Acceptable Morita Contexts for Semigroups

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×𝐴→𝐴,(𝑠,𝑎)↦𝑠𝑎, such that 𝑠(𝑠′𝑎)=(𝑠𝑠′)𝑎 for all 𝑠,𝑠′∈𝑆 and 𝑎∈𝐴. 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  _𝑆𝐴_𝑇 is a set 𝐴 which is both left 𝑆-act and right 𝑇-act and (𝑠𝑎)𝑡=𝑠(𝑎𝑡) for all 𝑠∈𝑆, 𝑡∈𝑇, 𝑎∈𝐴. Clearly 𝑆 is an (𝑆,𝑆)-biact with respect to actions defined by multiplication in 𝑆. A left 𝑆-act morphism   𝜓∶_𝑆𝐴→_𝑆𝐵 is a mapping 𝜓∶𝐴→𝐵 such that 𝜓(𝑠𝑎)=𝑠𝜓(𝑎) for all 𝑠∈𝑆 and 𝑎∈𝐴. 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  𝐴⊗_𝑇𝐵 of acts 𝐴_𝑇 and _𝑇𝐵 is the quotient set (𝐴×𝐵)/𝜎 by the equivalence relation 𝜎 generated by the set {((𝑎𝑡,𝑏),(𝑎,𝑡𝑏))∣𝑎∈𝐴,𝑏∈𝐵,𝑡∈𝑇}. The 𝜎-class of (𝑎,𝑏)∈𝐴×𝐵 is denoted by 𝑎⊗𝑏. If _𝑆𝐴_𝑇 is an (𝑆,𝑇)-biact and _𝑇𝐵_𝑆 is a (𝑇,𝑆)-biact then the tensor product 𝐴⊗_𝑇𝐵 can be turned into an (𝑆,𝑇)-biact by setting 𝑠(𝑎⊗𝑏)∶=(𝑠𝑎)⊗𝑏 and (𝑎⊗𝑏)𝑡∶=𝑎⊗(𝑏𝑡), 𝑎∈𝐴, 𝑏∈𝐵, 𝑠∈𝑆, 𝑡∈𝑇.

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

Definition 1.1. A Morita context is a sextuple (𝑆,𝑇,_𝑆𝑃_𝑇,_𝑇𝑄_𝑆,𝜃,𝜙), where 𝑆 and 𝑇 are semigroups, _𝑆𝑃_𝑇 and _𝑇𝑄_𝑆 are biacts, and 𝜃∶_𝑆𝑃⊗_𝑇𝑄_𝑆⟶_𝑆𝑆_𝑆,𝜙∶_𝑇𝑄⊗_𝑆𝑃_𝑇⟶_𝑇𝑇_𝑇,(1.1) are biact morphisms such that, for every 𝑝,𝑝′∈𝑃 and 𝑞,𝑞′∈𝑄, 𝜃(𝑝⊗𝑞)𝑝′=𝑝𝜙𝑞⊗𝑝^,𝑞𝜃𝑝⊗𝑞^=𝜙(𝑞⊗𝑝)𝑞′.(1.2) Such a Morita context is called unitary if _𝑆𝑃_𝑇 and _𝑇𝑄_𝑆 are unitary biacts.

2. The Result

Let 𝑆 be a semigroup. We consider the category FAct_𝑆 of unitary closed right 𝑆-acts 𝐴_𝑆 that is acts for which the canonical right 𝑆-act morphism 𝜇_𝐴∶𝐴⊗_𝑆𝑆→𝐴, defined by 𝜇_𝐴(𝑎⊗𝑠)∶=𝑎𝑠,(2.1)𝑎∈𝐴, 𝑠∈𝑆, is an isomorphism. Since for a unitary act 𝐴_𝑆 the mapping 𝜇_𝐴 is obvioulsy surjective, the closedness of such 𝐴_𝑆 is equivalent to injectivity of 𝜇_𝐴.

Definition 2.1 (cf. Definition  3.6 of [6]). We say that a Morita context (𝑆,𝑇,_𝑆𝑃_𝑇,_𝑇𝑄_𝑆,𝜃,𝜙) is right acceptable if (1)for every sequence (𝑠_𝑚)_𝑚∈ℕ∈𝑆^ℕ there exists 𝑚₀∈ℕ such that 𝑠_𝑚0⋯𝑠₁∈im𝜃, (2)for every sequence (𝑡_𝑛)_𝑛∈ℕ∈𝑇^ℕ there exists 𝑛₀∈ℕ such that 𝑡_𝑛0⋯𝑡₁∈im𝜙.

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

Proof. Necessity. Let (𝑆,𝑇,_𝑆𝑃_𝑇,_𝑇𝑄_𝑆,𝜃,𝜙) be a right acceptable Morita context. Suppose that 𝜃 is not surjective. Then there is 𝑠∈𝑆⧵im𝜃. We may factorise it as 𝑠=𝑠^1𝑠₁. We also may find for every 𝑖∈ℕ elements 𝑠^𝑖+1,𝑠_𝑖+1∈𝑆 such that 𝑠^𝑖=𝑠^𝑖+1𝑠_𝑖+1. In particular, 𝑠=𝑠^𝑘𝑠_𝑘𝑠_𝑘−1⋯𝑠₁ for every 𝑘∈ℕ. By right acceptability, for the sequence (𝑠_𝑚)_𝑚∈ℕ there exists 𝑚₀∈ℕ such that 𝑠_𝑚0⋯𝑠₁=𝜃(𝑝⊗𝑞) for some 𝑝∈𝑃 and 𝑞∈𝑄. But then 𝑠=𝑠^𝑚₀𝑠_𝑚0⋯𝑠₁=𝑠^𝑚₀𝜃(𝑝⊗𝑞)=𝜃𝑠^𝑚₀𝑝⊗𝑞∈im𝜃,(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,_𝑆𝑃_𝑇,_𝑇𝑄_𝑆,𝜃,𝜙) right 𝐅𝐀𝐜𝐭-pure if for every 𝐴_𝑆∈FAct_𝑆 and 𝐵_𝑇∈FAct_𝑇 the mappings 11_𝐴⊗𝜃∶𝐴⊗𝑃⊗𝑄⟶𝐴⊗𝑆,𝑎⊗𝑝⊗𝑞⟼𝑎⊗𝜃(𝑝⊗𝑞),_𝐵⊗𝜙∶𝐵⊗𝑄⊗𝑃⟶𝐵⊗𝑇,𝑏⊗𝑞⊗𝑝⟼𝑏⊗𝜙(𝑞⊗𝑝),(2.3) are injective.
Next we show how to construct closed acts over semigroups.

Construction 1. Let 𝑆 be a semigroup and (𝑠_𝑚)_𝑚∈ℕ∈𝑆^ℕ. We consider the free right 𝑆-act 𝐹_𝑆∶=_ℕ𝑆=_𝑛∈ℕ({𝑛}×𝑆),(2.4) with the right 𝑆-action (𝑛,𝑠)𝑧∶=(𝑛,𝑠𝑧),(2.5) and its quotient act 𝑀_𝑆∶=𝐹/∼={[𝑘,𝑠]∣𝑘∈ℕ,𝑠∈𝑆},(2.6) where the right 𝑆-act congruence ∼ on 𝐹 is defined by (𝑘,𝑠)∼(𝑙,𝑧)⟺(∃𝑛∈ℕ)𝑛≥𝑘,𝑙∧𝑠_𝑛⋯𝑠_𝑘+1𝑠=𝑠_𝑛⋯𝑠_𝑙+1𝑧,(2.7)𝑘,𝑙∈ℕ, 𝑠,𝑧∈𝑆, and the equivalence class of a pair (𝑘,𝑠) by ∼ is denoted by [𝑘,𝑠]. Take [𝑘,𝑠]∈𝑀, where 𝑘∈ℕ, 𝑠∈𝑆. Since (𝑠_𝑘+2𝑠_𝑘+1)𝑠=𝑠_𝑘+2(𝑠_𝑘+1𝑠), we have (𝑘,𝑠)∼(𝑘+1,𝑠_𝑘+1𝑠), and hence [𝑘,𝑠]=𝑘+1,𝑠_𝑘+1𝑠=𝑘+1,𝑠_𝑘+1𝑠∈𝑀𝑆.(2.8) Consequently, 𝑀𝑆=𝑀 and 𝑀_𝑆 is unitary.
To prove the injectivity of the mapping 𝜇_𝑀∶𝑀⊗_𝑆𝑆→𝑀, suppose that 𝜇_𝑀([𝑘,𝑠]⊗𝑢)=𝜇_𝑀([𝑙,𝑧]⊗𝑣) where 𝑘,𝑙∈ℕ, 𝑠,𝑧,𝑢,𝑣∈𝑆. Then [𝑘,𝑠𝑢]=[𝑙,𝑧𝑣] and hence 𝑠_𝑛⋯𝑠_𝑘+1𝑠𝑢=𝑠_𝑛⋯𝑠_𝑙+1𝑧𝑣 for some 𝑛≥𝑘,𝑙. Consequently, [𝑘,𝑠]⊗𝑢==𝑛𝑛++11,,𝑠𝑠_𝑛+1𝑛+1𝑠_𝑛⋯𝑠_𝑘+1𝑠⊗𝑢=𝑛+1,𝑠_𝑛+1𝑠_𝑛⋯𝑠_𝑘+1𝑠⊗𝑢=𝑛+1,𝑠_𝑛+1⊗𝑠_𝑛⋯𝑠_𝑘+1𝑠𝑢=𝑛+1,𝑠_𝑛+1⊗𝑠_𝑛⋯𝑠_𝑙+1𝑧𝑣=[𝑙,𝑧]⊗𝑣,𝑠_𝑛⋯𝑠_𝑙+1𝑧⊗𝑣=𝑛+1,𝑠_𝑛+1𝑠_𝑛⋯𝑠_𝑙+1𝑧⊗𝑣(2.9) in 𝑀⊗_𝑆𝑆, so 𝜇_𝑀 is an isomorphism and 𝑀_𝑆∈FAct_𝑆.
Given a Morita context (𝑆,𝑇,_𝑆𝑃_𝑇,_𝑇𝑄_𝑆,𝜃,𝜙), one can always consider natural transformations 𝛾∶−⊗_𝑆𝑃⊗_𝑇𝑄⇒1_FAct𝑆 and 𝛿∶−⊗_𝑇𝑄⊗_𝑆𝑃⇒1_FAct𝑇 which are defined by 𝛾_𝐴∶=𝜇_𝐴∘1_𝐴⊗𝜃∶𝐴⊗_𝑆𝑃⊗_𝑇𝑄_𝑆⟶𝐴_𝑆𝛿_𝐵∶=𝜇_𝐵∘1_𝐵⊗𝜙∶𝐵⊗_𝑇𝑄⊗_𝑆𝑃_𝑇⟶𝐵_𝑇,(2.10)𝐴_𝑆∈FAct_𝑆, 𝐵_𝑇∈FAct_𝑇. Explicitly: 𝛿𝛾_𝐴(𝑎⊗𝑝⊗𝑞)=𝑎𝜃(𝑝⊗𝑞),_𝐵(𝑏⊗𝑞⊗𝑝)=𝑏𝜙(𝑞⊗𝑝),(2.11) where 𝑎∈𝐴, 𝑏∈𝐵, 𝑝∈𝑃, 𝑞∈𝑄. 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,_𝑆𝑃_𝑇,_𝑇𝑄_𝑆,𝜃,𝜙) be a Morita context. The following assertions are equivalent. (1)FAct_S^𝑠𝑃→^−⊗←_{𝑇𝑄−⊗}FAct_T are inverse equivalence functors with the natural isomorphisms induced by the context.(2)The context (𝑆,𝑇,_𝑆𝑃_𝑇,_𝑇𝑄_𝑆,𝜃,𝜙) is right acceptable and right FAct-pure.

Proof. (1)⇒(2). If 𝛾_𝐴,𝛿_𝐵 are isomorphisms then clearly 1_𝐴⊗𝜃 and 1_𝐵⊗𝜙 have to be injective, so the context is right FAct-pure.
Consider now a sequence (𝑠_𝑚)_𝑚∈ℕ∈𝑆^ℕ and 𝑀_𝑆 as in Construction 1. Using surjectivity of 𝛾_𝑀∶𝑀⊗𝑃⊗𝑄→𝑀 we can find 𝑝∈𝑃, 𝑞∈𝑄, 𝑘∈ℕ, 𝑠∈𝑆 such that 1,𝑠₁=𝛾_𝑀([𝑘,𝑠]⊗𝑝⊗𝑞)=[𝑘,𝑠]𝜃(𝑝⊗𝑞)=[𝑘,𝑠𝜃(𝑝⊗𝑞)].(2.12) Hence 𝑠_𝑛⋯𝑠₁=𝑠_𝑛⋯𝑠_𝑘+1𝑠𝜃(𝑝⊗𝑞)=𝜃𝑠_𝑛⋯𝑠_𝑘+1𝑠𝑝⊗𝑞∈im𝜃(2.13) for some 𝑛∈ℕ, 𝑛≥𝑘.
(2)⇒(1). Obviously 𝛾_𝐴 is injective if 𝐴_𝑆∈FAct_𝑆 and the context is right FAct-pure. To prove that 𝛾_𝐴 is surjective, take 𝑎∈𝐴. By unitarity of 𝐴_𝑆, there exist (𝑎_𝑚)_𝑚∈ℕ∈𝐴^ℕ, (𝑠_𝑚)_𝑚∈ℕ∈𝑆^ℕ such that 𝑎=𝑎₁𝑠₁ and 𝑎_𝑘=𝑎_𝑘+1𝑠_𝑘+1 for every 𝑘∈ℕ. Then for the sequence (𝑠_𝑚)_𝑚∈ℕ there exists 𝑛∈ℕ, 𝑝∈𝑃, 𝑞∈𝑄 such that 𝑠_𝑛⋯𝑠₁=𝜃(𝑝⊗𝑞)∈im𝜃. Hence 𝑎=𝑎₁𝑠₁=𝑎₂𝑠₂𝑠₁=⋯=𝑎_𝑛𝑠_𝑛𝑠_𝑛−1⋯𝑠₁=𝑎_𝑛𝜃(𝑝⊗𝑞)=𝛾_𝐴𝑎_𝑛⊗𝑝⊗𝑞.(2.14) For 𝛿_𝐵 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 FAct_S and FAct_T 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.