Generating Sets of an Infinite Semigroup of Transformations Preserving a Zig-zag Order

TL;DR

This paper investigates the structure of the semigroup of all transformations on natural numbers that preserve a zig-zag order, revealing it has no minimal generating set and providing specific infinite chains of generating sets.

Contribution

It extends previous finite case results to the infinite set of natural numbers, showing the non-existence of minimal generating sets for this semigroup.

Findings

No minimal generating sets exist for F_N.

Two infinite decreasing chains of generating sets are constructed.

Abstract

A zig-zag 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 zig-zag order was determined by Fenandes et al. in 2019. This paper deals with generating sets of the semigroup $F_{\mathbb{N}}$ of all full transformations on the set of all natural numbers preserving the zig-zag order. We prove that $F_{\mathbb{N}}$ has no minimal generating sets and present two particular infinite decreasing chains of generating sets of $F_{\mathbb{N}}$.