Fair semigroups and Morita equivalence

Laan, Valdis; Márki, László

doi:10.1007/s00233-015-9723-3

Cited by 5 publications

(22 citation statements)

References 10 publications

(28 reference statements)

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“…The following two examples show that for a semigroup S, the categories FAct S and NAct S are in general incomparable. What we know is that FAct S = NAct S if S is a semigroup with common weak local units (this follows from Proposition 4 and Lemma 4 of [25]). However, even for semigroups with local units, these categories need not coincide.…”

Section: Firm Acts and Semigroupsmentioning

confidence: 94%

“…We denote the category of all unitary right S-acts by UAct S . A semigroup S is said to be fair if every subact of a unitary right S-act and every subact of a unitary left S-act is unitary, see [25]. Definition 2.1.…”

Section: Firm Acts and Semigroupsmentioning

confidence: 99%

“…Work in this direction has been inspired by papers on Morita equivalence on nonunital rings by the Spanish school, mainly by Marín and his coauthors, continuing the development started by García and Simón [17]. Laan and Márki [25] considered so-called fair semigroups -this class corresponds to the class of xst-rings considered by García and Marín [15], based on previous work by Xu, Shum, and Turner-Smith [39] (whence the name of these rings). Among many other results, it is shown in [25] that every finite monogenic semigroup is Morita equivalent to its group part, thus we have examples of non-factorisable semigroups which are Morita equivalent to groups, and strong Morita equivalence is impossible between them.…”

Section: Introductionmentioning

confidence: 99%

“…In the main line of the present paper we consider the same class of acts as was done in [27], [23] and [25]. There they were called 'closed acts' -however, here we call them 'firm acts'.…”

Section: Introductionmentioning

confidence: 99%

“…[25, Proposition 3.12]). Let S and T be fair semigroups such that U (S) and U (T ) have common weak local units.…”

mentioning

confidence: 99%

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Morita equivalence of semigroups revisited: Firm semigroups

2018

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We define firm semigroups and firm acts as non-additive analogues of firm rings and firm modules. Using the categories of firm acts we develop Morita theory for firm semigroups. We show that equivalence functors between categories of firm acts over two firm semigroups have to be tensor multiplication functors. Our main result states that the categories of firm right acts over two firm semigroups are equivalent if and only if these semigroups are strongly Morita equivalent, which means that they are contained in a unitary Morita context with bijective mappings. We also investigate other categories of acts which have been used earlier to develop Morita equivalence. The main tool in our work is adjoint functors. We prove that over firm semigroups all the considered categories are equivalent to the category of firm acts. All this suggests that firm semigroups and firm acts are the natural environment to study Morita equivalence of semigroups.

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Section: Firm Acts and Semigroupsmentioning

confidence: 94%

Section: Firm Acts and Semigroupsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

“…In the main line of the present paper we consider the same class of acts as was done in [27], [23] and [25]. There they were called 'closed acts' -however, here we call them 'firm acts'.…”

Section: Introductionmentioning

confidence: 99%

“…[25, Proposition 3.12]). Let S and T be fair semigroups such that U (S) and U (T ) have common weak local units.…”

mentioning

confidence: 99%

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Morita equivalence of semigroups revisited: Firm semigroups

2018

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Morita equivalence of finite semigroups

2021

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Globalization of partial actions of semigroups

Kudryavtseva

Laan

2023

Semigroup Forum

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We propose two universal constructions of globalization of a partial action of a semigroup on a set, satisfying certain conditions which arise in Morita theory of semigroups. One of the constructions is based on the tensor product of a partial semigroup act with the semigroup and generalizes the globalization construction of strong partial actions of monoids due to Megrelishvili and Schröder. It produces the initial object in an appropriate category of globalizations of a given partial action. The other construction involves $${\textrm{Hom}}$$ Hom -sets and is novel even in the monoid setting. It produces the terminal object in an appropriate category of globalizations. While in the group case the results of the two constructions are isomorphic, they can be far different in the monoid case.

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Fair semigroups and Morita equivalence

Abstract: Abstract. In analogy to the xst-rings studied by García and Marín, we define fair semigroups and investigate Morita equivalence for a subclass of them. In particular, we present examples for semigroups which are Morita equivalent but not strongly Morita equivalent.

Cited by 5 publications

References 10 publications

Morita equivalence of semigroups revisited: Firm semigroups

Morita equivalence of semigroups revisited: Firm semigroups

Morita equivalence of finite semigroups

Globalization of partial actions of semigroups

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