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Thunderbird
The Algebra and Logic seminar is a research and learning seminar organized by the algebra and logic research group of the Department of Mathematics at the University of Colorado at Boulder. The scope of the seminar includes all topics with links to algebra, logic, or their applications, like general algebra, logic, model theory, category theory, set theory, set-theoretic topology, or theoretical computer science. If you have questions regarding this seminar, please contact Keith Kearnes or Peter Mayr.
A strong Maltsev class of varieties is a class of varieties for which there exists a finitely presented variety (i.e. a variety in a finite signature satisfying finitely many identities) whose interpretability class is a lower bound to every variety in the interpretability order, while a Maltsev class of varieties is a class for which there exists a countably infinite descending chain of finitely presented varieties and each variety in the Maltsev class is bounded below by some of the chain. A Maltsev term is the original example of a strong Maltsev condition and the identities that define it define the class of congruence permutable varieties. On the other hand, the classes of congruence distributive and congruence modular varieties are not strong Maltsev classes, but rather Maltsev classes (the terms for distributivity were discovered by Jónsson and the terms for modularity were discovered by Day). There has been for some years a fertile area of research which connects Universal Algebra to the complexity theory of constraint satisfaction. While many of these results are largely important for complexity theory, some of them have been important for algebra. In particular, Siggers discovered that the class of locally finite Taylor varieties is captured by a finite package of identities. In 2015, Olsak made the surprising discovery that in fact the class of all Taylor algebras is strong Maltsev. A natural next question was: what about the class of congruence meet semidistributive varieties? This is an important class in fixed template finite domain CSP, since the class of locally finite congruence meet semidistributive varieties is exactly the class of varieties which correspond to CSP templates which can be solved with local consistency methods. Actually, Kozik, Krokhin, Valeriote, and Willard proved that locally finite congruence meet semidistributive varieties are characterized by a finite package of identities. We show that, unlike the class of Taylor varieties, this finiteness does not lift to the general case, that is, there does not exist a strong Maltsev condition that captures every congruence meet semidistirbutive variety.
The class of congruence meet semidistributive varieties is not strong Maltsev
Given a semigroup S and elements s,t in S, write s~t if s=pr and t=rp, for some p,r in S (with 1 adjoined). This relation, and its transitive closure, both known as "primary conjugacy", has been extensively used and studied in connection with many algebraic objects, including groups, rings, and C*-algebras. It is the standard tool for either measuring or forcing commutativity. We discuss a natural generalization, defined by s~t whenever s=p_{1}...p_{n} and t=p_{f(1)}...p_{f(n)}, for some p_{1},...,p_{n} in S (with 1 adjoined) and permutation f of {1,...,n}, together with its transitive closure, which we call the "permutation" relation. The permutation relation is actually the congruence generated by the primary conjugacy, and is the least commutative congruence on any semigroup. We explore general properties of the permutation relation, discuss it in the context of groups and rings, compare it to various known semigroup conjugacy relations, and describe its equivalence classes in different semigroups.
Conjugacy and least commutative congruences in semigroups Sponsored by the Meyer Fund
The Yang--Baxter equation (YBE) is a fundamental relation in mathematical physics that has its roots in the investigation of many-particle interactions in two-dimensional quantum dynamical systems, where it can be interpreted as governing integrability properties. When a system presents a boundary that interacts with the objects in motion, the Reflection Equation (RE) - as first studied by Cherednik and Sklyanin in 1984 and 1988, respectively - is a condition that ensures the persistence of integrability. To determine a uniform procedure that produces special solutions, in 1990 Drinfel'd suggested studying the set-theoretical YBE, thus paving the way for the introduction of new combinatorial tools in the subject. In recent years, Caudrelier and Zhang initiated the investigation of the set-theoretic formulation of the RE, taking inspiration from the study of soliton collisions on the half-line. The aim of this talk is to provide a self-contained overview of the set-theoretic YBE with a particular attention to the role played by self-distributive structures. Moreover, we study the interplay between bijective non-degenerate set-theoretic solutions of the YBE and their reflections, with a focus on solutions of derived type. Based on a joint work with M. Mazzotta and P. Stefanelli.
Set-theoretic solutions of the Yang-Baxter equation and their reflections Sponsored by the Meyer Fund
This talk will be about idempotent semifields, which, from a purely algebraic perspective are just lattice-ordered groups formulated in a restricted language with the group multiplication, neutral element, and lattice-join operation. We will see that although countably infinitely many equational theories of lattice-ordered groups have a finite equational basis, no non-trivial equational theory of idempotent semifields has this property. On the other hand, as in the case of lattice-ordered groups, there are continuum-many equational theories of classes of idempotent semifields. Finally, we will relate the problem of deciding equations in the class of idempotent semifields to the problem of deciding if there exists a right order on a free group satisfying a given finite set of inequalities, and show that these problems are, respectively, co-NP-complete and NP-complete. This is joint work with Simon Santschi.
The class of congruence meet semidistributive varieties is not strong Maltsev