Universal Algebra and Logic

Spring 2007 Seminars

Time: Tuesday, April 24. (SC 1312, 4:10 – 5:30 p.m.)
Speaker: Ciro Russo, University of Salerno, Italy
Title: An order-theoretical approach to image processing
Abstract: We show that an order-theoretical and algebraic approach to some techniques of image compression – that make use of the theory of fuzzy relation equations – gives rise to interesting results in the theory of modules over residuated lattices. Such results have, in their turn, effects on the applications. Moreover, this approach yields new methods and techniques, and strongly motivates a deep investigation of the mathematical concepts involved.

Time: Tuesday, April 17. (SC 1312, 4:10 – 5:30 p.m.)
Speaker: Jaroslav Jezek, Charles University, Czech Republic
Title: Definability for semilattices.
Abstract: We prove that the lattice of universal classes of semilattices has only the identical automorphism and its finitely generated and finitely axiomatizable elements are definable.

Time: Tuesday, April 10. (SC 1312, 4:10 – 5:30 p.m.)
Speaker: George Metcalfe, Vanderbilt University
Title: Herbrand’s Theorem and Skolemization by Approximation for First-Order Lukasiewicz Logic
Abstract: Herbrand’s theorem and Skolemization are key results for first-order classical logic. In this talk I will explain how they transfer to the case of infinite-valued first-order Lukasiewicz logic. We will see that while the usual Herbrand theorem fails, an approximate Herbrand theorem can be proved and used to establish Skolemization.

Time: Tuesday, April 3. (SC 1312, 4:10 – 5:30 p.m.)
Speaker: Anton Klyachko, Moscow State University.
Title: Groupification of universal algebras
Abstract: We show that many universal algebras admit group structures such that all operations can be expressed via multiplication, inverse, and constants.

Time: Tuesday, March 20. (SC 1312, 4:10 – 5:30 p.m.)
Speaker: Yuri Bahturin, Memorial University, Canada.
Title: Large Lie algebras and Burnside type problems
Abstract: I will speak on the results of our joint work with Alexander Olshanskii. I will define restricted Lie algebras and show that some problems about Lie algebras in characteristic p>0 can be reduced to restricted Lie algebras. The advantage of restricted Lie algebras is that their properties are closer to groups. This allows us to use group theoretic approaches, in particular the approach of Olshanskii – Osin that allowed to construct new examples of Burnside type groups. We introduce and study large restricted Lie algebras, that is, algebras each of which contains a subalgebra of finite codimension that maps onto a nonabelian free restricted algebra. We describe a procedure that allows us to construct finitely generated restricted nil algebras that are direct limits of large algebras. This yields apparently new examples of finitely generated nil (=Engel) Lie algebras.

Time: Tuesday, February 20. (SC 1312, 4:10 – 5:30 p.m.)
Speaker: Gabor Kun, RWTH Aachen University, Germany.
Title: NP by forbidden lists
Abstract: We present three definitions of the class NP in terms of homomorphisms of colored digraphs: in terms of injective homomorphisms, full homomorphisms and colorings of pairs, respectively. We apply this to special syntactically defined subclasses of NP. The most of the applications of our viewpoint are about the relationship of the class of Constraint Satisfaction Problems (CSP) and Monotone Monadic SNP (MMSNP) defined by Feder and Vardi. Our setting streamlines the analysis of these languages. We give a characterization (with simple proof) of MMSNP languages which are actually CSP languages. We show that an MMSNP language of digraphs restricted to a bounded expansion class (a generalization of bounded degree and minor closed classes) is actually a CSP language.

Time: Tuesday, February 13. (SC 1312, 4:10 – 5:30 p.m.)
Speaker: Ralph McKenzie, Vanderbilt University
Title: Avoidable distributive lattices and nicely structured ordered sets

Time: Tuesday, January 30. (SC 1312, 4:10 – 5:30 p.m.)
Speaker: Dan Guralnik, Vanderbilt University
Title: Reflections on memory, language, logic and non-positively curved cube complexes
Abstract: In this talk I will try to share with you some of my naive ideas about how one could try to bring our understanding of natural languages, cognitive models of memory, ideas from Ethology — the study of animal behaviour — and information theory under the umbrella of a formal mathematical theory describing the connections between the above. The talk will have two parts. In the first, I will introduce the mathematical tool — the Sageev-Roller duality between partially-ordered sets with an involution (poc-sets) and cubical complexes of non-positive curvature (cubings, for short). In the second part, I will discuss the application.

Time: Tuesday, January 23. (SC 1312, 4:10 – 5:30 p.m.)
Speaker: Wieslaw Dziobiak, University of Puertro Rico, Mayaguez
Title: One More Proof of Willard’s Finite Basis Theorem
Abstract: Willard’s finite basis theorem states that the set of universally valid equations in a congruence meet semi-distributive and residually very finite variety of algebras of finite signature has a finite equational base (R. Willard, JSL 65 (2000), 187-200). The goal of the talk is to present a short proof of Willard’s theorem. The crucial ingredient of the presented proof is a consequnce of a result that is contained in the paper by W. Dziobiak, M. Maroti, R. McKenzie, and A. Nurakunov, Fund. Math. (to appear). Other known new proofs of Willard’s theorem are contained in the papers by K. Baker, G. McNulty, and Ju. Wang (AU 52 (2004), 289-302) and M. Maroti and R. McKenzie (SL 78 (2004), 393-320).

Time: Tuesday, February 6. (SC 1312, 4:10 – 5:30 p.m.)
Speaker: Jaroslav Jezek, Charles University, Czech Republic
Title: Avoidable posets, semilattices and lattices
Abstract: A finite structure A is said to be avoidable in a class of structures K if there exists an infinite set S of finite structures from K such that A belongs to S and no structure from S can be embedded into another structure from S. We will speak about avoidable and unavoidable posets, semilattices and lattices.

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

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