On the semigroup of monoid endomorphisms of the semigroup $\mathscr{C}_{+}(a,b)$

TL;DR

This paper characterizes the monoid endomorphisms of a specific submonoid of the bicyclic monoid, focusing on those generated by congruences and injective endomorphisms, advancing understanding of its algebraic structure.

Contribution

It provides a detailed description of the monoid endomorphisms of al_{+}(a,b), highlighting those generated by congruences and injective endomorphisms, which was not previously known.

Findings

Characterization of monoid endomorphisms of al_{+}(a,b)

Identification of endomorphisms generated by congruences

Description of injective monoid endomorphisms

Abstract

Let $\mathscr{C}_{+}(a,b)$ be the submonoid of the bicyclic monoid which is studied in \cite{Makanjuola-Umar=1997}. We describe monoid endomorphisms of the semigroup $\mathscr{C}_{+}(a,b)$ which are generated by the family of all congruences of the bicyclic monoid and all injective monoid endomorphisms of $\mathscr{C}_{+}(a,b)$.