Sandwich semigroups in locally small categories I: Foundations

Fix (not necessarily distinct) objects \(i\) and \(j\) of a locally small category \(S\), and write \(S_{ij}\) for the set of all morphisms \(i\to j\). 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 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 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 \(S_{ij}^a\), Reg\((S_{ij}^a)\) and \(\mathbb E(S_{ij}^a)\); we give lower bounds for these ranks, and in the case of 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.