Howson property and finitely generated intersection problem for monogenic inverse semigroups

Abstract

An algebraic structure is said to have the Howson property if the intersection of any two finitely generated subalgebras is finitely generated. We explore the Howson property in the context of monogenic inverse semigroups. It is known, due to work of Jones and Trotter (1989) and Jones (2016), that every monogenic inverse semigroup has the Howson property considered as an inverse semigroup, i.e. with respect to its inverse subsemigroups. In this paper, we consider monogenic inverse semigroups qua semigroups, i.e. we consider all their subsemigroups. We prove that every monogenic inverse semigroup possesses the Howson property in this broader sense, with the sole exception of the monogenic free inverse semigroup. For this exceptional case, we show that the problem of determining whether the intersection of two finitely generated subsemigroups is finitely generated is algorithmically decidable.