Sandwich semigroups in locally small categories I: foundations

Abstract

© 2018, Springer Nature Switzerland AG. Fix (not necessarily distinct) objects i and j of a locally small category S, and write Sij for the set of all morphisms i→ j. Fix a morphism a∈ Sji, and define an operation ⋆ a on Sij by x⋆ ay= xay for all x, y∈ Sij. Then (Sij, ⋆ a) is a semigroup, known as a sandwich semigroup, and denoted by Sija. 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 Sija and the whole category S. We then identify a natural condition on a, called sandwich regularity, under which the set Reg(Sija) of all regular elements of Sija is a subsemigroup of Sija. Under this condition, we carefully analyse the structure of the semigroup Reg(Sija), relating it via pullback products to certain regular subsemigroups of Sii and Sjj, and to a certain regular sandwich monoid defined on a subset of Sji; among other things, this allows us to also describe the idempotent-generated subsemigroup E(Sija) of Sija. We also study combinatorial invariants such as the rank (minimal size of a generating set) of the semigroups Sija, Reg(Sija) and E(Sija); we give lower bounds for these ranks, and in the case of Reg(Sija) and E(Sija) 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.