Congruences on direct products of transformation and matrix monoids
| Autor: | Wolfram Bentz, João Araújo, Gracinda M. S. Gomes |
|---|---|
| Rok vydání: | 2018 |
| Předmět: |
Monoid
Algebra and Number Theory 010102 general mathematics Multiplicative function Inverse Field (mathematics) 0102 computer and information sciences Congruence relation 01 natural sciences Injective function Combinatorics Matrix (mathematics) 010201 computation theory & mathematics Mathematics::Category Theory 0101 mathematics Finite set Mathematics |
| Zdroj: | Semigroup Forum. 97:384-416 |
| ISSN: | 1432-2137 0037-1912 |
| DOI: | 10.1007/s00233-018-9931-8 |
| Popis: | Mal $$'$$ cev described the congruences of the monoid $$\mathcal {T}_n$$ of all full transformations on a finite set $$X_n=\{1, \dots ,n\}$$ . Since then, congruences have been characterized in various other monoids of (partial) transformations on $$X_n$$ , such as the symmetric inverse monoid $$\mathcal {I}_n$$ of all injective partial transformations, or the monoid $$\mathcal {PT}_n$$ of all partial transformations. The first aim of this paper is to describe the congruences of the direct products $$Q_m\times P_n$$ , where Q and P belong to $$\{\mathcal {T}, \mathcal {PT},\mathcal {I}\}$$ . Mal $$'$$ cev also provided a similar description of the congruences on the multiplicative monoid $$F_n$$ of all $$n\times n$$ matrices with entries in a field F; our second aim is to provide a description of the principal congruences of $$F_m \times F_n$$ . The paper finishes with some comments on the congruences of products of more than two transformation semigroups, and on a number of related open problems. |
| Databáze: | OpenAIRE |
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