Research Article | Open Access
Valdis Laan, "Acceptable Morita Contexts for Semigroups", International Scholarly Research Notices, vol. 2012, Article ID 725627, 5 pages, 2012. https://doi.org/10.5402/2012/725627
Acceptable Morita Contexts for Semigroups
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.
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  or Theorem 1.1 of ). It is known (see Proposition 1 of ) that if two semigroups are contained in a unitary Morita context with surjective mappings (in such case they are called strongly Morita equivalent ), 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 . 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 , 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 be a semigroup. A left -act is a set equipped with a left -action , such that for all and . We write if is a left -act. Similarly right acts over semigroups are defined. If and are semigroups then an -biact is a set which is both left -act and right -act and for all , , . Clearly is an -biact with respect to actions defined by multiplication in . A left -act morphism is a mapping such that for all and . Similarly morphisms of right acts are defined. A biact morphism has to preserve both actions. Left -acts (right -acts, -biacts) and their morphisms form a category.
The tensor product of acts and is the quotient set by the equivalence relation generated by the set . The -class of is denoted by . If is an -biact and is a -biact then the tensor product can be turned into an -biact by setting and , , , , .
A left -act over a semigroup is unitary if . A biact over semigroups and is unitary if and .
Definition 1.1. A Morita context is a sextuple , where and are semigroups, and are biacts, and are biact morphisms such that, for every and , Such a Morita context is called unitary if and are unitary biacts.
2. The Result
Let be a semigroup. We consider the category of unitary closed right -acts that is acts for which the canonical right -act morphism , defined by , , is an isomorphism. Since for a unitary act the mapping is obvioulsy surjective, the closedness of such is equivalent to injectivity of .
Definition 2.1 (cf. Definition 3.6 of ). We say that a Morita context is right acceptable if (1)for every sequence there exists such that , (2)for every sequence there exists such that .
Lemma 2.2. Let and be factorisable semigroups. Then a Morita context is right acceptable if and only if and are surjective.
Proof. Necessity. Let be a right acceptable Morita context. Suppose that is not surjective. Then there is . We may factorise it as . We also may find for every elements such that . In particular, for every . By right acceptability, for the sequence there exists such that for some and . But then
a contradiction. Similarly we obtain a contradiction if is not surjective.
Sufficiency is clear.
Definition 2.3 (cf. , page 289). We call a Morita context right -pure if for every and the mappings
Next we show how to construct closed acts over semigroups.
Construction 1. Let be a semigroup and . We consider the free right -act
with the right -action
and its quotient act
where the right -act congruence on is defined by
, , and the equivalence class of a pair by is denoted by . Take , where , . Since , we have , and hence
Consequently, and is unitary.
To prove the injectivity of the mapping , suppose that where , . Then and hence for some . Consequently, in , so is an isomorphism and .
Given a Morita context , one can always consider natural transformations and which are defined by , . Explicitly: where , , , . We say that the natural transformations and are induced by the context. The following is an analogue of Theorem 3.10 of .
Theorem 2.4. Let be a Morita context. The following assertions are equivalent. (1)FActSFActT are inverse equivalence functors with the natural isomorphisms induced by the context.(2)The context is right acceptable and right FAct-pure.
Proof. . If are isomorphisms then clearly and have to be injective, so the context is right -pure.
Consider now a sequence and as in Construction 1. Using surjectivity of we can find , , , such that Hence for some , .
. Obviously is injective if and the context is right -pure. To prove that is surjective, take . By unitarity of , there exist , such that and for every . Then for the sequence there exists , , such that . Hence For the proof is similar.
Corollary 2.5. If for semigroups and 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 , which states that if factorisable semigroups and are strongly Morita equivalent then they are Morita equivalent.
This research was supported by the Estonian Science Foundation grant no. 8394 and Estonian Targeted Financing Project SF0180039s08.
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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.