Authors: Dolinka, Igor
Đurđev, Ivana 
East, James
Affiliations: Mathematics 
Title: Sandwich semigroups in diagram categories
Journal: International Journal of Algebra and Computation
Issue Date: 13-Aug-2021
Rank: ~M23
ISSN: 0218-1967
DOI: 10.1142/S021819672150048X
This paper concerns a number of diagram categories, namely the partition, planar partition, Brauer, partial Brauer, Motzkin and Temperley-Lieb categories. If K denotes any of these categories, and if σ ϵ Knm is a fixed morphism, then an associative operation ∗σ may be defined on Kmn by α ∗σ β = α;σβ. The resulting semigroup Kσmn= (Kmn, ∗σ) 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.
Keywords: Brauer categories | Diagram categories | Motzkin categories | Partition categories | Sandwich semigroups | Temperley-Lieb categories
Publisher: World Scientific
Project: Algebraic, logical and combinatorial methods with applications in theoretical computer science 
Numerical Linear Algebra and Discrete Structures 

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