This paper deals with the monoid P F N of all partial transformations on N preserving a zig-zag order on N. We determine the relative rank of P F N modulo a set containing all idempotents and all surjections in P F N . Moreover, we show that all transformations in P F N with finite rank can be generated by the idempotents with finite rank and the full transformation γ0 with infinite rank, where γ0 maps each natural number n to n + 2.

Semigroups of order-preserving transformations have been extensively studied for finite chains. We study the monoid OP N of all order-preserving partial transformations on the set N of natural numbers, where the partial order is a fence (also called zigzag poset). The monoid OP N is not regular. In this paper, we determine particular maximal regular subsemigroups of OP N and show that OP N has infinitely many maximal regular subsemigroups.

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.

The concept of linear terms of a given type [Formula: see text] (terms in which each variable occurs at most once) was introduced by M. Couceiro and E. Lethonen in [Galois theory for sets of operations closed under permutation, cylindrification and composition, Algebr. Univ. 67 (2012) 273–297, doi:10.1007/s00012-012-0184-1] (see also [T. Changphas, K. Denecke and B. Pibaljommee, Linear terms and linear hypersubstitutions, SEAMS Bull. Math. 40 (2016) (to be published).]). In this paper, we introduce binary partial operations [Formula: see text] and [Formula: see text] on the set [Formula: see text] of all linear terms of type [Formula: see text] Then we characterize regular elements and Green’s relations in these partial algebras.

A zigzag order is like a directed path, only with alternating directions. A generating set of minimal size for the semigroup of all full transformations on a finite set preserving the zigzag order was determined by Fenandes et al. in 2019. This paper deals with generating sets of the semigroup F N of all full transformations on the set of all natural numbers preserving the zigzag order. We prove that F N has no minimal generating sets and present two particular infinite decreasing chains of generating sets of F N .