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

**Academic Editor:**K. Fujii

#### 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 βgoodβ (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., Theoremsββ7.3 andβ8.3 of [1] or Theoremββ1.1 of [2]). It is known (see Propositionββ1 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 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*ββ 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*ββ 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*ββ 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. Definitionββ3.6 of [6]). * 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. [7], page 289). *We call a Morita context ** right ****-pure** if for every and the mappings
are injective.

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 Theoremββ3.10 of [6].

Theorem 2.4. *Let be a Morita context. The following assertions are equivalent. *(1)*FAct _{S}FAct_{T} 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 FAct _{S} and FAct_{T} are equivalent. *

This corollary may be considered as a generalisation of Theoremββ3 of [4], which states that if factorisable semigroups and 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.

#### References

- S. Talwar, βMorita equivalence for semigroups,β
*Australian Mathematical Society A*, vol. 59, no. 1, pp. 81β111, 1995. View at: Publisher Site | Google Scholar | Zentralblatt MATH - M. V. Lawson, βMorita equivalence of semigroups with local units,β
*Journal of Pure and Applied Algebra*, vol. 215, no. 4, pp. 455β470, 2011. View at: Publisher Site | Google Scholar | Zentralblatt MATH - V. Laan and L. Márki, βStrong Morita equivalence of semigroups with local units,β
*Journal of Pure and Applied Algebra*, vol. 215, no. 10, pp. 2538β2546, 2011. View at: Publisher Site | Google Scholar - S. Talwar, βStrong Morita equivalence and a generalisation of the Rees theorem,β
*Journal of Algebra*, vol. 181, no. 2, pp. 371β394, 1996. View at: Publisher Site | Google Scholar | Zentralblatt MATH - Y. Q. Chen and K. P. Shum, βMorita equivalence for factorisable semigroups,β
*Acta Mathematica Sinica*, vol. 17, no. 3, pp. 437β454, 2001. View at: Publisher Site | Google Scholar | Zentralblatt MATH - L. Marín, βMorita equivalence based on contexts for various categories of modules over associative rings,β
*Journal of Pure and Applied Algebra*, vol. 133, no. 1-2, pp. 219β232, 1998. View at: Publisher Site | Google Scholar | Zentralblatt MATH - J. M. Howie,
*Fundamentals of Semigroup Theory*, Clarendon Press, Oxford, UK, 1995.

#### Copyright

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.