Sandwich semigroups in locally small categories I: foundations

Abstract

Fix (not necessarily distinct) objects i and j of a locally small category S, and write $$S_{ij}$$ for the set of all morphisms $$i\rightarrow j$$ . Fix a morphism $$a\in S_{ji}$$ , and define an operation $$\star _a$$ on $$S_{ij}$$ by $$x\star _ay=xay$$ for all $$x,y\in S_{ij}$$ . Then $$(S_{ij},\star _a)$$ is a semigroup, known as a sandwich semigroup, and denoted by $$S_{ij}^a$$ . This article develops a general theory of sandwich semigroups in locally small categories. We begin with structural issues such as regularity, Green’s relations and stability, focusing on the relationships between these properties on $$S_{ij}^a$$ and the whole category S. We then identify a natural condition on a, called sandwich regularity, under which the set $${\text {Reg}}(S_{ij}^a)$$ of all regular elements of $$S_{ij}^a$$ is a subsemigroup of $$S_{ij}^a$$ . Under this condition, we carefully analyse the structure of the semigroup $${\text {Reg}}(S_{ij}^a)$$ , relating it via pullback products to certain regular subsemigroups of $$S_{ii}$$ and $$S_{jj}$$ , and to a certain regular sandwich monoid defined on a subset of $$S_{ji}$$ ; among other things, this allows us to also describe the idempotent-generated subsemigroup $$\mathbb E(S_{ij}^a)$$ of $$S_{ij}^a$$ . We also study combinatorial invariants such as the rank (minimal size of a generating set) of the semigroups $$S_{ij}^a$$ , $${\text {Reg}}(S_{ij}^a)$$ and $$\mathbb E(S_{ij}^a)$$ ; we give lower bounds for these ranks, and in the case of $${\text {Reg}}(S_{ij}^a)$$ and $$\mathbb E(S_{ij}^a)$$ show that the bounds are sharp under a certain condition we call MI-domination. Applications to concrete categories of transformations and partial transformations are given in Part II.