Locally adequate semigroup algebras

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 ℛ-trivial monoids are basic; furthermore the projective indecomposable modules have been described, see [10–12]. 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 ℬ¯ 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 ℬ¯ 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 K_a or K_a(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; b ∈ S

It is clear that ℒ (resp., ℛ) is a right (resp., left) congruence on S and D⊆J. A semigroup S is called regular if every ℒ-class and every 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,

where J^*(a) is the smallest ideal containing a which is saturated by ℒ* and ℛ*.

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

We say a semigroup is abundant if each ℒ*-class and each 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 ℛ* (resp., ℒ*).

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^*a†∈Ra*(S)∩E(S) and a*∈La*(S)∩E(S), there is an isomorphism

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 ↦ tmt) is an (S, T)-isomorphism from S ⊗_SM (resp., M ⊗_TT) 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 Mλμ=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 0x = x0 = 0 for every element x of {0}∪ ∪{Mλμ: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 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_sjb)_it makes M be a semigroup, which we denote by 𝓜0(Mλμ;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, a₂ ∈ M_λμ, b, b₁, b₂ ∈ M_μκ, then ab₁ = ab₂ implies b₁ = b₂; a₁b = a_2b implies a₁ = a₂;

(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.

Let M=𝓜0(Mλμ;J,Λ,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 H1λ1λ*=Tλ is a cancellative monoid with an identity e_λ(λ ∈ Γ).

Recall that a Munn algebra is an algebra ℳ(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=𝓜0(G;J,Λ;P) be a completely 0-simple semigroup. It is known that R0[M]≅𝓜(R[G];J,Λ;P), see [22, Lemma 5.17].

Let M=𝓜0(Mλμ;J,Λ,Γ;P) be a PA blocked Rees matrix semigroup. Define the generalized Munn algebraℳ(R[Mμλ];J,Λ,Γ;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_isP (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].

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 𝓜0(Mλμ;J,J,Γ;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 R₀ [S], we mean the factor algebra R[S]/R[θ].

If a = ∑ r_is_i ∈ R₀ [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 R₀ [S], because it is a R-basis of R₀ and 0-closed (S² ⊆ 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 I₁ ⊕ … ⊕ 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 {e₁, …, e_n} of primitive orthogonal idempotents of A such that I_i = e_iA (i = 1, …, n).

Assume that A is a R-algebra with an identity and {e₁,…, e_n} is a complete set of primitive orthogonal idempotents of A. The algebra A is called basic if e_iA ≇ e_jA, for all i ≠ j.

The basic algebra associated toA is the algebra A^b = e_AAe_A, where eA=ej1+⋯+ejm, and ej1,⋯,ejm are chosen such that ejtA,1≤t≤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.

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}. Es is said to be pseudofinite if

1. for e, f ∈ E with e > f, there exists an element g such that e covers g and e > g ≥ f;

2. 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) ∈ R₀[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 R₀ [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 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 R₀[S], and σ(e), e ∈ E(S), are called the Rukolaĭne idempotents of R₀[S].

Proof. (i) By the hypothesis that S is IC abundant, there exists a semigroup isomorphism αa: 〈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 α_a (a^†) = a* and (i) holds.

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

□

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).

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 ℬ¯ of R₀[S] by means of the Rukolaĭne idempotents defined in Section 3. Then we provide some properties of the semigroup ℬ¯∪{0}.

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 a*^ commute, f = e_i1 … e_ij is an idempotent and f < a*. Now aℒ*a* implies 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 ℬ¯ is a multiplicative basis of R₀[S].

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

Suppose that E(S)∩La*∩Rb*≠0. Take e∈E(S)La* and f∈E(S)∩Rb*. Then ā = aσ(e) and b¯=σ(f)b. Note that σ(e)e = σ(e) and fσ(f) = σ(f).

If ef = θ, then ef = 0 in R₀[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, S(e,f)≠0. Thus there exists a nonzero idempotent 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 g ℛ*f and g​ ℒ*e. So g∈Le*∩Rf*∩E(S)=0, which is a contradiction. Without loss of generality, assume that eg < e. Then there exists h_g ∈ ê such that eg ≤ h_g since E(S) is pseudofinite. It follows that

Therefore

□

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

Let x_l be a maximal element of {x₁, x₂, ... , x_n} under the natural partial order ≤ on S. By (3) suppose that x¯1=xt+∑i=1ktrilbil with r_il ≠ 0 and b_il < x_l for i = 1, ... , k_l, l = 1, 2, ..., n. Then

Since S \ {θ} is a basis of R₀[S] and r_l ≠ 0, there exists at least an element b_ij for some j ≠ l and some i such that b_ij = x_i. Thus x_l = b_ij ≤ x_j, which is a contradiction. Therefore ℬ¯ is linearly independent. □

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

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 = {e₁, e₂, . . . , e_n} with n ≤ 1. Then f ≤ e if and only if f ≤ e_s for some 1 ≤ 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 e₁, e₂, ... , e_n such that f ≤ e_t1 ··· e_tm. Then f¯ appears in the sum

with coefficient (−1)m+Cm1(−1)m−1+Cm2(−1)m−2+⋯+Cmm−1(−1)=−1. Thus

and e=∑f≤ef¯. □

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 f∈Lc* and af = c. We claim that such an idempotent f is unique. Suppose that g is another such an idempotent. Then g​ ℒ* f, and hence 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.

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_af. 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 {b ∈ S* | b ≤ a} is finite. Hence the element ∑θ≠b≤ab¯∈R0[S] is well defined. In what follows, we write ∑b≤ab¯ instead of ∑θ≠b≤ab¯.

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

Now

Let f ∈ E(S)* with f ≤ e and f∉Laf*. Since e_af ℒ* af and af · 1 = af · f, we have e_af · 1 = e_af · f, that is, e_af = e_af f. Note that E(eSe) is a semilattice. Then e_af f = fe_af since e_af, f ∈ E(eSe). Thus e_af ≤ f. But e_af ≠ f because f∉Laf*, hence e_af < f. By the fact E(eSe) is finite that there exists h∈f^ such that e_af ≤ h. Hence

Therefore ∑f≤e,f∉Laf*af¯=0. It follows that

Summing up, we have

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

Let

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

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¯=c¯. By (3), there exist a₁, · · ·, a_s, c₁, · · ·, c_t ∈ S* with a₁, · · ·, a_s < a, c₁, · · ·, c_t < c such that a¯=a+∑i=1sriai and c¯=c+∑i=1tr′ici for some r1,…,rs,r′1,…,r′t∈R*. Thus

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

Proof. Let e ∈ E(S)*. Note e∈Le*∩Re*. 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

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

Proof. Note that La*∩Rz*∩E(S)=La**∩Rz*∩E(S) for any z ∈ S. 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 ax¯=a¯x¯=a¯y¯=ay¯. Hence ax = ay. Which together with a*ℒ*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¯* ℒ* a¯. The case for ℛ* is a dual. □

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

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

Assume that a ℒ*(S) b. Then a* ℒ*(S) b* and so a*b* = a*, b*a* = b*. Clearly, a*∈Lb**∩Ra** and b*∈La**∩Rb**. It follows from Theorem 4.1 that a¯*b¯*=a*b*¯=a¯* and b¯*a¯*=b*a*¯=b¯*, that is, a¯* ℒ*(S¯)b¯*. It follows from Lemma 4.11 that a¯​ ℒ*(S¯) a¯* ℒ*(S¯) b¯* ℒ*(S¯)b¯. Conversely, suppose that a¯ ℒ*(S¯) b¯. Then by Lemma 4.11a¯* ℒ*(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* ℒ* (S)b*, hence a* ℒ*(S) b. □