Right ideal Howson semigroups

Abstract

We define a semigroup S to be right ideal Howson if the intersection of any two finitely generated right ideals, or, equivalently, any two principal right ideals, is again finitely generated. We give many examples of such semigroups, including right coherent monoids, finitely aligned semigroups, and inverse semigroups. We investigate the closure of the class of right ideal Howson semigroups under algebraic constructions. For any n 2 N 0 we give a presentation of a right ideal Howson semigroup possessing an intersection of principal right ideals that requires exactly n generators that is, in a particular sense, universal. We give analogous presentations for commutative, and for commutative cancellative, (right) ideal Howson semigroups. Keywords Semigroup Á Monoid Á One-sided ideal Á Presentations 1 Introduction An algebra exhibits the Howson property if the intersection of two finitely generated subalgebras is also finitely generated. This term is in honour of the author of [15], who showed that the intersection of finitely generated subgroups of free groups is finitely generated. There have been a number of investigations of the Howson property for other classes of algebras. In particular, the Howson property for inverse semigroups has been studied by several authors such as Jones and Trotter [16, 17], Lawson and Vdovina [20] and Silva and Soares [25]. By contrast to the situation for Communicated by Mark V. Lawson. The first author acknowledges the support of EPSRC in the form of a Ph.D. studentship.