯ 0 = D α ¯ 0 = 0 ( G α ; J α , Λ α ; P α )
is a completely 0-simple semigroup. The result follows from Proposition 6.6 immediately. □

Let G be a finite group and K be a field with characteristic p. If p − |G|, then K[G] is semisimple and conversely (Maschke’s Theorem). If this is the case, K[G] is representation-finite since semisimple algebra is representation- finite. If p⃒|G|, K[G] is representation-finite if and only if the Sylow p-subgroups Gp of G are cyclic (Higman’s Theorem [28]). Therefore, K[G] is representation-finite if and only if either p ⃒ |G|, or all the Sylow p-subgroups Gp of G are cyclic.

Now the next result follows from Corollary 6.7 directly.

Corollary 6.8.

Let S be a locally inverse semigroup with E(S) locally finite and all its maximal subgroups of finite order. Suppose that K0[S] contains an identity. Then K0[S] is representation-finite if and only if for each α S / D withp⃒|Gα|, the Sylow p-subgroups(Gα)pof Gα are all cyclic.

# Acknowledgement

This research is partially supported by the National Natural Science Foundation of China (no. 11371177, 11401275), and the Fundamental Research Funds for the Central Universities of China (no. lzujbky-2015-78).

The authors would like to thank the anonymous referees for their valuable comments and suggestions.

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