Universal Algebra and Logic

Spring 2011 Seminars

Time: Tuesday, 26 April 2011, 4:10 PM
Place: SC 1310
Speaker: William Young (Vanderbilt University)
Title: A Comparison of two Categorical Equivalences 

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.

Time: Tuesday, 19 April 2011, 4:10 PM
Place: SC 1310
Speaker: Abderezak Ould Houcine (University of Mons)
Joint Meeting with Group Theory and Topology Seminar
Title: Homogeneity in nonabelian free groups 

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.

Time: Wednesday, 13 April 2011, 4:10 PM
Place: SC 1310
Joint Meeting with Group Theory and Topology Seminar
Speaker: Mikhail Volkov (Ural State University, Russia)
Title: Trakhtman’s Road Coloring Theorem I 

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.

Time: Tuesday, 29 March 2011, 4:10 PM
Place: SC 1310
Speaker: Luca Spada (Università di Salerno)
Title: Duality, projectivity, and unification in Lukasiewicz infinite-valued propositional logic
Abstract: I will show that the unification type of Lukasiewicz (infinite-valued propositional) logic and of its equivalent algebraic semantics, the variety of MV-algebras, is nullary. The proof rests upon Ghilardi’s algebraic characterisation of unification types in terms of projective objects, recent developments in the study of finitely generated projective MV-algebras, the categorical duality between finitely presented MV-algebras and rational polyhedra and, finally, a homotopy-theoretic argument exploiting the lifting properties of the universal covering space of the circle.

Time: Tuesday, 22 March 2011, 4:10 PM
Place: SC 1310
Speaker: Simone Bova (Vanderbilt University)
Title: Complexity of Equations over Ordered Residuated Structures 

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.

Time: Tuesday, 15 March 2011, 4:10 PM
Place: SC 1310
Speaker: Ciro Russo (Università di Salerno)
Title: MV-topologies: an approach to fuzzy topologies through MV-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

Time: Tuesday, 1 March 2011, 4:10 PM
Place: SC 1310
Speaker: Matthew Smedberg (Vanderbilt University)
Title: Subdirectly irreducible finite left-distributive groupoids 

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.

Time: Tuesday, 22 February 2011, 4:10 PM
Place: SC 1310
Speaker: Costas Tsinakis (Vanderbilt University)
Title: An Abstract View of Logical Consequence Relations: A Gentle Introduction 

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.

Time: Tuesday, 15 February 2011, 4:10 PM 

No Meeting this week

Time: Tuesday, 8 February 2011, 4:10 PM
Place: SC 1310
Speaker: Steve Tschantz (Vanderbilt University)
Title: The automorphism group of a square is never Z4 

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.

Time: Tuesday, 1 February 2011, 4:10 PM
Place: SC 1310
Speaker: Alexander Wires (Vanderbilt University)
Title: Perfect Regraphs and Finite Lattice Representability 

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.

Time: Tuesday, 25 January 2011, 4:10 PM
Place: SC 1310
Speaker: Ralph McKenzie (Vanderbilt University)
Title: Definability in Substructure Orderings I 

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.

Time: Friday, 14 January 2011, 4:10 PM
Place: SC1210
Title: Organizational Meeting

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Department of Mathematics
Vanderbilt University
Stevenson Center 1326
Nashville, TN 37240

Phone: (615) 322-6672
Fax: (615) 343-0215