Covers for monoids. (English) Zbl 1081.20069
A monoid \(M\) is said to be an extension of a submonoid \(T\) by a group \(G\) if there is a homomorphism \(\varphi\colon M\to G\) such that \(T=\varphi^{-1}(1)\). Given a monoid \(M\) and a submonoid \(T\), if there is a monoid \(\widehat M\) with a homomorphism \(\theta\colon\widehat M\to M\) such that \(\widehat M\) is an extension of a submonoid \(\widehat T\) by a group and \(\theta\) maps \(\widehat T\) isomorphically onto \(T\), then such a monoid \(\widehat M\) is said to be a \(T\)-cover of \(M\).
The main results in the paper are the following: (1) several characterizations of when a monoid is an extension of a given submonoid by a group (Theorem 4.5); (2) a sufficient condition, in terms of a monoid \(M\) and its submonoid \(T\), for \(M\) to admit a \(T\)-cover (Theorem 5.1).
These rather general and remarkable results, which extend considerably many results in the area and for whose precise statements the reader is referred to the paper, are then specialized to particular classes of monoids of interest, namely: (1) monoids which have a minimum group congruence; (2) \(E\)-dense monoids; (3) regular monoids; and (4) finite monoids. – Group actions on categories play an important role in the characterizations and proofs of the main results.
The main results in the paper are the following: (1) several characterizations of when a monoid is an extension of a given submonoid by a group (Theorem 4.5); (2) a sufficient condition, in terms of a monoid \(M\) and its submonoid \(T\), for \(M\) to admit a \(T\)-cover (Theorem 5.1).
These rather general and remarkable results, which extend considerably many results in the area and for whose precise statements the reader is referred to the paper, are then specialized to particular classes of monoids of interest, namely: (1) monoids which have a minimum group congruence; (2) \(E\)-dense monoids; (3) regular monoids; and (4) finite monoids. – Group actions on categories play an important role in the characterizations and proofs of the main results.
Reviewer: Jorge Almeida (Porto)
MSC:
| 20M10 | General structure theory for semigroups |
| 20M50 | Connections of semigroups with homological algebra and category theory |
Keywords:
covers; group actions on categories; groupoids; congruences; \(E\)-dense monoids; regular monoids; finite monoids; extensionsReferences:
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