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 $\mathcal{K}$ be an equivalence relation on *S*. The $\mathcal{K}$-class containing an element *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 $\mathcal{K}$-classes of *S* by ${(S/\mathcal{K})}^{*}$.

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*
$$\begin{array}{lll}a\text{\hspace{0.17em}}\mathcal{L}\text{\hspace{0.17em}}b\hfill & \iff \hfill & {S}^{1}a={S}^{1}b,\hfill \\ a\text{\hspace{0.17em}}\mathcal{R}\text{\hspace{0.17em}}b\hfill & \iff \hfill & a\text{\hspace{0.17em}}{S}^{1}=b{S}^{1},\hfill \\ a\text{\hspace{0.17em}}\mathcal{J}\text{\hspace{0.17em}}b\hfill & \iff \hfill & {S}^{1}a{S}^{1}={S}^{1}b{S}^{1},\hfill \\ \mathcal{H}\hfill & =\hfill & \mathcal{L}\cap \mathcal{R},\hfill \\ \mathcal{D}\hfill & =\hfill & \mathcal{L}\vee \mathcal{R}.\hfill \end{array}$$

It is clear that $\mathcal{L}$ (resp., $\mathcal{R}$) is a right (resp., left) congruence on *S* and $\mathcal{D}\subseteq \mathcal{J}$. A semigroup *S* is called *regular* if every $\mathcal{L}$-class and every $\mathcal{R}$-class contain idempotents. The regularity of a semigroup *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*,
$$\begin{array}{lll}a\text{\hspace{0.17em}}{\mathcal{L}}^{*}\text{\hspace{0.17em}}b\hfill & \iff \hfill & (\forall x,\text{\hspace{0.17em}}y\in {S}^{1})(ax=ay\leftrightarrow bx=by),\hfill \\ a\text{\hspace{0.17em}}{\mathcal{R}}^{*}\text{\hspace{0.17em}}b\hfill & \iff \hfill & (\forall x,\text{\hspace{0.17em}}y\in {S}^{1})(xa=ya\leftrightarrow xb=yb,\hfill \\ a\text{\hspace{0.17em}}\mathcal{J}*\text{\hspace{0.17em}}b\hfill & \iff \hfill & {J}^{*}(a)=J*(b),\hfill \\ {\mathcal{H}}^{*}={\mathcal{L}}^{*}\wedge {\mathcal{R}}^{*}\hfill & \text{and}\hfill & {\mathcal{D}}^{*}={\mathcal{L}}^{*}\vee {\mathcal{R}}^{*},\hfill \end{array}$$
where *J*^{*}(*a*) is the smallest ideal containing a which is saturated by ${\mathcal{L}}^{*}$ and ${\mathcal{R}}^{*}$.

Clearly, ${\mathcal{L}}^{*}$ (resp., ${\mathcal{R}}^{*}$) is a right (resp., left) congruence on *S*. It is easy to see that ${\mathcal{L}\subseteq \mathcal{L}}^{*}$ (resp., ${\mathcal{R}\subseteq \mathcal{R}}^{*}$), and for *a*, *b* ∈ Reg (*S*), $a\mathcal{L}b$ (resp., $a\mathcal{R}b$) if and only if $a{\mathcal{L}}^{*}b$ (resp., $a{\mathcal{R}}^{*}b$). So $\mathcal{L}={\mathcal{L}}^{*},\mathcal{R}={\mathcal{R}}^{*}$ and $\mathcal{J}={\mathcal{J}}^{*}$ on regular semigroups.

We say a semigroup is *abundant* if each ${\mathcal{L}}^{*}$-class and each ${\mathcal{R}}^{*}$-class of it contains an idempotent. It is clear that regular semigroups are abundant semigroups.

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

Define two partial orders ≤_{r} and ≤_{l} on *S* [16] by
$$\begin{array}{lll}a{\le}_{r}b\hfill & \iff \hfill & {R}_{a}^{*}\le {R}_{b}^{*}\text{\hspace{0.17em}}\text{and}\text{\hspace{0.17em}}a={a}^{\u2020}b\text{\hspace{0.17em}}\text{for}\text{\hspace{0.17em}}\text{some}\text{\hspace{0.17em}}{a}^{\u2020},\hfill \\ a{\le}_{l}b\hfill & \iff \hfill & {L}_{a}^{*}\le {L}_{b}^{*}\text{\hspace{0.17em}}\text{and}\text{\hspace{0.17em}}a=b{a}^{*}\text{\hspace{0.17em}}\text{for}\text{\hspace{0.17em}}\text{some}\text{\hspace{0.17em}}{a}^{*},\hfill \end{array}$$

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*^{*} ${a}^{\u2020}\in {R}_{a}^{*}(S)\cap E(S)$ and ${a}^{*}\in {L}_{a}^{*}(S)\cap E(S)$, there is an isomorphism
$${\alpha}_{a}:\u3008{a}^{\u2020}\u3009\to \u3008{a}^{*}\u3009,\text{\hspace{0.17em}}\text{with}\text{\hspace{0.17em}}xa=a{\alpha}_{a}(x),$$
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*) = {*g* ∈ *V* (*ef*) ∩ *E*(*S*) | *ge* = *fg* = *g*} of idempotents *e* and *f* is non-empty, and takes the form
$$S(e,f)=\{g\in E(S)|\text{\hspace{0.17em}}ge=fg=g,\text{\hspace{0.17em}}egf=ef\}.$$

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 ${M}_{\lambda \mu}=\overline{)0}$ or *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 $\left\{0\right\}\cup \text{\hspace{0.17em}}{\displaystyle \cup \{{M}_{\lambda \mu}:\lambda ,\mu \in \Gamma \}}$.

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 ${\U0001d4dc}^{0}({M}_{\lambda \mu};J,\Lambda ,\Gamma ;P)$ 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*, *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* $M={\U0001d4dc}^{0}({M}_{\lambda \mu};J,\Lambda ,T;P)$ *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* ${\mathcal{R}}^{*}$-*related if and only if i* = *j*;

*(iv) the non-zero elements* (*a*)_{is} *and* (*b*)_{jt} *of M are* ${\mathcal{L}}^{*}$-*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* $\mathcal{D}$-*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 $M={\U0001d4dc}^{0}({M}_{\lambda \mu};J,\Lambda ,T;P)$ be a PA blocked Rees matrix semigroup. Then we can always assume that there exists 1_{λ} ∈ *J*_{λ} 1_{λ} ∈ *J*_{λ} ∩ Λ_{λ} such that ${H}_{{1}_{\lambda}{1}_{\lambda}}^{*}={T}_{\lambda}$ is a cancellative monoid with an identity *e*_{λ}(*λ* ∈ Γ).

Recall that a *Munn algebra* is an algebra $\mathcal{M}(A;I,\Lambda ;p)$ 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 $M={\U0001d4dc}^{0}(G;J,\Lambda ;P)$ be a completely 0-simple semigroup. It is known that ${R}_{0}[M]\cong \U0001d4dc(R[G];J,\Lambda ;P)$, see [22, Lemma 5.17].

Let $M={\U0001d4dc}^{0}({M}_{\lambda \mu};J,\Lambda ,\Gamma ;P)$ be a PA blocked Rees matrix semigroup. Define the *generalized Munn algebra* $\mathcal{M}(R[{M}_{\mu \lambda}];J,\Lambda ,\Gamma ;P)$ *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].

${R}_{0}[M]\cong \U0001d4dc(R[{M}_{\mu \lambda}];J,\Lambda ,\Gamma ;P)$.

If $({a}_{is})\in \mathcal{M}(R[{M}_{\mu \lambda}];J,\Lambda ,\Gamma ;P)$ has only one nonzero entry *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 ${\U0001d4dc}^{0}({M}_{\mu \lambda};J,\Lambda ,\Gamma ;P)$; furthermore, there is variability in the sandwich matrix *P* on the choice of data in constructing the isomorphism.

We can simply take ${r}_{{1}_{\lambda}}={q}_{{1}_{\lambda}}={e}_{\lambda}$, and thus for all $\lambda \in \Gamma ,{p}_{{1}_{\lambda}{1}_{\lambda}}={q}_{{1}_{\lambda}}{r}_{{1}_{\lambda}}={e}_{\lambda}$, 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 ${0\text{-}\mathcal{J}}^{*}$-*simple* if *S*^{2} ≠ {*θ*} and *S*, {*θ*} are the only ${\mathcal{J}}^{*}$-classes of *S*. It is known that a primitive abundant semigroup is a 0-direct union of primitive abundant ${0\text{-}\mathcal{J}}^{*}$-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 *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 ${0\text{-}\mathcal{J}}^{*}$ -simple primitive adequate semigroup, or just a ${0\text{-}\mathcal{J}}^{*}$ -simple PA blocked Rees matrix semigroup ${\U0001d4dc}^{0}({M}_{\lambda \mu};J,J,\Gamma ;P)$ with the properties that the sandwich matrix *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 ${e}_{A}={e}_{{j}_{1}}+\cdots +{e}_{{j}_{m}}$, and ${e}_{{j}_{1}},\cdots ,{e}_{{j}_{m}}$ are chosen such that ${e}_{{j}_{t}}A,1\le t\le m$, are all the non-isomorphic projective indecomposable right *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.

## Comments (0)