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ISRN Algebra
Volume 2012 (2012), Article ID 725627, 5 pages
http://dx.doi.org/10.5402/2012/725627
Research Article

Acceptable Morita Contexts for Semigroups

Institute of Mathematics, Faculty of Mathematics and Computer Science, University of Tartu, Ülikooli 18, 50090 Tartu, Estonia

Received 20 March 2012; Accepted 29 April 2012

Academic Editors: K. Fujii, M. Jinzenji, F. Panaite, and J. van de Leur

Copyright © 2012 Valdis Laan. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

This short note deals with Morita equivalence of (arbitrary) semigroups. We give a necessary and sufficient condition for a Morita context containing two semigroups S and T to induce an equivalence between the category of closed right S-acts and the category of closed right T-acts.

1. Preliminaries

Morita equivalence of two semigroups is usually defined by requiring that certain categories of acts over these semigroups are equivalent. If the semigroups are sufficiently “good” (in particular, if they are monoids) then the equivalence of those categories is equivalent to the existence of a unitary Morita context with surjective mappings (see, e.g., Theorems  7.3 and 8.3 of [1] or Theorem  1.1 of [2]). It is known (see Proposition  1 of [3]) that if two semigroups are contained in a unitary Morita context with surjective mappings (in such case they are called strongly Morita equivalent [4]), then these semigroups have to be factorisable, meaning that each of their elements can be written as a product of two elements. This suggests that perhaps the class of factorisable semigroups is the largest one to admit a meaningful Morita theory, and that theory is developed in [5]. Still in ring theory there are some articles that go beyond that limit and consider Morita equivalence of arbitrary associative rings. Our inspiration is taken from [6], where a connection between the equivalence of certain module categories over two associative rings and the existence of a certain Morita context is established. We shall prove an analogue of that result for semigroups.

Let 𝑆 be a semigroup. A left 𝑆-act is a set 𝐴 equipped with a left 𝑆-action 𝑆×𝐴𝐴,(𝑠,𝑎)𝑠𝑎, such that 𝑠(𝑠𝑎)=(𝑠𝑠)𝑎 for all 𝑠,𝑠𝑆 and 𝑎𝐴. We write 𝑆𝐴 if 𝐴 is a left 𝑆-act. Similarly right acts over semigroups are defined. If 𝑆 and 𝑇 are semigroups then an (𝑆,𝑇)-biact  𝑆𝐴𝑇 is a set 𝐴 which is both left 𝑆-act and right 𝑇-act and (𝑠𝑎)𝑡=𝑠(𝑎𝑡) for all 𝑠𝑆, 𝑡𝑇, 𝑎𝐴. Clearly 𝑆 is an (𝑆,𝑆)-biact with respect to actions defined by multiplication in 𝑆. A left 𝑆-act morphism   𝜓𝑆𝐴𝑆𝐵 is a mapping 𝜓𝐴𝐵 such that 𝜓(𝑠𝑎)=𝑠𝜓(𝑎) for all 𝑠𝑆 and 𝑎𝐴. Similarly morphisms of right acts are defined. A biact morphism has to preserve both actions. Left 𝑆-acts (right 𝑇-acts, (𝑆,𝑇)-biacts) and their morphisms form a category.

The tensor product  𝐴𝑇𝐵 of acts 𝐴𝑇 and 𝑇𝐵 is the quotient set (𝐴×𝐵)/𝜎 by the equivalence relation 𝜎 generated by the set {((𝑎𝑡,𝑏),(𝑎,𝑡𝑏))𝑎𝐴,𝑏𝐵,𝑡𝑇}. The 𝜎-class of (𝑎,𝑏)𝐴×𝐵 is denoted by 𝑎𝑏. If 𝑆𝐴𝑇 is an (𝑆,𝑇)-biact and 𝑇𝐵𝑆 is a (𝑇,𝑆)-biact then the tensor product 𝐴𝑇𝐵 can be turned into an (𝑆,𝑇)-biact by setting 𝑠(𝑎𝑏)=(𝑠𝑎)𝑏 and (𝑎𝑏)𝑡=𝑎(𝑏𝑡), 𝑎𝐴, 𝑏𝐵, 𝑠𝑆, 𝑡𝑇.

A left 𝑆-act 𝑆𝐴 over a semigroup 𝑆 is unitary if 𝑆𝐴=𝐴. A biact 𝑆𝐴𝑇 over semigroups 𝑆 and 𝑇 is unitary if 𝑆𝐴=𝐴 and 𝐴𝑇=𝐴.

Definition 1.1. A Morita context is a sextuple (𝑆,𝑇,𝑆𝑃𝑇,𝑇𝑄𝑆,𝜃,𝜙), where 𝑆 and 𝑇 are semigroups, 𝑆𝑃𝑇 and 𝑇𝑄𝑆 are biacts, and 𝜃𝑆𝑃𝑇𝑄𝑆𝑆𝑆𝑆,𝜙𝑇𝑄𝑆𝑃𝑇𝑇𝑇𝑇,(1.1) are biact morphisms such that, for every 𝑝,𝑝𝑃 and 𝑞,𝑞𝑄, 𝜃(𝑝𝑞)𝑝=𝑝𝜙𝑞𝑝,𝑞𝜃𝑝𝑞=𝜙(𝑞𝑝)𝑞.(1.2) Such a Morita context is called unitary if 𝑆𝑃𝑇 and 𝑇𝑄𝑆 are unitary biacts.

2. The Result

Let 𝑆 be a semigroup. We consider the category FAct𝑆 of unitary closed right 𝑆-acts 𝐴𝑆 that is acts for which the canonical right 𝑆-act morphism 𝜇𝐴𝐴𝑆𝑆𝐴, defined by 𝜇𝐴(𝑎𝑠)=𝑎𝑠,(2.1)𝑎𝐴, 𝑠𝑆, is an isomorphism. Since for a unitary act 𝐴𝑆 the mapping 𝜇𝐴 is obvioulsy surjective, the closedness of such 𝐴𝑆 is equivalent to injectivity of 𝜇𝐴.

Definition 2.1 (cf. Definition  3.6 of [6]). We say that a Morita context (𝑆,𝑇,𝑆𝑃𝑇,𝑇𝑄𝑆,𝜃,𝜙) is right acceptable if (1)for every sequence (𝑠𝑚)𝑚𝑆 there exists 𝑚0 such that 𝑠𝑚0𝑠1im𝜃, (2)for every sequence (𝑡𝑛)𝑛𝑇 there exists 𝑛0 such that 𝑡𝑛0𝑡1im𝜙.

Lemma 2.2. Let 𝑆 and 𝑇 be factorisable semigroups. Then a Morita context (𝑆,𝑇,𝑆𝑃𝑇,𝑇𝑄𝑆,𝜃,𝜙) is right acceptable if and only if 𝜃 and 𝜙 are surjective.

Proof. Necessity. Let (𝑆,𝑇,𝑆𝑃𝑇,𝑇𝑄𝑆,𝜃,𝜙) be a right acceptable Morita context. Suppose that 𝜃 is not surjective. Then there is 𝑠𝑆im𝜃. We may factorise it as 𝑠=𝑠1𝑠1. We also may find for every 𝑖 elements 𝑠𝑖+1,𝑠𝑖+1𝑆 such that 𝑠𝑖=𝑠𝑖+1𝑠𝑖+1. In particular, 𝑠=𝑠𝑘𝑠𝑘𝑠𝑘1𝑠1 for every 𝑘. By right acceptability, for the sequence (𝑠𝑚)𝑚 there exists 𝑚0 such that 𝑠𝑚0𝑠1=𝜃(𝑝𝑞) for some 𝑝𝑃 and 𝑞𝑄. But then 𝑠=𝑠𝑚0𝑠𝑚0𝑠1=𝑠𝑚0𝜃𝑠(𝑝𝑞)=𝜃𝑚0𝑝𝑞im𝜃,(2.2) a contradiction. Similarly we obtain a contradiction if 𝜙 is not surjective.
Sufficiency is clear.

Definition 2.3 (cf. [7], page 289). We call a Morita context (𝑆,𝑇,𝑆𝑃𝑇,𝑇𝑄𝑆,𝜃,𝜙) right 𝐅𝐀𝐜𝐭-pure if for every 𝐴𝑆FAct𝑆 and 𝐵𝑇FAct𝑇 the mappings 1𝐴1𝜃𝐴𝑃𝑄𝐴𝑆,𝑎𝑝𝑞𝑎𝜃(𝑝𝑞),𝐵𝜙𝐵𝑄𝑃𝐵𝑇,𝑏𝑞𝑝𝑏𝜙(𝑞𝑝),(2.3) are injective.
Next we show how to construct closed acts over semigroups.

Construction 1. Let 𝑆 be a semigroup and (𝑠𝑚)𝑚𝑆. We consider the free right 𝑆-act 𝐹𝑆=𝑆=𝑛({𝑛}×𝑆),(2.4) with the right 𝑆-action (𝑛,𝑠)𝑧=(𝑛,𝑠𝑧),(2.5) and its quotient act 𝑀𝑆[]=𝐹/={𝑘,𝑠𝑘,𝑠𝑆},(2.6) where the right 𝑆-act congruence on 𝐹 is defined by (𝑘,𝑠)(𝑙,𝑧)(𝑛)𝑛𝑘,𝑙𝑠𝑛𝑠𝑘+1𝑠=𝑠𝑛𝑠𝑙+1𝑧,(2.7)𝑘,𝑙, 𝑠,𝑧𝑆, and the equivalence class of a pair (𝑘,𝑠) by is denoted by [𝑘,𝑠]. Take [𝑘,𝑠]𝑀, where 𝑘, 𝑠𝑆. Since (𝑠𝑘+2𝑠𝑘+1)𝑠=𝑠𝑘+2(𝑠𝑘+1𝑠), we have (𝑘,𝑠)(𝑘+1,𝑠𝑘+1𝑠), and hence []=𝑘,𝑠𝑘+1,𝑠𝑘+1𝑠=𝑘+1,𝑠𝑘+1𝑠𝑀𝑆.(2.8) Consequently, 𝑀𝑆=𝑀 and 𝑀𝑆 is unitary.
To prove the injectivity of the mapping 𝜇𝑀𝑀𝑆𝑆𝑀, suppose that 𝜇𝑀([𝑘,𝑠]𝑢)=𝜇𝑀([𝑙,𝑧]𝑣) where 𝑘,𝑙, 𝑠,𝑧,𝑢,𝑣𝑆. Then [𝑘,𝑠𝑢]=[𝑙,𝑧𝑣] and hence 𝑠𝑛𝑠𝑘+1𝑠𝑢=𝑠𝑛𝑠𝑙+1𝑧𝑣 for some 𝑛𝑘,𝑙. Consequently, []𝑘,𝑠𝑢=𝑛+1,𝑠𝑛+1𝑠𝑛𝑠𝑘+1𝑠𝑢=𝑛+1,𝑠𝑛+1𝑠𝑛𝑠𝑘+1=𝑠𝑢𝑛+1,𝑠𝑛+1𝑠𝑛𝑠𝑘+1𝑠𝑢=𝑛+1,𝑠𝑛+1𝑠𝑛𝑠𝑙+1=𝑧𝑣𝑛+1,𝑠𝑛+1𝑠𝑛𝑠𝑙+1𝑧𝑣=𝑛+1,𝑠𝑛+1𝑠𝑛𝑠𝑙+1𝑧=[]𝑣𝑙,𝑧𝑣,(2.9) in 𝑀𝑆𝑆, so 𝜇𝑀 is an isomorphism and 𝑀𝑆FAct𝑆.
Given a Morita context (𝑆,𝑇,𝑆𝑃𝑇,𝑇𝑄𝑆,𝜃,𝜙), one can always consider natural transformations 𝛾𝑆𝑃𝑇𝑄1FAct𝑆 and 𝛿𝑇𝑄𝑆𝑃1FAct𝑇 which are defined by 𝛾𝐴=𝜇𝐴1𝐴𝜃𝐴𝑆𝑃𝑇𝑄𝑆𝐴𝑆𝛿𝐵=𝜇𝐵1𝐵𝜙𝐵𝑇𝑄𝑆𝑃𝑇𝐵𝑇,(2.10)𝐴𝑆FAct𝑆, 𝐵𝑇FAct𝑇. Explicitly: 𝛾𝐴𝛿(𝑎𝑝𝑞)=𝑎𝜃(𝑝𝑞),𝐵(𝑏𝑞𝑝)=𝑏𝜙(𝑞𝑝),(2.11) where 𝑎𝐴, 𝑏𝐵, 𝑝𝑃, 𝑞𝑄. We say that the natural transformations 𝛾 and 𝛿 are induced by the context. The following is an analogue of Theorem  3.10 of [6].

Theorem 2.4. Let (𝑆,𝑇,𝑆𝑃𝑇,𝑇𝑄𝑆,𝜃,𝜙) be a Morita context. The following assertions are equivalent. (1)FActS𝑠𝑃𝑇𝑄FActT are inverse equivalence functors with the natural isomorphisms induced by the context.(2)The context (𝑆,𝑇,𝑆𝑃𝑇,𝑇𝑄𝑆,𝜃,𝜙) is right acceptable and right FAct-pure.

Proof. (1)(2). If 𝛾𝐴,𝛿𝐵 are isomorphisms then clearly 1𝐴𝜃 and 1𝐵𝜙 have to be injective, so the context is right FAct-pure.
Consider now a sequence (𝑠𝑚)𝑚𝑆 and 𝑀𝑆 as in Construction 1. Using surjectivity of 𝛾𝑀𝑀𝑃𝑄𝑀 we can find 𝑝𝑃, 𝑞𝑄, 𝑘, 𝑠𝑆 such that 1,𝑠1=𝛾𝑀([][]𝜃[]𝑘,𝑠𝑝𝑞)=𝑘,𝑠(𝑝𝑞)=𝑘,𝑠𝜃(𝑝𝑞).(2.12) Hence 𝑠𝑛𝑠1=𝑠𝑛𝑠𝑘+1𝑠𝑠𝜃(𝑝𝑞)=𝜃𝑛𝑠𝑘+1𝑠𝑝𝑞im𝜃(2.13) for some 𝑛, 𝑛𝑘.
(2)(1). Obviously 𝛾𝐴 is injective if 𝐴𝑆FAct𝑆 and the context is right FAct-pure. To prove that 𝛾𝐴 is surjective, take 𝑎𝐴. By unitarity of 𝐴𝑆, there exist (𝑎𝑚)𝑚𝐴, (𝑠𝑚)𝑚𝑆 such that 𝑎=𝑎1𝑠1 and 𝑎𝑘=𝑎𝑘+1𝑠𝑘+1 for every 𝑘. Then for the sequence (𝑠𝑚)𝑚 there exists 𝑛, 𝑝𝑃, 𝑞𝑄 such that 𝑠𝑛𝑠1=𝜃(𝑝𝑞)im𝜃. Hence 𝑎=𝑎1𝑠1=𝑎2𝑠2𝑠1==𝑎𝑛𝑠𝑛𝑠𝑛1𝑠1=𝑎𝑛𝜃(𝑝𝑞)=𝛾𝐴𝑎𝑛𝑝𝑞.(2.14) For 𝛿𝐵 the proof is similar.

Corollary 2.5. If for semigroups 𝑆 and 𝑇 there exists a right acceptable and right FAct-pure Morita context then the categories FActS and FActT are equivalent.

This corollary may be considered as a generalisation of Theorem  3 of [4], which states that if factorisable semigroups 𝑆 and 𝑇 are strongly Morita equivalent then they are Morita equivalent.

Acknowledgments

This research was supported by the Estonian Science Foundation grant no. 8394 and Estonian Targeted Financing Project SF0180039s08.

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