Product decompositions of semigroups induced by action pairs. (English) Zbl 1532.20063
Summary: This paper concerns a class of semigroups that arise as products \(US\), associated to what we call ‘action pairs’. Here \(U\) and \(S\) are subsemigroups of a common monoid and, roughly speaking, \(S\) has an action on the monoid completion \(U^1\) that is suitably compatible with the product in the over-monoid.
The semigroups encapsulated by the action pair construction include many natural classes such as inverse semigroups and (left) restriction semigroups, as well as many important concrete examples such as transformational wreath products, linear monoids, (partial) endomorphism monoids of independence algebras, and the singular ideals of many of these. Action pairs provide a unified framework for systematically studying such semigroups, within which we build a suite of tools to ensure an understanding of them. We then apply our abstract results to many special cases of interest.
The first part of the paper constitutes a detailed structural analysis of semigroups arising from action pairs. We show that any such semigroup \(US\) is a quotient of a semidirect product \(U\rtimes S\), and we classify all congruences on semidirect products that correspond to action pairs. We also prove several covering and embedding theorems, each of which naturally extends celebrated results of McAlister on proper (also called \(E\)-unitary) inverse semigroups.
The second part of the paper concerns presentations by generators and relations for semigroups arising from action pairs. We develop a substantial body of general results and techniques that allow us to build presentations for \(US\) out of presentations for the constituents \(U\) and \(S\) in many cases, and then apply these to several examples, including those listed above. Due to the broad applicability of the action pair construction, many results in the literature are special cases of our more general ones.
The semigroups encapsulated by the action pair construction include many natural classes such as inverse semigroups and (left) restriction semigroups, as well as many important concrete examples such as transformational wreath products, linear monoids, (partial) endomorphism monoids of independence algebras, and the singular ideals of many of these. Action pairs provide a unified framework for systematically studying such semigroups, within which we build a suite of tools to ensure an understanding of them. We then apply our abstract results to many special cases of interest.
The first part of the paper constitutes a detailed structural analysis of semigroups arising from action pairs. We show that any such semigroup \(US\) is a quotient of a semidirect product \(U\rtimes S\), and we classify all congruences on semidirect products that correspond to action pairs. We also prove several covering and embedding theorems, each of which naturally extends celebrated results of McAlister on proper (also called \(E\)-unitary) inverse semigroups.
The second part of the paper concerns presentations by generators and relations for semigroups arising from action pairs. We develop a substantial body of general results and techniques that allow us to build presentations for \(US\) out of presentations for the constituents \(U\) and \(S\) in many cases, and then apply these to several examples, including those listed above. Due to the broad applicability of the action pair construction, many results in the literature are special cases of our more general ones.
MSC:
| 20M10 | General structure theory for semigroups |
| 20M05 | Free semigroups, generators and relations, word problems |
| 20M20 | Semigroups of transformations, relations, partitions, etc. |
| 20M30 | Representation of semigroups; actions of semigroups on sets |
Keywords:
actions; semidirect products; covers; embeddings; presentations; inverse semigroups; left restriction semigroups; independence algebras; endomorphism monoids; singular endomorphism semigroups; transformation semigroups; wreath productsReferences:
| [1] | 3. Maximal subalgebras . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . |
| [2] | Finite-codimensional subalgebras . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . |
| [3] | 5. Generating (finitary) automorphisms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . |
| [4] | Action pairs and subsemigroups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . |
| [5] | 7. Congruence conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . |
| [6] | Presentations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . |
| [7] | Wreath products and free acts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . |
| [8] | 1. Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . |
| [9] | 2. Action pairs and subsemigroups I . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . |
| [10] | 3. Congruence conditions I . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . |
| [11] | 4. Presentations I . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . |
| [12] | 5. Action pairs and subsemigroups II . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . |
| [13] | 6. Congruence conditions II . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . |
| [14] | 7. Presentations II . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . |
| [15] | Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References |
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