A general approach for generating sets of certain subsemigroups of monotone maps

Abstract

Let [Formula: see text] and [Formula: see text] be the monoids of all (full) transformations and of all partial transformations, on a finite chain [Formula: see text] under its natural order, respectively. Moreover, let [Formula: see text] ([Formula: see text]) be the subsemigroup of [Formula: see text] ([Formula: see text]) consists of all monotone transformations (monotone partial transformations) with height less than or equal to [Formula: see text] for [Formula: see text] ([Formula: see text]). In this paper, we develop a new and general approach to find a (minimal) generating set of [Formula: see text] ([Formula: see text]).