## 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

## 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

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 *R*_{0} [*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 *R*_{0} [*S*]. Section 4 involves constructing a multiplicative basis *R*_{0} [*S*], see Theorem 4.8, and developing basic properties of the semigroup *R*_{0} [*S*], is decomposed into a direct product of primitive abundant *R*_{0}[*S*] contains an identity, the multiplicative basis *R*_{0} [*S*]. Theorem 6.5 translates the problem involving the projective indecomposable modules of *R*_{0} [*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 *S*. The *a* of the semigroup *S* will be denoted by *K*_{a} or *K*_{a}(*S*) in case of ambiguity. We denote the set of nonzero *S* by

Denote by *S*^{1} the semigroup obtained from a semigroup *S* by adding an identity if *S* has no identity, otherwise, let *S*^{1} = *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*; *b* ∈ *S*

It is clear that *S* and *S* is called *regular* if every *S* can be characterized by the property that the set *V*(*a*) = {*a*′ ∈ *S* | *aa*′ *a* = *a*, *a*′ *aa*′ = *a*′} is nonempty for each *a* ∈ *S*.

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

where *J*^{*}(*a*) is the smallest ideal containing a which is saturated by

Clearly, *S*. It is easy to see that *a*, *b* ∈ Reg (*S*),

We say a semigroup is *abundant* if each

Let *S* be an abundant semigroup and *a* ∈ *S*^{*}. We use *a*^{†} (resp., *a*^{*}) to denote a typical idempotent related to *a* by

Define two partial orders ≤_{r} and ≤_{l} on *S* [16] by

The *natural partial order* ≤ on *S* is defined to be ≤=≤_{r} ∩ ≤_{l}. We have an alternative characterisation of ≤: for *x*, *y* ∈ *S*, *x* ≤ *y* if and only if there exist *e*, *f* ∈ *E*(*S*) such that *x* = *ey* = *yf*.

Let *S* be an abundant semigroup and *e* ∈ *E*(*S*)^{*}. Define *ω*(*e*) = {*f* ∈ *E*(*S*) | *f* ≤ *e*}. 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 *a* ∈ *S*^{*}

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*) = {*g* ∈ *V* (*ef*) ∩ *E*(*S*) | *ge* = *fg* = *g*} of idempotents *e* and *f* is non-empty, and takes the form

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* (*e* ∈ *E*(*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 *s* ⊗ *m* ↦ *sm* (resp., *m* ↦ *t**mt*) is an (*S*, *T*)-isomorphism from *S* ⊗_{S}*M* (resp., *M* ⊗_{T}*T*) 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*) = {*J*_{λ}: *λ* ∈ Γ}, *P*(Λ) = {Λ_{λ}: *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 *j* ∈ *J*_{λ} and *s* ∈ Λ_{μ}. The (*λ*, *λ*)-blocks are called the *diagonal blocks* of the matrix.

For each pair (*λ*, *μ*) ∈ Γ × Γ, let *M*_{λμ} be a set such that for each *λ*, *M*_{λλ} = *T*_{λ} is a monoid and for *λ* ≠ *λ*, either *M*_{λμ} is a (*T*_{λ}, *T*_{μ})-bisystem. Moreover, we impose the following condition on {*M*_{λμ} :*λ*, *μ* ∈ Γ}.

(M) For all *λ*, *μ*, *ν* ∈ Γ, if *M*_{λμ}, *M*_{μλ} are both non-empty, then *M*_{λμ} is non-empty and there is a (*T*_{λ}, *T*_{ν})-homomorphism *φ*_{λμν}: *M*_{λμ} ⊗ *M*_{μν} such that if *λ* = *μ* or *μ* = *ν*, then *φ*_{λμν} is the canonical isomorphism and such that the square

is commutative.

Here, for *a* ∈ *M*_{λμ}, *b* ∈ *M*_{μν}, we denote (*a* ⊗ *b*) *φ*_{λνμ} by *ab*. On the other hand, let 0 (zero) be a symbol not in any *M*_{λμ} with the convention that 0*x* = *x*0 = 0 for every element *x* of

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 *a* ∈ *M*_{λ, μ}, and the zero matrix (denoted by *θ*). Define a Λ × *J* sandwich matrix *P* = (*p*_{si}) where a nonzero entry in the (*λ*, *μ*)-block of *P* is a member of *M*_{λμ}.

Let *A* = (*a*)_{is}, *B* = (*b*)_{jt} ∈ *M*, by condition (M), the product *A* ∘ *B* = *APB* = (*ap*_{sj}*b*)_{it} makes *M* be a semigroup, which we denote by *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*, *a*_{1}, *a*_{2} ∈ *M*_{λμ}, *b*, *b*_{1}, *b*_{2} ∈ *M*_{μκ}, then *ab*_{1} = *ab*_{2} implies *b*_{1} = *b*_{2}; *a*_{1}*b* = *a*_{2}_{b} implies *a*_{1} = *a*_{2};

(U) For each *λ* ∈ Γ and each *s* ∈ Λ_{λ} (resp., *j* ∈ *J*_{λ}), there is a member *j* of *J*_{λ} (resp., *s* ∈ Λ_{λ}) such that *p*_{sj} is a unit in *M*_{λλ};

(R) If *M*_{λμ}, Mare both non-empty where *λ* ≠ *μ*, then *aba* ≠ *a* for all *a* ∈ *M*_{λμ}. *b* ∈ *M*_{λμ}.

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

([14, Proposition 2.4]). *Let**be a PA blocked Rees matrix semigroup. Then*

*(i) a non-zero element* (*a*)_{is}*of M is an idempotent if and only if there is an element λ* ∈ Γ *such that* (*i*, *s*) ∈ *J*_{λ} × Λ_{λ}*and a is a unit in T*_{λ}*with inverse p*_{si};

*(ii) all nonzero idempotents of M are primitive*;

*(iii) the non-zero elements* (*a*)_{is}*and* (*b*)_{jt}*of M are**related if and only if i* = *j*;

*(iv) the non-zero elements* (*a*)_{is}*and* (*b*)_{jt}*of M are**related if and only if s* = *t*;

*(v) M is abundant*;

*(vi) the non-zero idempotents e* = (*a*)_{is}*and f* = (*b*)_{jt}*of M with* (*i*, *s*) ∈ *J*_{λ} Ø Λ_{λ}*and* )*j*, *t*) ∈ (*i*, *s*) ∈ *J*_{μ} Ø Λ_{μ}*are**related if and only if λ* = *μ*;

*(vii) the non-zero element* (*a*)_{is}*of M is regular if and only if there is an element λ* ∈ Γ *such that* (*i*, *s*) ∈ *J*_{λ} Ø Λ_{λ}*and a is a unit in T*_{λ}.

Let _{λ} ∈ *J*_{λ} 1_{λ} ∈ *J*_{λ} ∩ Λ_{λ} such that *e*_{λ}(*λ* ∈ Γ).

Recall that a *Munn algebra* is an algebra *A* [21] such that each row and each column of the sandwich matrix *P* contains a unit of *A*. Let

Let *generalized Munn algebra**of M* to be the vector space consisting of all the *J* Ø Λ-matrices (*a*_{is}) with only finitely many nonzero entries, where *a*_{is} ∈ *R*[*M*_{λμ}] if (*i*, *s*) ∈ *J*_{λ} Ø Λ_{λ}, with multiplication defined by the formula (*a*_{is}) ∘ (*b*_{jt}) = (*a*_{is}*P* (*b*_{jt}).

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].

If *a*_{jt}, we will write (*j*, *a*, *t*) or (*a*)_{jt} instead of (*a*_{is}).

Now we recall the definition of primitive abundant semigroups. Let *S* be an abundant semigroup. If *e* ∈ *E*(*S*)* is minimal under the natural order ≤ defined on *S*, *e* is said to be *primitive*. It is known that an idempotent *e* ∈ *S* is primitive if and only if e has the property that for each idempotent *f* ∈ *E*(*S*), *fe* = *ef* = *f* ≠ *θ* ⇒) *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 *P* on the choice of data in constructing the isomorphism.

We can simply take

A semigroup *S* is called *simple* if *S*^{2} ≠ {*θ*} and *S*, {*θ*} are the only *S*. It is known that a primitive abundant semigroup is a 0-direct union of primitive abundant *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 *e*_{1} …, *e*_{n} of *S* with *e*_{1} = *e*, *e*_{n} = *f* such that for each *i* = *1*, …, *n* − *1*, one of *e*_{i}*Se*_{i + 1}, *e*_{i + 1}*Se*_{i} is nonzero.

Obviously, a Brandt semigroup is a weak Brandt semigroup.

By [14, Corollary 5.6], a weak Brandt semigroup is just a *P* is diagonal and *p*_{jj} is equal to the identity *e*_{λ} of the monoid *M*_{λλ} for each *λ* ∈ Γ and each *j* ∈ *J*_{λ}.

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 *R*_{0} [*S*], we mean the factor algebra *R*[*S*]/*R*[*θ*].

If *a* = ∑ *r*_{i}*s*_{i} ∈ *R*_{0} [*S*], then the set supp *a* = {*s*_{i} ∈ *S*\{*θ*}| *r*_{i} ≠ 0} is called the *support* of a.

Obviously, *S*\{*θ*} is a *multiplicative basis* of the contracted semigroup algebra *R*_{0} [*S*], because it is a *R*-basis of *R*_{0} and 0-closed (*S*^{2} ⊆ *S* ∪ {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* = *N* ⊕ *L*, where *N* and *L* are nonzero right *A*-modules.

An idempotent *e* ∈ *A* 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 *A*_{A} is a direct sum *I*_{1} ⊕ … ⊕ *I*_{n} 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 {*e*_{1}, …, *e*_{n}} of primitive orthogonal idempotents of *A* such that *I*_{i} = *e*_{i}*A* (*i* = 1, …, *n*).

Assume that *A* is a *R*-algebra with an identity and {*e*_{1},…, *e*_{n}} is a complete set of primitive orthogonal idempotents of *A*. The algebra *A* is called *basic* if *e*_{i}*A* ≇ *e*_{j}*A*, for all *i* ≠ *j*.

The *basic algebra associated to**A* is the algebra *A*^{b} = *e*_{A}*Ae*_{A}, where *A*-modules.

It is known that *A*^{b} is basic and mod *A*^{b} ≇ 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*, *f* ∈ *E*. Then *f* is said to be *maximal under e* [25] or *e covers f* [4] if *e* > *f* and there is no *g* ∈ *E* such that *e* > *g* > *f*. Denote by ê the set {*f* ∈ *E*: *e* covers *f*}. *E*s is said to be *pseudofinite* if

for

*e*,*f*∈*E*with*e*>*f*, there exists an element*g*such that*e*covers*g*and*e*>*g*≥*f*;*ê*is a finite set for each*e*∈*E*.

It is clear that finite semilattices are pseudofinite.

Let *S* be a finite inverse semigroup and *e* ∈ *E*(*S*)* Rukolaĭne [3] defined an element *σ*(*e*) ∈ *R*_{0}[*S*] by

where {*e*_{i1}, ... , *e _{ij}*} takes over all non-empty subset of

*ê*. He proved that the set {

*σ*(

*e*) |

*e*∈

*E*(

*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*) ∈ *R*_{0}[*S*] is well defined for each *e* ∈ *E*(*S*)*. In this case, Munn [4] gave an obvious alternative definition of *σ*(*e*) as

It is shown that {*σ*(*e*) |*e* ∈ *E*(*S*)*} is a set of orthogonal idempotents of *R*_{0}[*S*], and *σ*(*e*), *e* ∈ *E*(*S*), are called the *Rukolaĭne idempotents* of *R*_{0}[*S*].

*Let S be an inverse semigroup with E*(*S*) *pseudofinite. If e* ∈ *S is a minimal nonzero idempotent, that is e covers θ*, *then**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 *e* ∈ *E*(*S*).

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

We shall show that *σ*(*e*) is an idempotent of *R*_{0}[*S*] for each *e* ∈ *E*(*S*). To this aim, we need the following results.

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

*(i) α _{a}*(

*a*

^{†}) =

*a**,

*where α*(

_{a}is the isomorphism from w*a*

^{†})

*to w*(

*a**);

*(ii) if we still denote by α _{a} the extension of α_{a} to R*

_{0}[

*w*(

*a*

^{†})

*by R-linearly, then α*(

_{a}*σ*(

*a*

^{†})) =

*σ*(

*a**).

*Proof*. (i) By the hypothesis that *S* is IC abundant, there exists a semigroup isomorphism *S* is locally adequate, *a*^{†} and *a**. It follows that *α _{a}* (

*a*

^{†}) =

*a** and (i) holds.

(ii) Note that *α _{a}* is a semilattice isomorphism. It follows from the definition of

*α*(

_{a}*a*

^{†}) = a

^{*}implies that

□

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

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

*(ii) σ*(*a*^{†})*a* = *aσ*(*a**) *for each a* ∈ *S**;

*(iii) for a* ∈ *S**, *σ*(*h*)*a* = *aσ*(*f*).

*Proof*. (i) Let *g*, *h* ∈ *ê*. Note that *ê* ⊆ *E*(*eSe*). Since *S* is locally adequate, *E*(*eSe*) is a semilattice, hence (*e* − *g*)^{2} = *e* − *g* and (*e* − *g*) commutes with (*e* − *h*). It follows that

This shows that *σ*(*e*) is an idempotent. The rest is obvious.

(ii) By Lemma 3.2 (ii), we have *α _{a}*(

*σ*(

*a*

^{†})) =

*σ*(

*a*). It follows that

as required.

(iii) It follows directly from (ii).

(*i*) *If S is an adequate semigroup, then* {*σ*(*e*) | *e* ∈ *E*(*S*)*} ⊆ *R*_{0}[*S*] *is a set of pairwise orthogonal idempotents. Indeed, let e*, *f* ∈ *E*(*S*)*, *and there is no loss of generality in assuming e* ≰ *f*. *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* > *ef* ∈ *E*(*S*). *If ef* = *θ* (*in S*), *this is trivial*. *If ef* ≠ *θ*, *there exists an idempotent g* ∈ *ê such that g* > *ef*. *Then*

*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*) (*e* ∈ *E*(*S*)*) *are not pairwise orthogonal. To see this, let**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 p*_{21} = 0 *and p _{ij}* =

*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*

*Consequently*, {

*σ*(

*e*) |

*e*∈

*E*(

*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 *R*_{0}[*S*] by means of the Rukolaĭne idempotents defined in Section 3. Then we provide some properties of the semigroup

For each *a* ∈ *S**, in view of Lemma 3.3 (ii) and (iii),

and *σ*(*a*^{†})*aσ* (*a**) does not depend on the choice of the elements *a** and *a*^{†}. Denote

Then by (1) we have

Note that *e*_{i1} … *e*_{ij} ≤ *e*_{it} < *a** for *t* = 1, ... , *j*. Then *ae*_{i1} … *e*_{ij} ≤ *aa** = *a* since ≤ is compatible with the multiplication of *S*. Moreover, *ae*_{i1} … *e _{ij}* <

*a*. Otherwise, suppose that

*ae*

_{i1}…

*e*=

_{ij}*a*. Since the elements of

*f*=

*e*

_{i1}…

*e*is an idempotent and

_{ij}*f*<

*a**. Now

*a**

*f*=

*a*, hence

*f*=

*a**

*f*=

*a**. This is a contradiction. Therefore

In particular, we have *ā* ≠ 0 for each *a* ∈ *S**. Now let

We will show that *R*_{0}[*S*].

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

*In particular,*ℬ ¯ is0

*-closed.*

*Proof*. Suppose that *S*, we have *ab* ≠ 0. On the other hand, *ā* = *aσ*(*g*) and

Suppose that *ā* = *aσ*(*e*) and *σ*(*e*)*e* = *σ*(*e*) and *fσ*(*f*) = *σ*(*f*).

If *ef* = *θ*, then *ef* = 0 in *R*_{0}[*S*], and hence

If *ef* ≠ *θ*, then *θ* … *S*(*e*, *f*) = {*g* ∈ *E*(*S*) | *ge* = *fg* = *g*, *egf* = *ef*}. Since *S* satisfies the regularity condition, *g* ∈ *S*(*e*, *f*) and *eg*, *gf* ∈ *E*(*S*). Moreover, *eg* ≤ *e* and *gf* ≤ *f*. We claim that either *gf* < *f* or *eg* < *e*. Otherwise, suppose that *gf* = *f* and *eg* = *e*. Then *eg* < *e*. Then there exists *h _{g}* ∈

*ê*such that

*eg*≤

*h*since

_{g}*E*(

*S*) is pseudofinite. It follows that

Therefore

□

*For e*, *f* ∈ *E*(*S**), *either**or*

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

*To prove the reverse inclusion, suppose that h* ∈ *S*(*e*, *f*), *we shall show that**Note that h* ∈ *V*(*ef*). *Then*

*It follows from he*=

*h that ee*

_{1}=

*e*

_{1}=

*e*

_{1}

*e*.

*Since*

*we have*

*Thus*

*and e*

_{1}=

*e*

_{1}

*e*=

*e*.

*Hence*

*Similarly, we may show that*

*Therefore*

*and*

*We have shown that*

*Consequently*,

*Let S be a locally adequate concordant semigroup with E*(*S*) *locally pseudofinite. Then**is linearly independent in R*_{0}[*S*].

*Proof*. Suppose to the contrary that *R*_{0}[*S*]. Then there exist an nonzero integer *n*, distinct *r*_{1}, . . . , *r _{n}* ∈

*R*\{0} such that

Let *x _{l}* be a maximal element of {

*x*

_{1},

*x*

_{2}, ... ,

*x*} under the natural partial order ≤ on

_{n}*S*. By (3) suppose that

*r*≠ 0 and

_{il}*b*<

_{il}*x*for

_{l}*i*= 1, ... ,

*k*,

_{l}*l*= 1, 2, ...,

*n*. Then

Since *S* \ {*θ*} is a basis of *R*_{0}[*S*] and *r _{l}* ≠ 0, there exists at least an element

*b*for some

_{ij}*j*≠

*l*and some

*i*such that

*b*=

_{ij}*x*. Thus

_{i}*x*=

_{l}*b*≤

_{ij}*x*, which is a contradiction. Therefore

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

([16, Proposition 2.5]). *Let S be an abundant semigroup and a, c* ∈ *S**. *Then c* ≤ *a if and only if there exists an idempotent f* ∈ *ω*(*a**) *such that*

Let *S* be an abundant semigroup. By Lemma 4.4, if *b* ≤ *g* ∈ *E*(*S*), then *b* ∈ *E*(*S*). For each *e* ∈ *E*(*S*), if *ω*(*e*) = *E*(*eSe*) is finite, then the element

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

*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 *f* ≥ *e* if and only if *f* ∈ *E*(*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* = {*e*_{1}, *e*_{2}, . . . , *e _{n}*} with

*n*≤ 1. Then

*f*≤

*e*if and only if

*f*≤

*e*for some 1 ≤

_{s}*S*≤

*n*. By (1) we have

where {*e*_{i1}, . . . ,*e _{ij}*} takes over all the non-empty subset of

*ê*. It follows from the hypothesis that

Fix some *f* < *e*. Let *e*_{t1} ... *e _{tm}* be a smallest (under the natural partial order) product of

*e*

_{1},

*e*

_{2}, ... ,

*e*such that

_{n}*f*≤

*e*

_{t1}···

*e*. Then

_{tm}
with coefficient

and

Let *S* be a locally adequate IC abundant semigroup and *a*, *c* ∈ *S** with *c* ≤ *a*. Then by Lemma 4.4 there exists an idempotent *f* ∈ *ω*(*a**) such that *af* = *c*. We claim that such an idempotent *f* is unique. Suppose that g is another such an idempotent. Then *fg* = *f*, *gf* = *g*. Since *f*, *g* ≤ *a**, we have *f*, *g* ∈ *a** *Sa**. It follows that *gf* = *fg*, and so that *g* = *f*. Denote by *e _{c}* such unique idempotent.

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

*is a bijection;*

*(ii)* {*b* ∈ *S** | *b* ≤ *a*} = {*ae _{af}* |

*θ*≠

*f*≤

*e*}.

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

We shall show that *ϕ* and *ψ* are mutually inverse. Let *b* ≤ *a*. Then *b* = *ae _{b}* and so

*e*=

_{b}*e*∈ {

_{aeb}*e*|

_{af}*f*≤

*e*}. Thus

*ψϕ*(

*b*) =

*ψ*(

*e*) =

_{b}*ae*=

_{b}*b*. On the other hand, let

*f*≤

*e*. Since

*ae*=

_{af}*af*, we have

*ϕψ*(

*e*) =

_{af}*ϕψ*(

*e*) =

_{af}*ϕ*=

_{aeaf}*e*. Consequently

_{af}*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 {*b* ∈ *S** | *b* ≤ *a*} is finite. Hence the element

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

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

Now

Let *f* ∈ *E*(*S*)* with *f* ≤ *e* and *e _{af}* ℒ*

*af*and

*af*· 1 =

*af · f*, we have

*e*· 1 =

_{af}*e*, that is,

_{af}· f*e*=

_{af}*e*. Note that

_{af}f*E*(

*eSe*) is a semilattice. Then

*e*=

_{af}f*fe*since

_{af}*e*∈

_{af}, f*E*(eSe). Thus

*e*≤

_{af}*f*. But

*e*≠

_{af}*f*because

*e*<

_{af}*f*. By the fact

*E*(eSe) is finite that there exists

*e*≤

_{af}*h*. Hence

Therefore

Summing up, we have

*Let S be a locally adequate concordant semigroup with E*(*S*) *locally finite. Then**is a multiplicative basis of**R*_{0}[*S*] *with multiplication given by*

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

Let

Then, by Theorem 4.8, *R*_{0}[*S*] such that

In order to study *R*_{0}[*S*] better via *E*(*S*) locally pseudofinite.

*The map**given by a* ↦ *ā and θ* ↦ 0, *where a* ∈ *S**, *is a bijection.*

*Proof.* Obviously, *ϕ* is surjective. It suffices to show that *ϕ* is injective. Suppose to the contrary that there exist *a*, *c* ∈ *S* such that *a* ≠ *c* and *a*_{1}, · · ·, *a _{s}*,

*c*

_{1}, · · ·,

*c*∈

_{t}*S** with

*a*

_{1}, · · ·,

*a*<

_{s}*a*,

*c*

_{1}, · · ·,

*c*<

_{t}*c*such that

Because *S** is a basis of *R*_{0}[*S*] and *a* ≠ *c*, a must cancel with some *c _{i}*, hence

*a*=

*c*<

_{i}*c*. Similarly,

*c*=

*a*<

_{j}*a*for some

*a*. Now

_{j}*a*<

*c*<

*a*, a contradiction. Therefore

*ϕ*is injective.

*Proof*. Let *e* ∈ *E*(S)*. Note

by Theorem 4.1. By Lemma 4.9 we have *a*^{2} = *a*, that is *a* ∈ *E*(*S*)*. Hence

*Let a* ∈ *S**. Then *and*

*Proof.* Note that *z* ∈ *S*. Then by Theorem 4.1 *ax* = *ay*. Which together with *a***ℒ***a* implies that *a***x* = *a***y*. Therefore *ℛ** is a dual. □

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

*Let**a*, *b* ∈ *S*. *Then*

*(i)*

*(ii)*

*(iii)*

*Proof.* Note that

Assume that *a***b** = *a**, *b***a** = *b**. Clearly, *a***b** = *a** and *b***a** = *b**. Thus *a** *ℒ** (*S*)*b**, hence *a** *ℒ**(*S*) *b*. □

Let

where *a* ∈* *S**, is a bijection. Throughout this paper, we identify the set *Y* (resp., *I*, *L*). For each *a* ∈ *S**, then we denote by

As a direct consequence of Lemma 4.12, we have

*For each a* ∈ *S, we have*

*(i)**resp., *;

*(ii)**is an ideal of*

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

*Let e* ∈ *E*(*S*)*. *Then*

*Proof.* By Lemma 4.12, it is easy to see that the map *ϕ* defined in Lemma 4.9 sends *ϕ* to *ϕ _{e}* is a bijection. Let

*a*,

Hence *ϕ _{e}* is an isomorphism, as required. □

*is primitive abundant.*

*Proof.* That

Recall a semigroup *T* with zero *θ* is called a 0-*direct union of semigroups T _{α}* (

*α*∈

*X*) if

*T*= ∪

_{α∈X}

*T*and

_{α}*T*=

_{α}T_{β}*T*∩

_{α}*T*= {

_{β}*θ*} for all

*α*≠

*β*.

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

*(ii)**is a* 0-*direct union of*

*(iii) On*

*Proof.* (i) Let

It follows that

(ii) Note that *α* ≠ *β* follows from Theorem 4.1. Therefore

(iii) Let *J**(*ā*) is an ideal of *ā* which is saturated by *J**(*ā*) is the smallest ideal containing *ā* which is saturated by

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

It is clear that *tr**(*T*) is a disjoint union 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.

*Let S be a locally adequate concordant semigroup with E*(*S*) *locally pseudofinite. Then**is a multiplicative subsemigroup of R*_{0}[*S*] *and a good homomorphism image of tr**(*S*). *Indeed, from Lemmas 4.9, 4.10 and 4.12, one can deduce that**is isomorphic to the semigroup obtained from tr**(*S*)^{0}*by 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 *R*_{0}[*S*} in last section. As an application, we provide a direct product decomposition for *R*_{0}[*S*] in this section.

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

*is a direct product decomposition of R*

_{0}[

*S*],

*where*

*and*

*are all non-zero*

*classes of*

*Proof.* Since *R*_{0}[*S*], we have

Next we consider the case *R*_{0}[*S*] containning an identity. The following result is essential for us.

([26, Theorem 1.4]). *Let S be a semigroup. If the semigroup ring R*_{0}[*S*] *contains an identity, then there exists a finite subset U of E*(*S*) *and for all s* ∈ *S, there exist e, f* ∈ *U such that s* = *es* = *sf*.

*Let S be a locally adequate concordant semigroup with E*(*S*) *locally finite. If R*_{0}[*S*] *contains an identity, then S as well as**has finitely many**classes* (*resp.*, *classes*, *classes*). *In particular, S as well as**has finitely many idempotents.*

*Proof.* By Lemma 4.12, we only need to consider the case of *R*_{0}[*S*] contains an identity. Then by Lemma 5.2 there exists a finite subset *U* of

This shows that

which is a contradiction. It follows from Lemma 3.3 [14] that

Since

*Let S be a locally adequate concordant semigroup with E*(*S*) *locally finite. If R*_{0}[*S*] *contains an identity, then*

*where*

*and*

*are all non-zero*

*classes of*

*Proof.* If *R*_{0}[*S*] contains an identity, then *S* as well as

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.

*Let S be an IC adequate semigroup with E*(*S*) *locally finite. Then**R*_{0}[*S*] *is a direct product of contracted weak Brandt semigroup algebras. Moreover,**R*_{0}[*S*] *contains an identity if and only if**and**are finite*.

*Proof.* Let *S* be an IC adequate semigroup with *E*(*S*) locally finite and let

where *S* is adequate, it follows from Lemma 4.12 that *R*_{0}[*S*] is a direct product of contracted weak Brandt semigroup algebras.

Suppose that *R*_{0}[*S*] contains an identity. Then by Lemma 5.3

where *α* ∈ *Y*. Let

for each *α* ∈ *Y*, where *P ^{α}* is a diagonal matrix with

*λ*∈ Γ

^{α}. Then the element

is well defined, where *I ^{α}*| × |

*I*| matrix with entry

^{α}*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.

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

*where G*,

_{α}is the maximal subgroup in D_{α}*I*(

_{α}*resp.*, Λ

_{α})

*is the set of the*ℛ -classes(

*resp.*,

*classes*)

*contained in D*Λ

_{α}, and P is a regular_{α}×

*I*

_{α}-matrix with entries in*for each*

*Proof.* It is clear that a regular

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

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

*where G*|

_{α}is the maximal subgroup in D_{α}and*I*|

_{α}*denotes the number of the*

*classes of D*

_{α}for each*Proof.* By hypothesis, Lemmas 4.10 and 4.12, we deduce that *G _{α}* is a maximal subgroup of

*I*is the set of

_{α}*P*is a diagonal

_{α}*I*×

_{α}*I*-matrix with (

_{α}*p*)

_{α}_{ii}is equal to the identity

*e*of

_{α}*G*for each

_{α}*i*∈

*I*. Furhtermore, by Lemma 4.14,

_{α}*G*is isomorphic to any maximal subgroup of S contained in

_{α}*D*; by Lemma 4.12,

_{α}*I*is the set of

_{α}*D*. Now it is easily verified that

_{α}## 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 *R*_{0}[*S*] contains an identity.

By Corollary 4.13, for *R*_{0}[*S*] and can be considered as a right *R*_{0}[*S*]-module for each *i* ∈ *i*.

We first give out a direct sum decomposition of *R*_{0}[*S*].

*If R*_{0}[*S*] *has an identity, then*

*is a finite direct sum decomposition of R*

_{0}[

*S*].

*Proof*. If *R*_{0}[*S*] contains an identity, then S as well as *I* is finite. Since *R*_{0}[*S*]-module *R*_{0}[*S*]_{R0[S]} is a direct sum of *R*_{0}[*S*]. □

By Lemma 4.16,

Next we investigate conditions under which the projective *R*_{0}[*S*]-modues

*Let α, β*, ∈ *Y, i* ∈ *J ^{α}; J* ∈

*J*.

^{β}*If*

*then α*=

*β*.

*Proof.* Let *R*_{0}[*S*]-module isomorphism. Suppose to the contrary that *α* ≠ *β*. Let *α* = *β*, as required. □

Let *β* ∈ *Y* and *R*_{0}[*S*]-module. By Theorem 5.1,

Thus we only need to consider *R*_{0}[*S*]-module if and only if M is an indecomposable right *α* ∈ *Y*).

Let *λ* ∈ Γ, *i*; *j* ∈ *J _{λ}*. For each

*μ*∈ Γ, define

Here *n _{iμ}* =

*n*. Since

_{jμ}*k, s*), (

*l, t*) in the (λ, μ)-block, we have

*M satisfies the row-block condition*if for all

*λ*≠

*ν*∈ Γ,

*i*∈

*J*and

_{λ}*j*∈

*J*, there exists

_{ν}*μ*∈ Γ such that

*n*≠

_{iμ}*n*.

_{jμ}*Let**be a *

*and*

*i, j, ∈ J*.

*(i) If i, j* ∈ *J _{λ} for some λ* ∈ Γ,

*then*

*(ii) If**then n _{iμ}* =

*n*;

_{jμ}for each μ ∈ Γ*(iii) If**satisfies the row-block condition, then**collects pairwise non-isomorphic projective right**modules*.

*Proof.* (i) Let *i, j* ∈ *J _{λ}* for some

*λ*∈ Γ. Then for any

*μ*∈ Γ,

*n*=

_{iμ}*n*, and hence we can define a map

_{jμ}*i, ā, s*) ↦ (

*j, ā, s*), where

*s*∈ ⋀

_{μ}and

*ā*∈

*M*, and extend

_{λμ}*R*-linearly. By definition,

*ψ*restricts to a bijection

*ψ*is a

*R*-module isomorphism from

*ψ*is a right

Therefore *ψ* is a right *R*_{0}[*S*]-module isomorphism, and (i) is proved.

(ii) Without loss of generality, suppose that *μ* ∈ Γ and *l* ∈ *J _{μ}* and an idempotent

It follows from the fact *k* ∈ *J*. Because *ψ* is a bijection from *n _{iμ}* =

*n*.

_{jμ}(iii) This follows from (i) and (ii). □

Let *i, j* ∈ *J*, according to Lemma 6.3, *λ* ∈ Γ such that *i, j* ∈ *J _{λ}*. Thus it suffices to find the non-isomorphic indecomposable direct summands of

*λ*∈ Γ.

Let *λ* ∈ Γ and let *f _{λ}*, 1, ···

*f*

_{λ,nλ}, ···

*f*

_{λ,nλ+mλ}, be a complete set of primitive orthogonal idempotents of

*R*[

*T*] such that

_{λ}*f*

_{λ,1}

*R*[

*T*], ···

_{λ}*f*

_{μ,nλ}*R*[

*T*] are all the non-isomorphic projective indecomposable right

_{λ}*R*[

*T*]-modules. Notice that

_{λ}*Let**class of**and*

*(i) For each pair u, υ* ∈ *R*[*T _{λ}*],

*the right R*[

*T*]

_{λ}*-modules uR*[

*T*] ≅

_{λ}*υR*[

*T*]

_{λ}*if and only if the right*

*-modules*

*(ii) Let f* ∈ *R*[*T*_{λ}] *be an idempotent. Then f R*[*T _{λ}*]

*is an indecomposable right R*[

*T*]-

_{λ}*module if and only if*

*is an indecomposable right*

*module*.

*Proof*. (i) Suppose that *ϕ*: *uR*[*T _{λ}*] →

*υR*[

*T*] is a right

_{λ}*R*[

*T*]-module isomorphism. Let

_{λ}*w*∈

*R*[

*T*] and

_{λ}_{λ},

*w,*1

_{λ})(

*i, y, s*) = (1

_{λ},

*w*(

*p*

_{1λ,i}

*y*),

*s*). If

*i*= 1

_{λ}, then

*p*

_{1λ,i}

*y*=

*e*=

_{λ}y*y*by our assumption on

*P*. Therefore

By condition (C) in the definition of PA blocked Rees matrix semigroups, for all *μ* ∈ Γ and *x*, *y* ∈ *M _{λμ}*, if

*wx*=

*wy*, then

*x*=

*y*in

*M*. Thus the

_{λμ}*R*-linear map

is well defined and is injective. We claim that
*ϕ* is surjective,

Conversely, suppose that
*w* ∈ *uR*[*T _{λ}*],

Thus we can define a map *ϕ* : *uR*[*T _{λ}*] →

*μR*[

*T*] by

_{λ}*ϕ*is a bijection. It thus suffices to show

*ϕ*(

*wx*) =

*ϕ*(

*w*)

*x*for all

*x*∈

*R*[

*T*]. Indeed,

_{λ}
which implies *ϕ*(*wx*) = *ϕ*(*w*)*x*, and (i) follows.

(ii) Clearly, *f*′ = (1_{λ}, *f*, 1_{λ}) is an idempotent of *R*_{0}[*M*]. We only need to show that
*f* ∈ Mult *R*[*T _{λ}*] is primitive. Indeed, let

*e*′ <

*f*′ if and only if there exists an idempotent

*e*∈

*R*[

*T*] such that

_{λ}*e*′ = (1

_{λ},

*e*, 1

_{λ}) 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.

*Let S be a locally adequate concordant semigroup with E*(*S*) *locally finite and**If*

*(i) for each α* ∈ *Y*,
*satisfies the row-block condition,*

*(ii) for each λ* ∈ Γ^{α},
*is a complete set of primitive orthogonal idempotents of**such that**are all the non-isomorphic projective indecomposable right**modules,*

*then the set**collects all the non-isomorphic projective indecomposable right R*_{0}[*S*]-*modules.*

*Proof.* Let *α* ∈ *Y* and *λ* ∈ Γ^{α}. By Lemma 6.4 and the hypotheses, the right
*q*,

As mentioned before, *M* is an indecomposable right
*M* is an indecomposable right *R*_{0}[*S*]-module. Consequently,
*R*_{0}[*S*]-modules, where the union takes over all *α* ∈ *Y*, *λ* ∈ Γ^{α} and

For each
^{α}| = 1 then the semigroup

*Let S be a locally adequate concordant semigroup with E*(*S*) *locally finite and for each**be a Rees matrix semigroup over a cancellative monoid T _{α}.*

*(i)*

*(ii)**R*_{0}[*S*] *is representation-finite if and only if for each α* ∈ *Y*, *R*[*T _{α}*]

*is representation-finite.*

*Proof.* (i) It is clear that *R*_{0}[*S*] satisfies the row-block condition. Let
*R*[*T _{α}*] such that

*R*[

*T*]-modules. Then

_{α}*R*[

*T*]

_{α}^{b}=

*e*

_{R[Tα]}. By Theorem 6.5, we have

*e*

_{R0[S]}= ∑

_{α ∈ Y}(1

_{α},

*e*

_{R[Tα]}, 1

_{α}), where 1

_{α}∈

*J*denote the element

_{α}*α*∈

*Y*. This, together with the fact

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

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

*Let S be a locally inverse semigroup with idempotents set E*(*S*) *locally finite. Suppose that R*_{0}[*S*] *contains an identity. Then R*_{0}[*S*] *is representation-finite if and only if R*_{[Gα]}*is representation-finite for each α* ∈

*Proof*. let *α* ∈ *Y*. Then

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 *p*⃒|*G*|, *K*[*G*] is representation-finite if and only if the Sylow *p*-subgroups *G _{p}* 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

*G*of

_{p}*G*are cyclic.

Now the next result follows from Corollary 6.7 directly.

*Let S be a locally inverse semigroup with E*(*S*) *locally finite and all its maximal subgroups of finite order. Suppose that K*_{0}[*S*] *contains an identity. Then K*_{0}[*S*] *is representation-finite if and only if for each**with**p*⃒|*G _{α}*|,

*the Sylow p-subgroups*(

*G*)

_{α}_{p}

*of G*

_{α}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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**Received:**2014-4-2

**Accepted:**2015-12-28

**Published Online:**2016-2-9

**Published in Print:**2016-1-1

© 2016 Ji and Luo, published by De Gruyter Open

This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 3.0 License.