Sandwich semigroups in locally small categories II: Transformations

Abstract

Fix sets $X$ and $Y$, and write ${\mathcal{P}\mathcal{T}}_{XY}$ for the set of all partial functions $X\to Y$. Fix a partial function $a:Y\to X$, and define the operation ${\star }_a$ on ${\mathcal{P}\mathcal{T}}_{XY}$ by $f{\star }_{a}g=fag$ for $f,g\in {\mathcal{P}\mathcal{T}}_{XY}$. The sandwich semigroup $({\mathcal{P}\mathcal{T}}_{XY},{\star }_a)$ is denoted ${\mathcal{P}\mathcal{T}}_{XY}^a$. We apply general results from Part I to thoroughly describe the structural and combinatorial properties of ${\mathcal{P}\mathcal{T}}_{XY}^a$, as well as its regular and idempotent-generated subsemigroups, Reg$({\mathcal{P}\mathcal{T}}_{XY}^a)$ and $\mathbb{E}({\mathcal{P}\mathcal{T}}_{XY}^a)$. After describing regularity, stability and Green's relations and preorders, we exhibit Reg$({\mathcal{P}\mathcal{T}}_{XY}^a)$ as a pullback product of certain regular subsemigroups of the (non-sandwich) partial transformation semigroups ${\mathcal{P}\mathcal{T}}_X$ and ${\mathcal{P}\mathcal{T}}_Y$, and as a kind of "inflation" of ${\mathcal{P}\mathcal{T}}_A$, 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 ${\mathcal{P}\mathcal{T}}_{XY}^a$, Reg$({\mathcal{P}\mathcal{T}}_{XY}^a)$ and $\mathbb{E}({\mathcal{P}\mathcal{T}}_{XY}^a)$. 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.

35 pages, 11 figures, 1 table. V2: updated according to referee report, expanded abstract, to appear in Algebra Universalis