Sandwich semigroups in locally small categories II: Transformations

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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. 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.