On injective endomorphisms of the semigroup $\boldsymbol{B}_{\omega}^{\mathscr{F}^3}$ with a three-element family $\mathscr{F}^3$ of inductive non-empty subsets of $\omega$

Abstract

Let $\mathscr{F}$ be an $\omega$-closed family of the subsets of the set of all non-negative integers $\omega$ and $\boldsymbol{B}_{\omega}^{\mathscr{F}}$ be the bicyclic extension generated by $\mathscr{F}$, which are defined in [Visnyk Lviv Univ. Ser. Mech.-Mat. 2020, 90, 5-19]. In [Visnyk Lviv Univ. Ser. Mech.-Mat. 2023, 95, 28-45] we describe injective monoid endomorphisms of the semigroup $\boldsymbol{B}_{\omega}^{\mathscr{F}^3}$ with the family $\mathscr{F}^3=\{[0),[1),[2)\}$ and show that for every injective monoid endomorphism $\varepsilon$ of $\boldsymbol{B}_{\omega}^{\mathscr{F}^3}$ there exists a positive integer $k$ such that $\varepsilon=\alpha_{[k]}$ and the mapping $\alpha_{[k]}\colon \boldsymbol{B}_{\omega}^{\mathscr{F}^3}\to\boldsymbol{B}_{\omega}^{\mathscr{F}^3}$ is defined by the formula \[(i,j,[p))\alpha_{[k]}=
\begin{cases}
(ki,kj,[p)), & \text{if}\; p\in\{0,1\},\\
(k(i+1)-1,k(j+1)-1,[2)), & \text{if}\; p=2,
\end{cases}
\] for all $i,j\in\omega$.

In this paper, we describe injective endomorphisms of the semigroup $\boldsymbol{B}_{\omega}^{\mathscr{F}^3}$. In particular, we construct injective endomorphisms $\varpi_3$ and $\lambda$ of $\boldsymbol{B}_{\omega}^{\mathscr{F}^3}$ and describe the subsemigroup $\left\langle\lambda,\varpi_3\right\rangle$ of the semigroup $\boldsymbol{End}_{\textsf{inj}}(\boldsymbol{B}_{\omega}^{\mathscr{F}^3})$ of all injective endomorphisms of the semigroup $\boldsymbol{B}_{\omega}^{\mathscr{F}^3}$, generated by $\varpi_3$ and $\lambda$. We show that for every injective endomorphism $\varepsilon$ of the semigroup $\boldsymbol{B}_{\omega}^{\mathscr{F}^3}$ there exists an injective endomorphism $\iota\in\left\langle\lambda,\varpi_3\right\rangle$ such that $\varepsilon=\alpha_{[k]}\circ\iota$ for some positive integer $k$, where $\alpha_{[k]}$ is an injective monoid endomorphism of $\boldsymbol{B}_{\omega}^{\mathscr{F}^3}$. Also, we prove that the submonoid of $\boldsymbol{End}_{\textsf{inj}}(\boldsymbol{B}_{\omega}^{\mathscr{F}^3})$, which is generated by the set $\big\{\alpha_{[k]}\colon k\in\mathbb{N}\big\}$ and by the endomorphism $\lambda$, is isomorphic to the semidirect product $(\mathbb{N},\cdot)\ltimes_{\mathfrak{h}}(\omega,+)$.