Congruences on infinite partition and partial Brauer monoids. (English) Zbl 1530.20191
For a semigroup \(S\), let \(\mbox{Cong\,}(S)\) denote the set of all congruences on \(S\). For an infinite set \(X\), let \(\mathcal{P}_{X}\) and \(\mathcal{PB}_{X}\) denote the partition monoid and the partial Brauer monoid on \(X\), respectively. In the first part, the complete descriptions of Green’s relations on both \(\mathcal{P}_{X}\) and \(\mathcal{PB}_{X}\) are given in Lemma 2.2. It is shown in Proposition 2.4 that \(\mathcal{P}_{X}\) and \(\mathcal{PB}_{X}\) are not isomorphic and then, the authors classify all of the congruences on \(\mathcal{P}_{X}\) and \(\mathcal{PB}_{X}\) in Theorem 3.1 by defining five basic congruences on them. The rest of the first part is all about the proof of Theorem 3.1.
The second part gives a detailed analysis of the algebraic and combinatorial structure of the congruence lattices \(\mbox{Cong\,}(\mathcal{P}_{X})\) and \(\mbox{Cong\,}(\mathcal{PB}_{X})\). The order relation in these lattices is characterised in Theorem 8.1. From Theorems 3.1 and 8.1, it is immediate that the lattices \(\mbox{Cong\,}(\mathcal{P}_{X})\) and \(\mbox{Cong\,}(\mathcal{PB}_{X})\) are isomorphic. Moreover, formulas for meets and joins are given in Theorem 8.3. It is shown in Theorems 10.1 and 10.3 that the congruence lattices of \(\mathcal{P}_{X}\) and \(\mathcal{PB}_{X}\) are distributive and well quasi-ordered. For each pair \((\alpha ,\beta)\) in \(\mathcal{P}_{X}\) and \(\mathcal{PB}_{X}\), the least congruence containing \((\alpha ,\beta)\) is classified in Theorem 11.1. The ranks of all the other congruences on \(\mathcal{P}_{X}\) and \(\mathcal{PB}_{X}\) are calculated in Theorems 11.3 and 11.5. Finally, the results obtained in this paper are compared with existing results on finite diagram monoids.
The second part gives a detailed analysis of the algebraic and combinatorial structure of the congruence lattices \(\mbox{Cong\,}(\mathcal{P}_{X})\) and \(\mbox{Cong\,}(\mathcal{PB}_{X})\). The order relation in these lattices is characterised in Theorem 8.1. From Theorems 3.1 and 8.1, it is immediate that the lattices \(\mbox{Cong\,}(\mathcal{P}_{X})\) and \(\mbox{Cong\,}(\mathcal{PB}_{X})\) are isomorphic. Moreover, formulas for meets and joins are given in Theorem 8.3. It is shown in Theorems 10.1 and 10.3 that the congruence lattices of \(\mathcal{P}_{X}\) and \(\mathcal{PB}_{X}\) are distributive and well quasi-ordered. For each pair \((\alpha ,\beta)\) in \(\mathcal{P}_{X}\) and \(\mathcal{PB}_{X}\), the least congruence containing \((\alpha ,\beta)\) is classified in Theorem 11.1. The ranks of all the other congruences on \(\mathcal{P}_{X}\) and \(\mathcal{PB}_{X}\) are calculated in Theorems 11.3 and 11.5. Finally, the results obtained in this paper are compared with existing results on finite diagram monoids.
Reviewer: Hayrullah Ayık (Adana)
MSC:
| 20M20 | Semigroups of transformations, relations, partitions, etc. |
| 20M10 | General structure theory for semigroups |
| 20M50 | Connections of semigroups with homological algebra and category theory |
| 18B40 | Groupoids, semigroupoids, semigroups, groups (viewed as categories) |
Keywords:
diagram monoids; partition monoids; partial Brauer monoids; congruences; well quasi-orderednessSoftware:
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