Authors: Dolinka, Igor
Đurđev, Ivana 
East, James
Honyam, Preeyanuch
Sangkhanan, Kritsada
Sanwong, Jintana
Sommanee, Worachead
Title: Sandwich semigroups in locally small categories II: transformations
Journal: Algebra Universalis
Volume: 79
Issue: 3
Issue Date: 1-Sep-2018
Rank: M22
ISSN: 0002-5240
DOI: 10.1007/s00012-018-0539-3
Fix sets X and Y, and write PTXY for the set of all partial functions X→ Y. Fix a partial function a: Y→ X, and define the operation ⋆ a on PTXY by f⋆ ag= fag for f, g∈ PTXY. The sandwich semigroup(PTXY, ⋆ a) is denoted PTXYa. We apply general results from Part I to thoroughly describe the structural and combinatorial properties of PTXYa, as well as its regular and idempotent-generated subsemigroups, Reg(PTXYa) and E(PTXYa). After describing regularity, stability and Green’s relations and preorders, we exhibit Reg(PTXYa) as a pullback product of certain regular subsemigroups of the (non-sandwich) partial transformation semigroups PTX and PTY, and as a kind of “inflation” of PTA, where A is the image of the sandwich element a. We also calculate the rank (minimal size of a generating set) and, where appropriate, the idempotent rank (minimal size of an idempotent generating set) of PTXYa, Reg(PTXYa) and E(PTXYa). The same program is also carried out for sandwich semigroups of totally defined functions and for injective partial functions. Several corollaries are obtained for various (non-sandwich) semigroups of (partial) transformations with restricted image, domain and/or kernel.
Keywords: Categories | Idempotent rank | Mid-identities | Partial semigroups | Rank | Sandwich semigroups | Transformation semigroups
Publisher: Springer Link
Project: Numerical Linear Algebra and Discrete Structures 
Algebraic, logical and combinatorial methods with applications in theoretical computer science 

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