On injective endomorphisms of the monoid $\boldsymbol{B}_ω^{\mathscr{F}^3}$ with a three-element family $\mathscr{F}^3$ of inductive non-empty subsets of $ω$
Oleg Gutik, Marko Serivka
TL;DR
The paper addresses the problem of classifying injective endomorphisms of the bicyclic extension semigroup B_ω^{F^3} built from a three-element inductive family in ω. It constructs explicit endomorphisms λ and varpi_3 and shows that any injective endomorphism ε decomposes as ε = α_{[k]} ∘ ι with some k and ι in ⟨λ, varpi_3⟩, yielding a precise factorization. The authors develop a detailed algebraic description of the endomorphism semigroup End_inj(B_ω^{F^3}) as a semidirect product generated by α_{[k]} and λ^m, augmented by relations involving varpi_3, and relate this to isomorphisms among subsemigroups such as B_ω^{F^3_{0,1}} ≅ B_ω^{F^2}. The results extend previous endomorphism classifications for smaller families and provide a comprehensive structural picture of injective endomorphisms in this inverse-semigroup framework.
Abstract
We describe injective endomorphisms of the semigroup $\boldsymbol{B}_ω^{\mathscr{F}^3}$ with a three-element family $\mathscr{F}^3$ of inductive non-empty subsets of $ω$. In particular we find endomorphisms $\varpi_3$ and $λ$ of $\boldsymbol{B}_ω^{\mathscr{F}^3}$ such that for every injective endomorphism $\varepsilon$ of the semigroup $\boldsymbol{B}_ω^{\mathscr{F}^3}$ there exists an injective endomorphism $ι\in\left\langleλ,\varpi_3\right\rangle$ such that $\varepsilon=α_{[k]}\circι$ for some positive integer $k$, where $α_{[k]}$ is an injective monoid endomorphism of $\boldsymbol{B}_ω^{\mathscr{F}^3}$.