Universal Algebra and Logic

Fall 2004 Seminars

Time: Thursday, December 9. (SC 1404, 2:30 p.m.)
Speaker: Ralph McKenzie
Title: Does there exist a non-trivial meet-semi-distributive algebraic lattice that has no non-trivial meet-prime element? (Part II)
Time: Tuesday, December 7. (SC 1403, 2:10 p.m.)
Speaker: Ralph McKenzie
Title: Does there exist a non-trivial meet-semi-distributive algebraic lattice that has no non-trivial meet-prime element?
Time: Thursday, December 2. (SC 1404, 2:30 p.m.)
Speaker: Ralph McKenzie
Title: Some special examples of algebraic lattices.
Time: Tuesday, November 30. (SC 1403, 2:10 p.m.)
Speaker: David Stanovsky
Title: Commutative idempotent residuated lattices.
Abstract: We will show a couple of interesting properties of the variety of residuated lattices with commutative and idempotent multiplication. In particular, they satisfy no non-trivial lattice equation, they contain a non-finitely based subvariety, they contain two minimal subvarieties and I can describe those with totally ordered lattice reduct.
Time: Thursday, November 18. (SC 1404, 2:30 p.m.)
Speaker: David Stanovsky
Title: Idempotent left distributive left quasigroups, Part II.
Abstract: We continue our analysis of left distributive left quasigroups with the idempotent case.
Time: Tuesday, November 16. (SC 1403, 2:10 p.m.)
Speaker: David Stanovsky
Title: Non-idempotent left distributive left quasigroups.
Abstract: Left distributive left quasigroups are binary algebras, where all left translations are automorphisms. The main goal of the talk is a description of subdirectly irreducible non-idempotent left distributive left quasigroups.
Time: Thursday, November 11. (SC 1404, 2:30 p.m.)
Speaker: Nikolaos Galatos
Title: Residuated lattices and substructural logics, Part IV.
Abstract: We present two deductive logical systems, one formulated in Gentzen and the second in Hilbert style. We show that these systems are equivalent and that they constitute equivalent algebraic semantics for the class of non-associative residuated lattices. Moreover, we show, in an algebraic way, that the Gentzen system has the cut-elimination property and the Hilbert system has the strong separation property.
Time: Tuesday, November 9. (SC 1403, 2:10 p.m.)
Speaker: Nikolaos Galatos
Title: Residuated lattices and substructural logics, Part III.
Abstract: We present two deductive logical systems, one formulated in Gentzen and the second in Hilbert style. We show that these systems are equivalent and that they constitute equivalent algebraic semantics for the class of non-associative residuated lattices. Moreover, we show, in an algebraic way, that the Gentzen system has the cut-elimination property and the Hilbert system has the strong separation property.
Time: Thursday, November 4. (SC 1404, 2:30 p.m.)
Speaker: Nikolaos Galatos
Title: Residuated lattices and substructural logics, Part II.
Abstract: We present two deductive logical systems, one formulated in Gentzen and the second in Hilbert style. We show that these systems are equivalent and that they constitute equivalent algebraic semantics for the class of non-associative residuated lattices. Moreover, we show, in an algebraic way, that the Gentzen system has the cut-elimination property and the Hilbert system has the strong separation property.
Time: Tuesday, November 2. (SC 1403, 2:10 p.m.)
Speaker: Nikolaos Galatos
Title: Residuated lattices and substructural logics.
Abstract: We present two deductive logical systems, one formulated in Gentzen and the second in Hilbert style. We show that these systems are equivalent and that they constitute equivalent algebraic semantics for the class of non-associative residuated lattices. Moreover, we show, in an algebraic way, that the Gentzen system has the cut-elimination property and the Hilbert system has the strong separation property.
Time: Thursday, October 28. (SC 1404, 2:30 p.m.)
Speaker: Bill Lampe
Title: Some congruence lattice representation problems.
Abstract: The focus will be on the congruence-lattices-of-algebras-having-a-one-element-subalgebra problem and how it relates to other congruence lattice representation problems.
Time: Tuesday, October 26. (SC 1403, 2:10 p.m.)
Speaker: Constantine Tsinakis
Title: An extension of W. C. Holland’s theorem to residuated lattices, Part II.
Abstract: We present sufficient conditions for a residuated lattice (RL) to be represented as an RL of residuated maps on a chain. In particular, we show that every GBL-algebra satisfying the pre-linearity law and every GMV algebra has such a representation. These results extend W. C. Holland’s corresponding result for lattice-ordered groups and require careful analysis of the algebraic closure system of convex subalgebras of an RL.
Time: Thursday, October 21. (SC 1404, 2:30 p.m.)
Speaker: Constantine Tsinakis
Title: An extension of W. C. Holland’s theorem to residuated lattices.
Abstract: Let U be the variety of residuated lattices satisfying the identities x/x=x\x=e and (x\y join y\x) meet e = e (the pre-linearity law). We prove that an algebra in U can be represented as a residuated lattice of residuated maps on a chain if and only if it satisfies the identity x(y meet z)w = xyw meet xzw. In particular, every GBL-algebra satisfying the pre-linearity law and every GMV algebra has such a representation. These results extend W. C. Holland’s corresponding result for lattice-ordered groups. The proof is based on the fact that the compact element of the algebraic closure system of convex subalgebras of an algebra in U form a relatively normal lattice.
Time: Thursday, October 14. (SC 1404, 2:30 p.m.)
Speaker: Miklos Maroti (joint work with Ralph McKenzie)
Title: Finite basis problems and results for quasivarieties.
Abstract: We will continue our presentation on finite basis theorems for quasivarieties.
Time: Tuesday, October 12. (SC 1403, 2:10 p.m.)
Speaker: Miklos Maroti (joint work with Ralph McKenzie)
Title: Finite basis problems and results for quasivarieties.
Abstract: We will continue our presentation on finite basis theorems for quasivarieties.
Time: Thursday, October 7. (SC 1404, 2:30 p.m.)
Speaker: Miklos Maroti (joint work with Ralph McKenzie)
Title: Finite basis problems and results for quasivarieties.
Abstract: We will continue our presentation on finite basis theorems for quasivarieties.
Time: Tuesday, October 5. (SC 1403, 2:10 p.m.)
Speaker: Miklos Maroti (joint work with Ralph McKenzie)
Title: Finite basis problems and results for quasivarieties.
Abstract: We will continue our presentation on finite basis theorems for quasivarieties.
Time: Thursday, September 30. (SC 1404, 2:30 p.m.)
Speaker: Miklos Maroti (joint work with Ralph McKenzie)
Title: Finite basis problems and results for quasivarieties.
Abstract: An element x of a lattice is pseudo-prime if whenever the meet of y and z is zero then either y of z is below x. We will show that for algebraic lattices, PCC is equivalent to the property that the meet of the pseudo-prime elements is zero. Then we will continue our presentation on finite basis theorems for quasivarieties.
Time: Tuesday, September 28. (SC 1403, 2:10 p.m.)
Speaker: Miklos Maroti (joint work with Ralph McKenzie)
Title: Finite basis problems and results for quasivarieties.
Abstract: We offer new proofs and generalizations of two important finite basis theorems of Don Pigozzi and Ross Willard. Our proofs rely on the study of several congruence properties for quasivarieties, including having pseudo-complemented congruence lattices (PCC) and the weak extension principle (WEP). We show that congruence meet semi-distributive varieties have PCC, and relatively congruence distributive quasivarieties have both the PCC and the WEP.
Time: Thursday, September 23. (SC 1404, 2:30 p.m.)
Speaker: Marcin Kozik
Title: Modeling of Turing machine computations in finite algebras.
Abstract: We present a way to model a computation of an arbitrary Turing machine inside a cartesian product of some finite algebra. We follow an approach of Ralph McKenzie and show that, under certain conditions, the set of elements of “full support” corresponds exactly to tape configurations for some computation of the Turing machine.
Time: Tuesday, September 21. (SC 1403, 2:10 p.m.)
Speaker: Ralph McKenzie (joint work with Marcel Jackson)
Title: Interpreting graph colourability in finite semigroups.
Abstract: Ralph McKenzie will be finishing his proof that for the 55-element semigroup S constructed in last Thursday’s seminar, the problem to determine of a finite semigroup if it is in HSP(S) is NP-hard.
Time: Thursday, September 16. (SC 1404, 2:30 p.m.)
Speaker: Ralph McKenzie (joint work with Marcel Jackson)
Title: Interpreting graph colourability in finite semigroups.
Abstract: We show that a number of natural membership problems for classes associated with finite semigroups are computationally difficult. In particular, we construct a 55-element semigroup S such that the finite membership problem for the variety of semigroups generated by S interprets the graph 3-colourability problem.
Time: Tuesday, September 14. (SC 1403, 2:10 p.m.)
Speaker: Eric Zenk
Title: Overview of “Subset systems and generalized distributive lattices”.
Abstract: A subset system is a rule which picks a family of subsets of each poset, with the feature that (order preserving) images of selected sets are selected. These can be used to describe (possibly infinitary) algebras where the only operations are taking meets and joins of selected sets. The talk will give several examples of subset systems, then briefly indicate the methods one uses to prove things like: the existence of free algebras in some of these categories, the existence and properties of quotients (surjective images), and the existence of limits (i.e., products and equalizers) in these categories.
Time: Thursday, September 9. (SC 1404, 2:30 p.m.)
Speaker: Constantine Tsinakis (joint work withAnnika Wille)
Title: Minimal Varieties of Involutive Residuated Lattices.
Abstract: We establish the existence uncountably many atoms in the subvariety lattice of the variety of involutive residuated lattices. The proof utilizes a construction used in the proof of the corresponding result for residuated lattices and is based on the fact that every residuated lattice with greatest element can be associated in a canonical way with an involutive residuated lattice.
Time: Tuesday, September 7. (SC 1403, 2:10 p.m.)
Speaker: Jaroslav Jezek (joint work with Petar Markovic and David Stanovsky)
Title: Homomorphic images of finite subdirectly irreducible unary algebras.
Abstract: We prove that a finite unary algebra with at least two operation symbols is a homomorphic image of a (finite) subdirectly irreducible algebra if and only if the intersection of all its subalgebras which have at least two elements is nonempty.
Time: Tuesday, September 7. (SC 1403, 2:10 p.m.)
Speaker: Will Funk
Title: Tutorial on residuated lattices
Abstract: We will begin with the presentation of the definition of the variety of residuated lattices and give some examples of well-known algebraic structures which can be regarded as residuated lattices. Some results concerning the congruence lattice of residuated lattices is discussed. Finally we give a brief overview of the relationship between congruences and convex normal subalgebras of residuated lattices.

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Contact Information

Department of Mathematics
Vanderbilt University
Stevenson Center 1326
Nashville, TN 37240
U.S.A.

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