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 $X\rightarrow Y$. Fix a partial function ${a:Y\rightarrow X}$, and define the operation $\star _a$ on $\mathcal P\mathcal T_{XY}$ by $f\star _ag=fag$ 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, ${\text {Reg}}(\mathcal P\mathcal T_{XY}^a)$ and $\mathbb E(\mathcal P\mathcal T_{XY}^a)$. After describing regularity, stability and Green’s relations and preorders, we exhibit ${\text {Reg}}(\mathcal P\mathcal T_{XY}^a)$ 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 $\mathcal P\mathcal T_{XY}^a$, ${\text {Reg}}(\mathcal P\mathcal T_{XY}^a)$ 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.