In this qualifying talk, I will first give a brief introduction to the study of residuated lattices, including the recently developed concept of a conucleus on a residuated lattice. Then, a functor will be defined from the category of residuated lattices with a conucleus to the category of residuated lattices. This functor will provide the backdrop for the rest of the talk. In particular, in two specific subcategories of residuated lattices, a free object with respect to this functor will be constructed in appropriate subcategories of residuated lattice with a conucleus. The adjunctions that result from these free objects will then be shown to restrict to categorical equivalences. Lastly, some possibilities for a generalization of these results will be considered.

Abstract : We show that any free group of finite rank is homogeneous, that is if two tuples have the same type then they correspond each other by an automorphism. We also show that any non-free two-generated torsion-free hyperbolic group is existentially homogeneous and prime. This gives in particular examples of prime groups which are not QF and answers a question of A. Nies.

We shall present a recent advance in the theory of finite automata: Avraam Trahtman’s proof of the so-called Road Coloring Conjecture by Adler, Goodwyn, and Weiss. The conjecture that admits a formulation in terms of recreational mathematics arose in symbolic dynamics and has important implications in coding theory. The proof is elementary in its essence but clever and enjoyable.

The equational theory of a variety of algebras is the set of equations identically satisfied by every algebra in the variety. If a variety enjoys the finite embeddability property (every finite partial subalgebra of a member embeds into a finite member of the variety), then the equational theory is decidable. In ongoing joint-work with Constantine Tsinakis, we prove through join completions the finite embeddability property for a vaste class of ordered residuated structures (generalizing previous work by Blok, van Alten and others). In this talk, we discuss the complexity bounds arising from the construction with respect to the equational theory of two case studies, namely, Heyting and prelinear Heyting algebras.

We introduce the concept of MV-topology, a generalization of general topology to fuzzy subsets in the framework of MV-algebras, and we prove a proper extension of Stone duality to, respectively, semisimple MV-algebras and a subcategory of MV-topologies which is the fuzzy version of Stone spaces.
Shanks Workshop on Algebra and Proof Theory, amplified by Frames and Category Theory at Vanderbilt University

11-13 March 2011

No meeting Tuesday, 8 March, due to Spring Break

I will present a partial classification of the family of subdirectly irreducible groupoids with one generator in the left-distributive variety, and will discuss how this classification project connects to an open problem at the intersection of universal algebra and set theory.

Equivalences and translations between logical consequence relations abound in logic. The aim of this talk is to propose a uniform treatment of various constructions and concepts connected with the study of logical consequence relations. The approach is of order-theoretic and categorical nature, and provides a roadmap for future research in abstract algebraic logic.

No Meeting this week

This talk will present joint work with Keith Kearnes, showing that for any finite algebra A, it is never the case that Aut(A2) is isomorphic to Z4.

Pudlak and Tuma used the notion of perfect regraphs to prove that every finite lattice may be represented as a sub-lattice of a finite partition lattice(Alg. Universalis 1980). It is not known if every finite lattice may be represented as the congruence lattice of a finite algebra; nevertheless, Pudlak and Tuma(1983) showed how perfect regraphs may be used to obtain the following result: For every finite lattice L there is an N such that if L can be represented as the congruence lattice of some finite algebra, then L may be so represented on any set A where card(A) > N.

I will concentrate on the ordered set of isomorphism types of finite distributive lattices, where s < t in this ordered set is defined to mean that where s, t are the isomorphism types of A, B, we have that A is embeddable into B. It turns out that many properties of finite distributive lattices are first order definable in this ordered set.