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

Oleg Gutik, Sher-Ali Penza

TL;DR

This paper analyzes endomorphisms of the submonoid $\mathscr{C}_{+}(a,b)$ of the bicyclic monoid. It first classifies endomorphisms that extend from bicyclic monoid homomorphisms, revealing two natural families $\lambda_k$ and $\sigma_{l,m}$ with explicit semigroup structures and an ideal, and then classifies injective endomorphisms as either $\lambda_{n,s}$ or $\lambda_{p,s}\varsigma^{n}$ with a concrete multiplication rule on index triples that embeds the injective endomorphism semigroup into $\mathbb{N}^3$; analogous results hold for $\mathscr{C}_{-}(a,b)$. The results provide a complete algebraic description of how endomorphisms act on these submonoids and their interaction with the bicyclic structure, including ideal and semidirect-product-like compositions.

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 $\mathscr{C}_{+}(a,b)$.

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

TL;DR

This paper analyzes endomorphisms of the submonoid of the bicyclic monoid. It first classifies endomorphisms that extend from bicyclic monoid homomorphisms, revealing two natural families and with explicit semigroup structures and an ideal, and then classifies injective endomorphisms as either or with a concrete multiplication rule on index triples that embeds the injective endomorphism semigroup into ; analogous results hold for . The results provide a complete algebraic description of how endomorphisms act on these submonoids and their interaction with the bicyclic structure, including ideal and semidirect-product-like compositions.

Abstract

Let be the submonoid of the bicyclic monoid which is studied in \cite{Makanjuola-Umar=1997}. We describe monoid endomorphisms of the semigroup which are generated by the family of all congruences of the bicyclic monoid and all injective monoid endomorphisms of .
Paper Structure (2 sections, 15 theorems, 41 equations)

This paper contains 2 sections, 15 theorems, 41 equations.

Table of Contents

  1. On monoid endomorphisms of $\mathscr{C}_{+}(a,b)$ which are restrictions of homomorphisms of the bicyclic monoid
  2. On monoid injective endomorphisms of $\mathscr{C}_{+}(a,b)$

Key Result

Proposition 1

The semigroup $\mathfrak{End}_{\langle\lambda\rangle}(\mathscr{C}_{+}(a,b))$ is isomorphic to the multiplicative semigroup $(\omega,*)$ of non-negative integers.

Theorems & Definitions (31)

  • Proposition 1
  • proof
  • Lemma 1
  • proof
  • Definition 1
  • Lemma 2
  • proof
  • Proposition 2
  • Proposition 3
  • Theorem 1
  • ...and 21 more