On the semigroup of endomorphisms of the semigroup $\boldsymbol{B}_ω^{\mathscr{F}^2}$ with the two-element family $\mathscr{F}^2$ of inductive nonempty subsets of $ω$
Oleg Gutik, Marko Serivka
TL;DR
The paper studies endomorphisms of the bicyclic extension B_omega^{F^2} built from a two-element inductive family F^2 of subsets of omega. It proves that every endomorphism factors uniquely as a monoid endomorphism composed with a power of a distinguished injective endomorphism pi, enabling a complete decomposition of End. It provides a full classification of endomorphisms: injective ones have two types alpha_{k,p} and beta_{k,p} (up to pi^n), non-injective non-annihilating ones are gamma_k pi^n and delta_k pi^n, and annihilating endomorphisms chi_{s,q} form a minimal ideal. The results yield End(B_omega^{F^2}) = End(B_omega^{F^2}) · <varpi>^1 and clarify Green's relations on these endomorphism semigroups, offering a precise structural portrait of the endomorphism semigroup for the bicyclic extension.
Abstract
We study the semigroup $\overline{\boldsymbol{End}}(\boldsymbol{B}_ω^{\mathscr{F}^2})$ of all endomorphisms of the bicyclic extension $\boldsymbol{B}_ω^{\mathscr{F}^2}$ with the two-element family $\mathscr{F}^2$ of inductive nonempty subsets of $ω$. The submonoid $\left\langle\varpi\right\rangle^1$ of $\overline{\boldsymbol{End}}(\boldsymbol{B}_ω^{\mathscr{F}^2})$ with the property that every element of the semigroup $\overline{\boldsymbol{End}}(\boldsymbol{B}_ω^{\mathscr{F}^2})$ has the unique representation as the product of the monoid endomorphism of $\boldsymbol{B}_ω^{\mathscr{F}^2}$ and the element of $\left\langle\varpi\right\rangle^1$ is constructed.