Congruences, embeddings, and automorphisms of the generalized bicyclic semigroups $B_\omega^{\mathscr{F}}$ for arbitrary $\omega$-closed families $\mathscr{F}$

For an arbitrary $\omega$-closed family $\mathscr{F}$ of subsets of the non-negative integers $\omega$, Gutik and Mykhalenych introduced an inverse semigroup $B_\omega^{\mathscr{F}}$ that simultaneously generalizes the bicyclic monoid, the countable semigroup of matrix units, and several related combinatorial inverse semigroups, and they characterized its Green relations, natural partial order, idempotents, and (0-)simplicity and (0-)bisimplicity in terms of $\mathscr{F}$. Apart from the minimal group congruence, which the founding paper describes uniformly for all $\omega$-closed families, the homomorphism theory of these semigroups is known only in restricted special cases: for families of inductive non-empty subsets the group congruences and the isomorphism/automorphism classification were described in a 2021 follow-up, and for the finite bounded-interval families $\mathscr{F}_n$ it is known that only Rees congruences occur. The open problem is the general case: describe the full congruence lattice $\operatorname{Con}(B_\omega^{\mathscr{F}})$ — equivalently, all homomorphic images — together with the embedding/isomorphism relation among the semigroups $B_\omega^{\mathscr{F}}$ and their automorphism groups, uniformly for every $\omega$-closed family $\mathscr{F}$, in terms of $\mathscr{F}$. No work resolving these questions for general $\mathscr{F}$ is known; the problem is open.

Background

An inverse semigroup is a semigroup $S$ in which every element $x$ has a unique inverse $x^{-1}$ satisfying $xx^{-1}x=x$ and $x^{-1}xx^{-1}=x^{-1}$; inverse semigroups model partial symmetries, and their idempotents ($x^2=x$) commute and form a semilattice $E(S)$ under multiplication, carrying a natural partial order $e\le f \iff ef=fe=e$. A semigroup is combinatorial if all its maximal subgroups are trivial. Green's relations $\mathscr{R},\mathscr{L},\mathscr{D},\mathscr{H},\mathscr{J}$ are the standard equivalence relations on a semigroup that classify elements by the principal one-sided or two-sided ideals they generate; a semigroup with a single $\mathscr{J}$-class is simple, and with a single $\mathscr{D}$-class (and a zero, in the 0-analogue) bisimple (0-bisimple). The bicyclic monoid $C(p,q)=\langle p,q\mid pq=1\rangle$ is a foundational example: it is a bisimple inverse monoid, concretely representable on $\omega\times\omega$ with $(i_1,j_1)(i_2,j_2)=(i_1-j_1+i_2,j_2)$ when $j_1\le i_2$ and $(i_1,j_1-i_2+j_2)$ otherwise, and by Andersen's classical theorem its absence characterizes complete 0-simplicity among (0-)simple semigroups with idempotents. A congruence on a semigroup $S$ is an equivalence relation stable under multiplication on both sides; each ideal $I$ determines the Rees congruence collapsing $I$ to one point, and a congruence is a group congruence when $S/\mathfrak{C}$ is a group. Classically, every non-trivial congruence on the bicyclic monoid is a group congruence (recorded in standard monographs and invoked in the work below), while the countable semigroup of matrix units (the inverse semigroup of $\omega\times\omega$ matrix units $e_{ij}$ with multiplication $e_{ij}e_{kl}=e_{il}$ if $j=k$ and $0$ otherwise, with a zero adjoined) is congruence-free. Gutik and Mykhalenych (2020) generalized these objects: call a family $\mathscr{F}\subseteq\mathscr{P}(\omega)$, where $\omega=\{0,1,2,\dots\}$, $\omega$-closed if $F_1\cap(-n+F_2)\in\mathscr{F}$ for all $n\in\omega$ and $F_1,F_2\in\mathscr{F}$, where $-n+F=\{k-n:k\in F\}$ is the translate of $F$ by $-n$. On $\omega\times\omega\times\mathscr{F}$ they define a bicyclic-type product $(i_1,j_1,F_1)(i_2,j_2,F_2)=(i_1-j_1+i_2,j_2,(j_1-i_2+F_1)\cap F_2)$ if $j_1\le i_2$, and $(i_1,j_1-i_2+j_2,F_1\cap(i_2-j_1+F_2))$ if $j_1>i_2$; if the empty set belongs to $\mathscr{F}$, the ideal $\{(i,j,\emptyset)\}$ is collapsed by a Rees quotient (the quotient by the Rees congruence of this ideal), and the result is the semigroup $B_\omega^{\mathscr{F}}$. They proved $B_\omega^{\mathscr{F}}$ is always a combinatorial inverse semigroup, described Green's relations, the natural partial order, and the idempotents, gave necessary-and-sufficient criteria in terms of $\mathscr{F}$ for simplicity, 0-simplicity, bisimplicity, 0-bisimplicity, existence of an identity, and for $B_\omega^{\mathscr{F}}$ being isomorphic to the bicyclic monoid or to the countable semigroup of matrix units, and determined the minimal group congruence $\sigma$ on $B_\omega^{\mathscr{F}}$ for arbitrary $\omega$-closed $\mathscr{F}$: $B_\omega^{\mathscr{F}}/\sigma$ is trivial when $\emptyset\in\mathscr{F}$, and when $\emptyset\notin\mathscr{F}$ one has $(i_1,j_1,F_1)\,\sigma\,(i_2,j_2,F_2)$ if and only if $i_1-j_1=i_2-j_2$, so that $B_\omega^{\mathscr{F}}/\sigma\cong\mathbb{Z}(+)$. Important examples of $\omega$-closed families are the inductive tails $[k)=\{k,k+1,k+2,\dots\}$ (a family of non-empty inductive subsets is a subfamily of these) and the families $\mathscr{F}_n$ of finite intervals bounded by $n$ generated by $\{0,1,\dots,n\}$. Subsequent work by the same research program determined the further homomorphism theory only for such restricted families, so for a general $\omega$-closed family no description of the full congruence lattice, the embeddings, or the automorphisms is known.

Problem Statement

For an arbitrary $\omega$-closed family $\mathscr{F}$ of subsets of $\omega$, determine the full congruence lattice $\operatorname{Con}(B_\omega^{\mathscr{F}})$, described intrinsically in terms of $\mathscr{F}$; equivalently, describe all homomorphic images of $B_\omega^{\mathscr{F}}$. This entails: (i) characterizing precisely which equivalence relations on $B_\omega^{\mathscr{F}}$ are congruences — in particular which are Rees congruences of ideals, which are group congruences, and whether congruences of any other kind exist for families beyond the known special cases; and, as the embedding- and symmetry-parts of the same homomorphism theory: (ii) determining for which $\omega$-closed families $\mathscr{F}_1$ and $\mathscr{F}_2$ the semigroup $B_\omega^{\mathscr{F}_1}$ embeds into, or is isomorphic to, $B_\omega^{\mathscr{F}_2}$; and (iii) computing the automorphism group of $B_\omega^{\mathscr{F}}$ for general $\mathscr{F}$. The common objective uniting these deliverables is a uniform structure and homomorphism theory of the semigroups $B_\omega^{\mathscr{F}}$ over all $\omega$-closed families, generalizing the classical facts that every non-trivial congruence on the bicyclic monoid is a group congruence and that the countable semigroup of matrix units is congruence-free; results restricted to particular classes of families (such as inductive families or the finite-interval families $\mathscr{F}_n$) settle only special cases of this problem.

References

1. O. Gutik, M. Mykhalenych, On some generalization of the bicyclic monoid, Visnyk Lviv. Univ. Ser. Mech.-Mat. 90 (2020), 5–19; preprint arXiv:2107.14118, DOI 10.48550/arXiv.2107.14118, https://arxiv.org/abs/2107.14118
2. O. Gutik, M. Mykhalenych, On group congruences on the semigroup $B_\omega^{\mathscr{F}}$ and its homomorphic retracts in the case when a family $\mathscr{F}$ consists of inductive non-empty subsets of $\omega$, Visn. L'viv. Univ. Ser. Mekh.-Mat. 91 (2021), 5–27; preprint arXiv:2108.09543, DOI 10.48550/arXiv.2108.09543, https://arxiv.org/abs/2108.09543
3. O. Gutik, O. Lysetska, On the semigroup $B_\omega^{\mathscr{F}}$ which is generated by the family $\mathscr{F}$ of atomic subsets of $\omega$, Visnyk Lviv. Univ. Ser. Mech.-Mat. 92 (2021), 34–50; preprint arXiv:2108.11354, DOI 10.48550/arXiv.2108.11354, https://arxiv.org/abs/2108.11354
4. O. Gutik, O. Popadiuk, On the semigroup $B_\omega^{\mathscr{F}_n}$ which is generated by the family $\mathscr{F}_n$ of finite bounded intervals of $\omega$, Carpathian Mathematical Publications 15 (2023), no. 2, 331–355, DOI 10.15330/cmp.15.2.331-355, https://doi.org/10.15330/cmp.15.2.331-355; preprint arXiv:2208.09155, DOI 10.48550/arXiv.2208.09155, https://arxiv.org/abs/2208.09155
5. O. Gutik, M. Mykhalenych, On a semitopological semigroup $B_\omega^{\mathscr{F}}$ when a family $\mathscr{F}$ consists of inductive non-empty subsets of $\omega$, Matematychni Studii 59 (2023), no. 1, 20–28, DOI 10.30970/ms.59.1.20-28, https://doi.org/10.30970/ms.59.1.20-28; preprint arXiv:2212.05522, DOI 10.48550/arXiv.2212.05522, https://arxiv.org/abs/2212.05522
6. O. Gutik, M. Mykhalenych, On automorphisms of the semigroup $B_\omega^{\mathscr{F}}$ in the case when the family $\mathscr{F}$ consists of nonempty inductive subsets of $\omega$, preprint arXiv:2306.07844, DOI 10.48550/arXiv.2306.07844, https://arxiv.org/abs/2306.07844
7. O. Gutik, O. Popadiuk, On the semigroup of injective endomorphisms of the semigroup $B_\omega^{\mathscr{F}_n}$ which is generated by the family $\mathscr{F}_n$ of initial finite intervals of $\omega$, Matematychni Metody ta Fizyko-Mekhanichni Polya 65 (2022), no. 1–2, 42–57; preprint arXiv:2209.08377, DOI 10.48550/arXiv.2209.08377, https://arxiv.org/abs/2209.08377
8. O. Gutik, I. Pozdniakova, On the semigroup of non-injective monoid endomorphisms of the semigroup $B_\omega^{\mathscr{F}}$ with the two-element family $\mathscr{F}$ of inductive nonempty subsets of $\omega$, Visnyk Lviv. Univ. Ser. Mech.-Mat. 95 (2023), 14–27, DOI 10.30970/vmm.2023.95.014-027, https://doi.org/10.30970/vmm.2023.95.014-027; preprint arXiv:2402.12386, DOI 10.48550/arXiv.2402.12386, https://arxiv.org/abs/2402.12386
9. O. Gutik, M. Serivka, On the semigroup of injective monoid endomorphisms of the monoid $B_\omega^{\mathscr{F}^3}$ with a three element family $\mathscr{F}^3$ of inductive nonempty subsets of $\omega$, Visnyk Lviv. Univ. Ser. Mech.-Mat. 95 (2023), 28–45, DOI 10.30970/vmm.2023.95.028-045, https://doi.org/10.30970/vmm.2023.95.028-045; preprint arXiv:2404.12414, DOI 10.48550/arXiv.2404.12414, https://arxiv.org/abs/2404.12414