Sandwich semigroups in diagram categories
| Autor: | Ivana Đurđev, James East, Igor Dolinka |
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| Rok vydání: | 2019 |
| Předmět: |
General Mathematics
Diagram Mathematics::Rings and Algebras Mathematics - Category Theory Mathematics - Rings and Algebras Group Theory (math.GR) Combinatorics Rings and Algebras (math.RA) FOS: Mathematics Mathematics - Combinatorics Partition (number theory) Category Theory (math.CT) Combinatorics (math.CO) Mathematics - Group Theory Mathematics |
| DOI: | 10.48550/arxiv.1910.10286 |
| Popis: | This paper concerns a number of diagram categories, namely the partition, planar partition, Brauer, partial Brauer, Motzkin and Temperley-Lieb categories. If $\mathcal K$ denotes any of these categories, and if $\sigma\in\mathcal K_{nm}$ is a fixed morphism, then an associative operation $\star_\sigma$ may be defined on $\mathcal K_{mn}$ by $\alpha\star_\sigma\beta=\alpha\sigma\beta$. The resulting semigroup $\mathcal K_{mn}^\sigma=(\mathcal K_{mn},\star_\sigma)$ is called a sandwich semigroup. We conduct a thorough investigation of these sandwich semigroups, with an emphasis on structural and combinatorial properties such as Green's relations and preorders, regularity, stability, mid-identities, ideal structure, (products of) idempotents, and minimal generation. It turns out that the Brauer category has many remarkable properties not shared by any of the other diagram categories we study. Because of these unique properties, we may completely classify isomorphism classes of sandwich semigroups in the Brauer category, calculate the rank (smallest size of a generating set) of an arbitrary sandwich semigroup, enumerate Green's classes and idempotents, and calculate ranks (and idempotent ranks, where appropriate) of the regular subsemigroup and its ideals, as well as the idempotent-generated subsemigroup. Several illustrative examples are considered throughout, partly to demonstrate the sometimes-subtle differences between the various diagram categories. Comment: 50 pages, 13 figures. The second named author dedicates his contribution to this work to the memory of his father Vojin, who passed away exactly 10 years ago |
| Databáze: | OpenAIRE |
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