Sandwich semigroups in locally small categories II: Transformations

Abstract

Fix sets $X$ and $Y$, and write $\mathcal{PT}_{XY}$ 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 $\mathcal{PT}_{XY}$ by $f\star_ag=fag$ for $f,g\in\mathcal{PT}_{XY}$. The sandwich semigroup $(\mathcal{PT}_{XY},\star_a)$ is denoted $\mathcal{PT}_{XY}^a$. We apply general results from Part I to thoroughly describe the structural and combinatorial properties of $\mathcal{PT}_{XY}^a$, as well as its regular and idempotent-generated subsemigroups, Reg$(\mathcal{PT}_{XY}^a)$ and $\mathbb E(\mathcal{PT}_{XY}^a)$. After describing regularity, stability and Green's relations and preorders, we exhibit Reg$(\mathcal{PT}_{XY}^a)$ as a pullback product of certain regular subsemigroups of the (non-sandwich) partial transformation semigroups $\mathcal{PT}_X$ and $\mathcal{PT}_Y$, and as a kind of "inflation" of $\mathcal{PT}_A$, 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 $\mathcal{PT}_{XY}^a$, Reg$(\mathcal{PT}_{XY}^a)$ and $\mathbb E(\mathcal{PT}_{XY}^a)$. 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.