Congruences on infinite semigroups of transformations preserving a zig-zag order

Abstract

The purpose of this paper is the study of congruences on semigroups of transformations on a countably infinite fence. We consider the monoid [Formula: see text] of all full transformations on the set [Formula: see text] of all natural numbers preserving the zig-zag order on [Formula: see text], as well as the monoid [Formula: see text] of all idempotent transformations in [Formula: see text] additionally preserving the usual linear order on [Formula: see text] We show that there are uncountably many congruences on [Formula: see text] and determine seven maximal congruences on [Formula: see text] which are all the maximal congruences containing a particular congruence on [Formula: see text] Moreover, we characterize all congruences on the monoid of all transformations in [Formula: see text] with infinite rank. For the semigroup of all transformations in [Formula: see text] with finite rank, we determine the Rees congruences.