A Note on Locally Inverse Semigroup Algebras

Abstract

Let 𝑅 be a commutative ring and 𝑆 a finite locally inverse semigroup. It is proved that the semigroup algebra 𝑅[𝑆] is isomorphic to the direct product of Munn algebras ℳ(𝑅[𝐺_𝐽],𝑚_𝐽,𝑛_𝐽;𝑃_𝐽) with 𝐽∈𝑆/𝒥, where 𝑚_𝐽 is the number of ℛ-classes in 𝐽, 𝑛_𝐽 the number of ℒ-classes in 𝐽, and 𝐺_𝐽 a maximum subgroup of 𝐽. As applications, we obtain the sufficient and necessary conditions for the semigroup algebra of a finite locally inverse semigroup to be semisimple.

1. Main Results

A regular semigroup 𝑆 is called a locally inverse semigroup if for all idempotent 𝑒∈𝑆, the local submonoid 𝑒𝑆𝑒 is an inverse semigroup under the multiplication of 𝑆. Inverse semigroups are locally inverse semigroups. Inverse semigroup algebras are a class of semigroup algebras which is widely investigated. One of fundamentally important results is that a finite inverse semigroup algebra is the direct product of full matrix algebras over group algebras of the maximum subgroups of this finite inverse semigroup. Consider that all local submonoids of a locally inverse semigroup are inverse semigroups, it is a very natural problem whether a finite locally inverse semigroup algebra has a similar representation to inverse semigroup algebras. This is the main topic of this note.

Let 𝒜 be an 𝑅-algebra. Let 𝑚 and 𝑛 be positive integers, and let 𝑃 be a fixed 𝑛×𝑚 matrix over 𝒜. Let ℳ∶=ℳ(𝒜;𝑚,𝑛;𝑃) be the vector space of all 𝑚×𝑛 matrices over 𝒜. Define a product ∘ in ℳ by 𝐴∘𝐵=𝐴𝑃𝐵(𝐴,𝐵∈ℳ), (1.1) where 𝐴𝑃𝐵 is the usual matrix product of 𝐴, 𝑃, and 𝐵. Then ℳ is an algebra over 𝑅. Following [1], we call ℳ the Munn 𝑚×𝑛  matrix algebra over  𝒜  with sandwich matrix  𝑃.

By a semisimple semigroup, we mean a semigroup each of whose principal factor is either a completely 0-simple semigroup or a completely simple semigroup. It is well known that a finite regular semigroup is semisimple. The Rees theorem tells us that any completely 0-simple semigroup (completely simple semigroup) is isomorphic to some Rees matrix semigroup ℳ⁰(𝐺,𝐼,Λ;𝑃) (ℳ(𝐺,𝐼,Λ;𝑃)), and vice versa (for Rees matrix semigroups, refer to [1]). In what follows, by the phrase “Let 𝑆=⋃_{𝐽∈𝑆/𝒥}ℳ⁰(𝐺_𝐽;𝐼_𝐽,Λ_𝐽;𝑃_𝐽) be a finite regular semigroup,” we mean that 𝑆 is a finite regular semigroup in which the principal factor of 𝑆 determined by the 𝒥-class 𝐽 is isomorphic to the Rees matrix semigroup ℳ⁰(𝐺_𝐽;𝐼_𝐽,Λ_𝐽;𝑃_𝐽) or ℳ(𝐺_𝐽;𝐼_𝐽,Λ_𝐽;𝑃_𝐽) for any 𝐽∈𝑆/𝒥.

The following is the main result of this paper.

Theorem 1.1. Let 𝑆=⋃_{𝐽∈𝑆/𝒥}ℳ⁰(𝐺_𝐽,𝐼_𝐽,Λ_𝐽;𝑃_𝐽) be a finite locally inverse semigroup. Then the semigroup algebra 𝑅[𝑆] is isomorphic to the direct product of ℳ(𝑅[𝐺_𝐽];|𝐼_𝐽|,|Λ_𝐽|;𝑃_𝐽) with 𝐽∈𝑆/𝒥.

Based on Theorem 1.1 and [1, Lemma 5.17, page 162, and Lemma 5.18, page 163], the following corollary is straightforward.

Corollary 1.2. Let 𝑆=⋃_{𝐽∈𝑆/𝒥}ℳ⁰(𝐺_𝐽,𝐼_𝐽,Λ_𝐽;𝑃_𝐽) be a finite locally inverse semigroup. Then the semigroup algebra 𝑅[𝑆] has an identity if and only if |𝐼_𝐽|=|Λ_𝐽| and 𝑃_𝐽 is invertible in the full matrix algebra 𝑀_|𝐼𝐽|(𝑅[𝐺_𝐽]) for all 𝐽∈𝑆/𝒥.

Reference [1, Lemma 5.18, page 163] told us that ℳ(𝑅[𝐺_𝐽],𝑚_𝐽,𝑛_𝐽;𝑃_𝐽) is isomorphic to the full matrix algebra 𝑀_𝑛𝐽(𝑅[𝐺_𝐽]) if ℳ(𝑅[𝐺_𝐽],𝑚_𝐽,𝑛_𝐽;𝑃_𝐽) has an identity. Now, we have the following.

Corollary 1.3. Let 𝑆=⋃_{𝐽∈𝑆/𝒥}ℳ⁰(𝐺_𝐽,𝐼_𝐽,Λ_𝐽;𝑃_𝐽) be a finite locally inverse semigroup. If 𝑅[𝑆] has an identity, then 𝑅[𝑆] is isomorphic to the direct product of the full matrix algebras 𝑀_|𝐼𝐽|(𝑅[𝐺_𝐽]) with 𝐽∈𝑆/𝒥.

The following corollary is a consequence of Corollary 1.3.

Corollary 1.4. Let 𝑆=⋃_{𝐽∈𝑆/𝒥}ℳ⁰(𝐺_𝐽,𝐼_𝐽,Λ_𝐽;𝑃_𝐽) be a finite locally inverse semigroup. Then the semigroup algebra 𝑅[𝑆] is semisimple if and only if for all 𝐽∈𝑆/𝒥,
(1)|𝐼_𝐽|=|Λ_𝐽|;(2)𝑃_𝐽 is invertible in the full matrix algebra 𝑀_|𝐼𝐽|(𝑅[𝐺_𝐽]);(3)𝑅[𝐺_𝐽] is semisimple.

2. Proof of Theorem 1.1

For our purpose, we have the Möbius inversion theorem [2].

Lemma 2.1. Let (𝑃,≤) be a locally finite partially ordered set (i.e., intervals are finite) in which each principal ideal has a maximum and 𝐺 be an Abelian group. Suppose that 𝑓∶𝑃→𝐺 is a function and define 𝑔∶𝑃→𝐺 by 𝑔(𝑥)=∑_𝑦≤𝑥𝑓(𝑦). Then 𝑓(𝑥)=∑_𝑦≤𝑥𝑔(𝑦)𝜇(𝑥,𝑦), where 𝜇 is a Möbius function.

Now assume that 𝑆 is a regular semigroup and 𝑎,𝑏∈𝑆. Define 𝑎≤𝑏⟺thereexist𝑒,𝑓∈𝐸(𝑆)suchthat𝑎=𝑒𝑏=𝑏𝑓. (2.1) Then ≤ is a partial order on 𝑆. Following [3], we call ≤ the natural partial order on 𝑆. Equivalently, 𝑎≤𝑏 if and only if for every (for some) 𝑓∈𝐸(𝑅_𝑏)(𝑓∈𝐸(𝐿_𝑏)), there exists 𝑒∈𝐸(𝑅_𝑎)(𝑒∈𝐸(𝐿_𝑎)) such that 𝑒≤𝑓 and 𝑎=𝑒𝑏(𝑎=𝑏𝑒). Moreover, Nambooripad [3, 4] proved that 𝑆 is a locally inverse semigroup if and only if the natural partial order ≤ is compatible with respect to the multiplication of 𝑆.

Lemma 2.2. Let 𝑆 be a locally inverse semigroup and 𝑎,𝑏∈𝑆. Then for any 𝑢≤𝑎𝑏, there exist 𝑥≤𝑎 and 𝑦≤𝑏 such that 𝑢=𝑥𝑦, 𝑥∈𝑅_𝑢 and 𝑦∈𝐿_𝑢.

Proof. For any 𝑒∈𝐸(𝑅_𝑎), we have 𝑒𝑎=𝑎 and 𝑒𝑎𝑏=𝑎𝑏. Let 𝑧 be an inverse of 𝑎𝑏. Clearly, 𝑎𝑏𝑧∈𝐸(𝑅_𝑎𝑏). Note that 𝑒𝑎𝑏𝑧=𝑎𝑏𝑧. It is easy to check that 𝑎𝑏𝑧𝑒∈𝐸(𝑆),𝑎𝑏𝑧𝑒≤𝑒, and 𝑎𝑏𝑧ℛ𝑎𝑏𝑧𝑒. Hence 𝑎𝑏𝑧𝑒ℛ𝑎𝑏 and there exists 𝑔∈𝐸(𝑆) such that 𝑢=𝑔𝑎𝑏 and 𝑔≤𝑎𝑏𝑧𝑒(≤𝑒). Thus 𝑔𝑎≤𝑎. On the other hand, since ℛ is a left congruence and since 𝑎𝑏𝑧𝑒ℛ𝑎𝑏, we have 𝑢=𝑔𝑎𝑏ℛ𝑔𝑎𝑏𝑧𝑒=𝑔; while since 𝑎ℛ𝑒, we have 𝑔𝑎ℛ𝑔𝑒=𝑔. These imply that 𝑢ℛ𝑔𝑎. Dually, we have ℎ∈𝐸(𝑆) such that 𝑢=𝑎𝑏ℎ, 𝑏ℎ≤𝑏 and 𝑢ℒ𝑏ℎ. Since 𝑢=𝑔𝑎𝑏=𝑎𝑏ℎ=𝑢ℎ=(𝑔𝑎)(𝑏ℎ), we know that 𝑔𝑎 and 𝑏ℎ are the required elements 𝑥 and 𝑦.

Define a multiplication ⊗ on 𝑆⁰=𝑆∪{0} by 𝑥⊗𝑦=𝑥𝑦if𝑥≠0,𝑦≠0,and𝑦,𝑥𝑦∈𝐽_𝑥;0otherwise, (2.2) where 𝑥𝑦 is the product of 𝑥 and 𝑦 in 𝑆. By the arguments of [4, page 9], (𝑆⁰,⊗) is a semigroup. We denote by 𝑆^⊗ the semigroup (𝑆⁰,⊗). For any 𝐽∈𝑆/𝒥, we denote 𝐽⁰=𝐽∪{0}. It is easy to check that (𝐽⁰,⊗) is a subsemigroup of 𝑆^⊗, which is isomorphic to the principal factor of 𝑆 determined by 𝐽. We will denote the semigroup (𝐽⁰,⊗) by 𝐽^⊗. By the definition of ⊗, it is easy to see that in the semigroup 𝑆^⊗,

(i)𝐽^⊗_𝑥⊗𝐽^⊗_𝑥⊆𝐽^⊗_𝑥 for all 𝑥∈𝑆;(ii)𝐽^⊗_𝑥⊗𝐽^⊗_𝑦=0 for all 𝑥,𝑦∈𝑆 such that 𝑥∉𝐽_𝑦. Thus 𝑅₀[𝑆^⊗] is the direct sum of the contracted semigroup algebras 𝑅₀[𝐽^⊗] with 𝐽∈𝑆/𝒥. Note that 𝐽^⊗ is isomorphic to some principal factor of 𝑆. We observe that 𝐽^⊗ is a completely 0-simple semigroup since 𝑆 is a semisimple semigroup, and thus 𝐽^⊗ is isomorphic to some Rees matrix semigroup ℳ⁰(𝐺_𝐽,𝐼_𝐽,Λ_𝐽;𝑃_𝐽). By a result of [1], 𝑅₀[ℳ⁰(𝐺_𝐽,𝐼_𝐽,Λ_𝐽;𝑃_𝐽)] is isomorphic to ℳ(𝑅[𝐺_𝐽],|𝐼_𝐽|,|Λ_𝐽|;𝑃_𝐽). Consequently, to verify Theorem 1.1, we need only to prove that 𝑅[𝑆] is isomorphic to 𝑅₀[𝑆^⊗].

For the convenience of description, we introduce the semigroup 𝑆. Put 𝑆={𝑥∣𝑥∈𝑆}∪{0}. Define a multiplication on 𝑆 as follows: 𝑥∗𝑦=𝑥⊗𝑦, (2.3) where we will identify 0 with 0. It is easy to see that 𝑆 is isomorphic to 𝑆^⊗. Hence the contracted semigroup algebra 𝑅₀[𝑆] is isomorphic to the contracted semigroup algebra 𝑅₀[𝑆^⊗]. For 𝐽∈𝑆/𝒥, we denote 𝐽={𝑥∣𝑥∈𝐽}∪{0}. It is easy to check that (𝐽,∗) is a subsemigroup of 𝑆 isomorphic to the semigroup 𝐽^⊗. So, for any 𝐽,𝐾∈𝑆/𝒥, we have 𝐽∗𝐾⊆𝐽if𝐾=𝐽,=0otherwise. (2.4)

For Theorem 1.1, it remains to prove the following lemma.

Lemma 2.3. 𝑅[𝑆]≅𝑅₀[𝑆].

Proof. We consider the mapping 𝜑∶𝑅[𝑆]→𝑅₀[𝑆] given on the basis by 𝜑(𝑠)=∑_𝑡≤𝑠𝑡(𝑠∈𝑆). Clearly, 𝜑 is well defined. Of course, 𝜑 and • may be regarded as the mappings of the ordered set (𝑆,≤) into the additive group of 𝑅₀[𝑆]. Now, by applying the Möbius inversion theorem to the mappings 𝜑 and •, we have 𝑠=_𝑡≤𝑠𝜑(𝑡)𝜇(𝑡,𝑠)=𝜑(_𝑡≤𝑠𝑡𝜇(𝑡,𝑠)), (2.5) where 𝜇 is the Möbius function for (𝑆,≤). Hence 𝜑 is surjective.
We will prove that 𝜑 is injective. For 𝛼₀=∑_𝑥∈𝑆𝑝⁰_𝑥𝑥∈𝑅[𝑆], we denote by supp(𝛼₀) the set {𝑥∈𝑆∣𝑝⁰_𝑥≠0} and by 𝑀(𝛼₀) the set of maximal elements in the set supp(𝛼₀) with respect to the partial order ≤. In recurrence, we define 𝛼_𝑛=𝛼_𝑛−1−∑_{𝑥∈𝑀(𝛼𝑛−1)}𝑝_𝑥^𝑛−1𝑥, where 𝛼_𝑛=∑_{𝑥∈supp(𝛼𝑛)}𝑝^𝑛_𝑥𝑥. Let 𝛽_𝑛=∑_{𝑥∈supp(𝛽𝑛)}𝑞^𝑛_𝑥𝑥 with 𝑛=0,1,2,…. If 𝜑(𝛼_𝑛)=𝜑(𝛽_𝑛), then by the definition of 𝜑, ∑_{𝑥∈𝑀(𝛼𝑛)}𝑝_𝑥𝑥+Γ_𝛼𝑛=𝜑(𝛼_𝑛)=𝜑(𝛽_𝑛)=∑_{𝑦∈𝑀(𝛽𝑛)}𝑞^𝑛_𝑦𝑦+Γ_𝛽𝑛, where Γ_𝛼𝑛=∑_{𝑥∈𝑀(𝛼𝑛)}∑_{𝑦∈𝑆,𝑦<𝑥}𝑝^𝑛_𝑦𝑦 and Γ_𝛽𝑛=∑_{𝑥∈𝑀(𝛽𝑛)}∑_{𝑦∈𝑆,𝑦<𝑥}𝑞^𝑛_𝑦𝑦, and hence ∑_{𝑥∈𝑀(𝛼𝑛)}𝑝^𝑛_𝑥𝑥=∑_{𝑥∈𝑀(𝛽𝑛)}𝑞^𝑛_𝑥𝑥, thus 𝑀(𝛼_𝑛)=𝑀(𝛽_𝑛) and 𝑝^𝑛_𝑥=𝑞^𝑛_𝑥 for any 𝑥∈𝑀(𝛼_𝑛). This can imply the following.
Fact 2.4. If 𝜑(𝛼_𝑛)=𝜑(𝛽_𝑛), then 𝑀(𝛼_𝑛)=𝑀(𝛽_𝑛) and by the definition of 𝜑, 𝜑(𝛼_𝑛+1)=𝜑(𝛽_𝑛+1).
By the definition of 𝜑, the following facts are immediate.
Fact 2.5. 𝛼_𝑛=𝛽_𝑛 if and only if 𝑀(𝛼_𝑛)=𝑀(𝛽_𝑛) and 𝛼_𝑛+1=𝛽_𝑛+1.
Fact 2.6. If 𝜑(𝛼_𝑛)=𝜑(𝛽_𝑛) and 𝑀(𝛼_𝑛)=supp(𝛼_𝑛),𝑀(𝛽_𝑛)=supp(𝛽_𝑛), then 𝛼_𝑛=𝛽_𝑛.
Note that |supp(𝛼₀)|<∞ and supp(𝛼_𝑛+1)⊆supp(𝛼_𝑛). We thus have a smallest integer 𝑘 such that 𝑀(𝛼_𝑘)=supp(𝛼_𝑘). Clearly, 𝛼_𝑘+1=0. This means that 𝑘 is the smallest integer 𝑡 such that 𝛼_𝑡+1=0. Similarly, there exists the smallest integer 𝑙 such that 𝛽_𝑙+1=0 and 𝑀(𝛽_𝑙)=supp(𝛽_𝑙). Now, assume 𝜑(𝛼₀)=𝜑(𝛽₀). By using Fact 2.4 repeatedly, 𝜑(𝛼₁)=𝜑(𝛽₁),𝜑(𝛼₂)=𝜑(𝛽₂),…,𝜑(𝛼_𝑘+1)=𝜑(𝛽_𝑘+1). (2.6) But 𝜑(𝛼_𝑘+1)=0, we have 𝜑(𝛽_𝑘+1)=0 and by the definition of 𝜑, 𝛽_𝑘+1=0. Thus 𝑘+1≥𝑙+1 by the minimality of 𝑙, and 𝑘≥𝑙. Similarly, 𝑙≥𝑘. Therefore 𝑘=𝑙. Since 𝜑(𝛼_𝑘)=𝜑(𝛽_𝑘), by Fact 2.6, we have 𝛼_𝑘=𝛽_𝑘 since 𝑀(𝛼_𝑘)=supp(𝛼_𝑘) and 𝑀(𝛽_𝑙)=supp(𝛽_𝑙). Again by the hypothesis 𝜑(𝛼₀)=𝜑(𝛽₀), and by Fact 2.4, 𝑀(𝛼₀)=𝑀(𝛽₀); and by (2.6), 𝑀(𝛼₁)=𝑀(𝛽₁),𝑀(𝛼₂)=𝑀(𝛽₂),…,𝑀(𝛼_𝑘)=𝑀(𝛽_𝑘). By Fact 2.5, 𝑀(𝛼_𝑘−1)=𝑀(𝛽_𝑘−1); and 𝛼_𝑘=𝛽_𝑘 imply 𝛼_𝑘−1=𝛽_𝑘−1; moreover, by using Fact 2.5 repeatedly, 𝛼_𝑘−2=𝛽_𝑘−2,…,𝛼₁=𝛽₁ and 𝛼₀=𝛽₀. We have now proved that 𝜑 is injective.
Finally, for any 𝑠,𝑡∈𝑆, by (2.4), we have 𝑠∗𝑡=𝑠𝑡if𝑠,𝑡∈𝐽_𝑠𝑡,0otherwise, (2.7) and by Lemma 2.2, 𝜑(𝑠)∗𝜑(𝑡)=(_𝑥≤𝑠𝑥)∗(_𝑦≤𝑡𝑦)=_{𝑥∈𝐽𝑠𝑡,𝑥≤𝑠𝑦∈𝐽𝑠𝑡,𝑦≤𝑡}𝑥∗𝑦=_{𝑥∈𝐽𝑠𝑡,𝑥≤𝑠𝑦∈𝐽𝑠𝑡,𝑦≤𝑡}𝑥𝑦. (2.8) Moreover, by Lemma 2.2, we have 𝜑(𝑠𝑡)=_{𝑢≤𝑠𝑡}𝑢=_{𝑥∈𝐽𝑠𝑡,𝑥≤𝑠𝑦∈𝐽𝑠𝑡,𝑦≤𝑡}𝑥𝑦=_{𝑥≤𝑠,𝑥∈𝐽𝑠𝑡𝑦≤𝑡,𝑦∈𝐽𝑠𝑡}𝑥∗𝑦=𝜑(𝑠)∗𝜑(𝑡). (2.9) Thus 𝜑 is a homomorphism of 𝑅[𝑆] into 𝑅₀[𝑆]. Consequently, 𝜑 is an isomorphism of 𝑅[𝑆] onto 𝑅₀[𝑆].

Acknowledgment

The research is supported by the NSF of Jiangxi Province, the SF of Education Department of Jiangxi Province, and the SF of Jiangxi Normal University.