Taylor's modularity conjecture and related problems for idempotent varieties

We provide a partial result on Taylor's modularity conjecture, and several related problems. Namely, we show that the interpretability join of two idempotent varieties that are not congruence modular is not congruence modular either, and we prove an analogue for idempotent varieties with a cube term. Also, similar results are proved for linear varieties and the properties of congruence modularity, having a cube term, congruence $n$-permutability for a fixed $n$, and satisfying a non-trivial congruence identity.Comment: 27 page

The Residual Character of Finitely Generated Varieties of Unary Algebras and Groupoids, and Applications to the Restricted Quackenbush Problem.

The crux of this thesis is to study the Restricted Quackenbush Problem: if B is a finite algebra of finite type and V(B) is residually finite, must V(B) be residually <N, for some positive integer N? We show that the Restricted Quackenbush Problem is answered affirmatively with respect to unary algebras, a result first shown by Baldwin and Berman in 1975. Then we turn our attention to groupoids and show that all groupoids that generate residually finite varieties must satisfy an identity of the form k(x)≈x∗x, where k(x) is an identity of type {∗} that is not equal to x∗x. Due to this result, we focus on a particular class of idempotent groupoids, called absorbing groupoids. We show that if a certain property holds, with respect to absorbing groupoids, then the Restricted Quackenbush Problem is answered affirmately.The original print copy of this thesis may be available here: http://wizard.unbc.ca/record=b180318

Relational Approach to Universal Algebra

Title: Relational Approach to Universal Algebra Author: Jakub Opršal Department: Department of Algebra Supervisor: doc. Libor Barto, Ph.D., Department of Algebra Abstract: We give some descriptions of certain algebraic properties using rela- tions and relational structures. In the first part, we focus on Neumann's lattice of interpretability types of varieties. First, we prove a characterization of vari- eties defined by linear identities, and we prove that some conditions cannot be characterized by linear identities. Next, we provide a partial result on Taylor's modularity conjecture, and we discuss several related problems. Namely, we show that the interpretability join of two idempotent varieties that are not congruence modular is not congruence modular either, and the analogue for idempotent va- rieties with a cube term. In the second part, we give a relational description of higher commutator operators, which were introduced by Bulatov, in varieties with a Mal'cev term. Furthermore, we use this result to prove that for every algebra with a Mal'cev term there exists a largest clone containing the Mal'cev operation and having the same congruence lattice and the same higher commu- tator operators as the original algebra, and to describe explicit (though infinite) set of identities describing supernilpotence..

Relační přístup k univerzální algebře

Title: Relational Approach to Universal Algebra Author: Jakub Opršal Department: Department of Algebra Supervisor: doc. Libor Barto, Ph.D., Department of Algebra Abstract: We give some descriptions of certain algebraic properties using rela- tions and relational structures. In the first part, we focus on Neumann's lattice of interpretability types of varieties. First, we prove a characterization of vari- eties defined by linear identities, and we prove that some conditions cannot be characterized by linear identities. Next, we provide a partial result on Taylor's modularity conjecture, and we discuss several related problems. Namely, we show that the interpretability join of two idempotent varieties that are not congruence modular is not congruence modular either, and the analogue for idempotent va- rieties with a cube term. In the second part, we give a relational description of higher commutator operators, which were introduced by Bulatov, in varieties with a Mal'cev term. Furthermore, we use this result to prove that for every algebra with a Mal'cev term there exists a largest clone containing the Mal'cev operation and having the same congruence lattice and the same higher commu- tator operators as the original algebra, and to describe explicit (though infinite) set of identities describing supernilpotence...Název práce: Relační přístup k universální algebře Autor: Jakub Opršal Katedra: Katedra algebry Vedoucí disertační práce: doc. Libor Barto, Ph.D., Katedra algebry Abstrakt: V této práci předkládáme popis některých algebraických vlastnostní pomocí relací a relačních struktur. V první části se zaměřujeme na Neumannův svaz interpretačních typů variet. Charakterizujeme variety definované lineárními rovnostmi a uvádíme příklad několika vlastností, které nejsou charakterizova- telné lineárními rovnostmi. Dále se věnujeme Taylorově domněnce o varietách s modulárními svazy kongruencí. Speciálně ukážeme, že interpretační spojení dvou idempotentních variet, které nemají modulární svazy kongruencí, samo nemá mo- dulární svazy kongruencí. Uvádíme i obdobný výsledek pro variety s krychlovým termem. V druhé části práce uvádíme popis Bulatovových vyšších komutátorů ve varietách s mal'cevským termem. Dále použijeme tento výsledek na to, abychom ukázali, že pro každou algebru s mal'cevskou operací existuje největší klon, který obsahuje tu samou mal'cevskou operaci, má stejný svaz kongruencí a jehož ko- mutátory se shodují s těmi v původní algebře. Nakonec uvádíme další aplikaci tohoto výsledku a to na...Katedra algebryDepartment of AlgebraFaculty of Mathematics and PhysicsMatematicko-fyzikální fakult