Semigroups of linear tree languages

Abstract

A linear tree language of type [Formula: see text] is a set of linear terms, terms in which each variable occurs at most once, of that type. We investigate a semigroup consisting of the collection of all linear tree languages such that products of any element in the collection are nonempty and the operation of the corresponding linear product especially idempotent elements, Green’s relations [Formula: see text], [Formula: see text], and [Formula: see text], and some of its subsemigroups. We discover that this semigroup is neither factorizable nor locally factorizable. We also study the linear product iteration and show that any iteration is idempotent in this semigroup. Moreover, we study a semigroup with the complement of the universe set of the above semigroup together with the same linear product operation.