The lattice of varieties of semigroups with completely regular square. II

T. A. Ershova; M. V. Volkov

Home/Archive/arXiv

arXiv · 2609.34588

The lattice of varieties of semigroups with completely regular square. II

T. A. Ershova·M. V. Volkov

View original source ↗Read source document ↗Export metadata ↓

Abstract

We solve (modulo groups) the word problem for free semigroup satisfying $xy = (xy)^n$. This enables us to show that the variety ${\cal {SCR}}_n$ given by this identity is not equal to the join of the varieties ${\cal{CR}}_n$ and ${\cal{SI}}$ defined by the identities $x = x^n$ and $xy = (xy)^2$ respectively. Therefore the result of (M.V.Volkov, T.A.Ershova, The lattice of varieties of semigroup with completely regular square, Monash Conference on Semigroup Theory in Honour of G. B. Preston, World Scientific, Singapore, 1991, 306-322) that the lattice $L({\cal {SCR}}_n)$ is modular does not follow from the earlier results concerning $L({\cal{CR}}_n)$ and $L({\cal{SI}})$. However the variety ${\cal{SO}}_n$ consisting of all semigroups from ${\cal {SCR}}_n$ whose idempotents form a subsemigroup turns out to be equal to the join of the varieties ${\cal O}_n$ of all orthogroups and ${\cal{SI}}$. We deduce from this a description of the lattice $L({\cal{SO}}_n)$ modulo groups.

Explore related subjects

math.GR

Keep this discovery

Explore connections, maps & timelines

BibTeX RIS

T. A. Ershova, M. V. Volkov. 2026-09-28. The lattice of varieties of semigroups with completely regular square. II. https://arxiv.org/abs/2609.34588

Cite the original work for its findings. Save a collection to share your selection of sources.

KEEP EXPLORING

Related papers

arXiv2026

Random Quotients of Free Products

We introduce a density model for random quotients of a free product of finitely generated groups. We prove that a random quotient in this model has the following properties with overwhelming probability: if the density is below $1/2$, the free factors embed into the random quotient and the random quotient is hyperbolic relative to the free factors. Further, there is a phase transition at $1/2$, with the random quotient being a finite group above this density. If the density is below $1/6$, the random quotient is cubulated relative to the free factors. Moreover, if the free factors are cubulated, then so is the random quotient.

math.GR↗

arXiv2026

Inherently nonfinitely based additively idempotent semirings

We establish a sufficient condition for an additively idempotent semiring to be inherently nonfinitely based: if its generated variety is locally finite and every Zimin word is minimal in the additive order, then it is contained in no finitely based locally finite variety. For every positive integer $n$, we construct an infinite finitely generated flat semiring satisfying all identities in at most $n$ variables of every ai-semiring with this minimality property. We also characterize Zimin minimality by the membership of a countable flat factor semiring $\Finf$ in the generated variety, and prove that$\V(\Finf)$ is locally finite and inherently nonfinitely based. As applications, we show that the six-element ai-semirings $\A$ and $\B$ are inherently nonfinitely based. In contrast, we prove that the six-element ai-semiring $\overline{A_2^1}$ is not inherently nonfinitely based, although its multiplicative reduct $A_2^1$ is inherently nonfinitely based. To the best of our knowledge, this is the first example of an ai-semiring whose multiplicative reduct is inherently nonfinitely based while the ai-semiring itself is not.

math.GR↗

arXiv2026

On the residual finiteness of the non-abelian tensor square $G \otimes G$, the non-abelian exterior square $G \wedge G$ and the weak commutativity construction $\X(G)$

We prove that if $G$ is a residually finite group, $\widehat{G}$ is its profinite completion and the map $H_2(G, \mathbb{Z}) $ $ \to H_2(\widehat{G}, \widehat{\mathbb{Z}}),$ induced by the canonical map $G \to \widehat{G}$, is injective, then the non-abelian exterior square $G \wedge G$ is residually finite. We show that if $G$ is a finitely presented centre-by-metabelian group then $ν(G)$, the non-abelian tensor product $G \otimes G$ and the non-abelian tensor square $G \wedge G$ are residually finite. Furthermore we prove that if $G$ be a finitely presented metabelian group then the weak commutative construction $\X(G)$ is residually finite. We discuss a criterion for the $q$-exterior square $G \wedge^q G$ to be residually finite.

math.GR↗