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.