arXiv subjects

Gergő Gyenizse

Publications and source records attributed to Gergő Gyenizse.

5 recordsLinked to original sources

arXiv2026

The filter of interpretability types of Hobby-McKenzie varieties is prime

We study the Hobby-McKenzie varieties that constitute a major class investigated thoroughly in the monograph The shape of congruence lattices by Kearnes and Kiss. We obtain new characterizations of the Hobby-McKenzie varieties via compatible reflexive ternary structures. Based on our findings, we prove that in the lattice of interpretability types of varieties, the filter of the interpretability types of Hobby-McKenzie varieties is prime.

math.CO↗

arXiv2026

Connectivity notions on compatible digraphs in equational classes

A digraph $\mathbb G$ is called weakly connected, strongly connected, and extremely connected if any two vertices of $\mathbb G$ are connected respectively by an oriented, a directed, and a symmetric path in $\mathbb G$. We investigate the algebraic properties of digraphs that force some of these connectivity notions to coincide. We prove that for digraphs with a Hobby-McKenzie polymorphism, the strong and the extreme components coincide. Conversely, if the strong and the extreme components of any compatible digraph in an equational class of algebras coincide, then the class must have a Hobby-McKenzie term. As a consequence, we obtain that an equational class $\mathcal V$ is $n$-permutable for some $n$ if and only if the weak components of any compatible reflexive digraph in $\mathcal V$ are extremely connected.

math.CO↗

arXiv2024

Taylor is prime

We study the Taylor varieties and obtain new characterizations of them via compatible reflexive digraphs. Based on our findings, we prove that in the lattice of interpretability types of varieties, the filter of the types of all Taylor varieties is prime.

math.CO↗

arXiv2021

Congruence permutability is prime

We give a combinatorial proof that congruence permutability is prime in the lattice of interpretability types of varieties. Thereby, we settle a 1984 conjecture of Garcia and Taylor.

math.CO↗

arXiv2018

Symmetric embeddings of free lattices into each other

By a 1941 result of Ph. M. Whitman, the free lattice FL(3) on three generators includes a sublattice $S$ that is isomorphic to the lattice FL($ω$)=FL($\aleph_0$) generated freely by denumerably many elements. The first author has recently "symmetrized" this classical result by constructing a sublattice $S\cong$ FL($ω)$ of FL(3) such that $S$ is SELFDUALLY POSITIONED in FL(3) in the sense that it is invariant under the natural dual automorphism of Fl(3) that keeps each of the three free generators fixed. Now we move to the furthest in terms of symmetry by constructing a selfdually positioned sublattice $S\cong$ FL$(ω)$ of FL(3) such that every element of $S$ is fixed by all automorphisms of FL(3). That is, in our terminology, we embed FL$(ω)$ into FL(3) in a TOTALLY SYMMETRIC way. Our main result determines all pairs $(κ,λ)$ of cardinals greater than 2 such that FL$(κ)$ is embeddable into FL$(λ)$ in a totally symmetric way. Also, we relax the stipulations on $S\cong$FL$κ$ by requiring only that $S$ is closed with respect to the automorphisms of FL$(λ)$, or $S$ is selfdually positioned and closed with respect to the automorphisms; we determine the corresponding pairs $(κ,λ)$ even in these two cases. We reaffirm some of our calculations with a computer program developed by the first author. This program is for the word problem of free lattices, it runs under Windows, and it is freely available.

math.RA↗