arXiv subjects

Olga B. Sapir

Publications and source records attributed to Olga B. Sapir.

6 recordsLinked to original sources

arXiv2026

Limit varieties of aperiodic monoids

A limit variety is a variety that is minimal with respect to being non-finitely based. We present a new limit variety of aperiodic monoid. We also show that if there exists any other limit variety of aperiodic monoids, then it is contained in the joint of the variety $\mathbb B^1$ of all idempotent monoids and certain finitely generated variety $\mathbb E^1$ with $\mathbb B^1 \wedge \mathbb E^1 = \mathbb L_2^1$, where $\mathbb L_2^1$ is the variety of left-zero monoids. Jackson and Lee proved that $\mathbb E^1$ is HFB, that is, its every subvariety is finitely based. We exend this result a step up the classical decomposition $\mathbb B^1=\bigcup_{i \ge 2} \mathbb L^1_i$ by showing that $\mathbb E^1 \vee \overline{\mathbb E^1} \vee \mathbb L^1_3$ is also HFB, where $\overline{\mathbb E^1}$ is the variety dual of $\mathbb E^1$.

math.GR↗

arXiv2025

Non-finitely related and finitely related monoids

We transform the method of Glasson into a sufficient condition under which a monoid is non-finitely related, add a new member to the collection of interlocking word-patterns, and use it to show that the monoid $M(ab^2a, a^2b^2)$ is non-finitely related. We also give a sufficient condition under which a monoid is finitely related and use it show that $M(a^2b^2)$ is finitely related. Together with the results of Glasson this completes the description of all finitely related monoids among the monoids of the form $M(W)$ where every word ${\bf u} \in W$ depends on two variables and every variable occurs twice in $\bf u$.

math.GR↗

arXiv2024

Strongly nonfinitely based monoids

We show that the 42-element monoid of all partial order preserving and extensive injections on the 4-element chain is not contained in any variety generated by a finitely based finite semigroup.

math.GR↗

arXiv2023

Limit varieties generated by finite non-J-trivial aperiodic monoids

Jackson and Lee proved that certain six-element monoid generates a hereditarily finitely based variety $\mathbb E^1$ whose lattice of subvarieties contains an infinite ascending chain. We identify syntactic monoids which generate finitely generated subvarieties of $\mathbb E^1$ and show that one of these finite monoids together with certain seven-element monoid generates a new limit variety.

math.GR↗

arXiv2023

Catalan monoids inherently nonfinitely based relative to finite $\mathcal{R}$-trivial semigroups

We show that the 42-element monoid of all partial order preserving and extensive injections on the 4-element chain is not contained in any variety generated by a finitely based finite $\mathcal{R}$-trivial semigroup. This provides unified proofs for several known facts and leads to a bunch of new results on the Finite Basis Problem for finite $\mathcal{R}$- and $\mathcal{J}$-trivial semigroups.

math.GR↗

arXiv2021

Limit varieties of $J$-trivial monoids

We show that limit varieties of monoids recently discovered by Gusev, Zhang and Luo and their subvarieties are generated by monoids of the form $M_τ(W)$ for certain congruences $τ$ on the free monoid. The construction $M_τ(W)$ is a generalization of widely used Dilworth-Perkins construction. Using this construction, we find explicit generators for Gusev limit varieties and give a short reproof to the fact that Zhang-Luo limit variety is non-finitely based.

math.GR↗