A note on Isomorphism theorems for semigroups of order-preserving transformations with restricted range

TL;DR

This paper addresses the open problem of determining necessary and sufficient conditions for isomorphism between semigroups of order-preserving transformations with restricted range over infinite domains, providing a proof strategy.

Contribution

It introduces a proof strategy to solve the open problem of characterizing isomorphisms between such semigroups.

Findings

Provides a proof strategy for the isomorphism problem

Addresses an open problem in semigroup theory

Lays groundwork for future classification of these semigroups

Abstract

Finding necessary and sufficient conditions for isomorphism between two semigroups of order-preserving transformations over an infinite domain with restricted range was an open problem in \cite{FHQS}. In this paper, we show a proof strategy to answer that question.