Universal Algebra and Logic
Fall 2007 Seminars
| Time: Tuesday, November 27. (SC 1308, 4:10 – 5:30 p.m.) Speaker: George Metcalfe, Vanderbilt University Title: Interpolation and amalgamation for abelian l-groups and MV-algebras Abstract: Algebra and Logic are closely related and often mutually beneficial companions. In this talk, I will show how the logical property of interpolation can be used as a stepping stone to the algebraic property of amalgamation. In particular, I will give a simple proof of the deductive interpolation property for abelian l-groups, that provides as nice by-products, not just amalgamation, but also generation of the variety by the integers Z, and decidability. This proof can also be extended to the variety of MV-algebras, the algebraic semantics for Lukasiewicz logic. |
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| Time: Tuesday, November 13. (SC 1308, 4:10 – 5:30 p.m.) Speaker: Constantine Tsinakis, Vanderbilt Universitybr /> Title: Residuated structures: Their Algebra and Logic (Part 2) Abstract: The focus of this talk is residuated lattices and their logical counterparts, the so called substructural logics. The talk has the following objectives: (1) Show how the algebraic theory of residuated lattices can produce powerful tools for the comparative study of substructural logics. (2) Demonstrate that the bridge algebraic logic builds is beneficial to both algebra and logic. I illustrate this fact with a detailed discussion of the connection between the algebraic property of amalgamation and the logical property of interpolation. |
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| Time: Tuesday, November 6. (SC 1308, 4:10 – 5:30 p.m.) Speaker: Constantine Tsinakis, Vanderbilt University Title: Residuated structures: Their Algebra and Logic Abstract: The focus of this talk is residuated lattices and their logical counterparts, the so called substructural logics. The talk has the following objectives: (1) Show how the algebraic theory of residuated lattices can produce powerful tools for the comparative study of substructural logics. (2) Demonstrate that the bridge algebraic logic builds is beneficial to both algebra and logic. I illustrate this fact with a detailed discussion of the connection between the algebraic property of amalgamation and the logical property of interpolation. |
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| Time: Tuesday, October 30. (SC 1308, 4:10 – 5:30 p.m.) Speaker: Ralph McKenzie, Vanderbilt Universitybr /> Title: Definability in substructure orderings (Part 2) |
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| Time: Tuesday, October 9. (SC 1308, 4:10 – 5:30 p.m.) Speaker: Ralph McKenzie, Vanderbilt University Title: Definability in substructure orderings Abstract: This is joint work with Jaroslav Jezek. Let P be the ordered set of isomorphism types of finite ordered sets, ordered by embeddability. We show that every isomorphism-invariant relation between posets that is first-order definable in the category of finite posets (where the morphisms are the monotone maps), is first-order definable up to duality in P. This implies, but is stronger than, the conclusion that every self-dual class of finite posets axiomatized by a second-order sentence in the language of posets is first-order definable in P. In particular, for every finite poset A, the set consisting of the isomorphism type of A and the isomorphism type of the dual of A, is first-order definable in the ordered set P. We prove analogous but slightly weaker results for the ordered sets consisting of isomorphism types of finite lattices (or semilattices, or distributive lattices, respectively) ordered by embeddability. In all cases, the individual isomorphism types are definable up to duality, and duality is the only automorphism of the ordered set of types. |
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| Time: Tuesday, October 2. (SC 1308, 4:10 – 5:30 p.m.) Speaker: Petar Markovic, University of Novi Sad, Serbia Title: Bounded width CSP: An overview Abstract: We will present one of the most general algorithms for solving the Constraint Satisfaction Problem in polynomial time, give a boundary on the cases where it may be attempted and prove it works in some of these cases. The attendees will be assumed to have heard the previous lecture in this seminar, though some definitions will be repeated. |
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| Time: Tuesday, September 18. (SC 1312, 4:10 – 5:30 p.m.) Speaker: Petar Markovic, University of Novi Sad, Serbia Title: Complexity of the constraint satisfaction problem, an universal-algebraic approach Abstract: This lecture will review the currect state of knowledge on a major problem of computational complexity: whether any (restricted) constraint satisfaction problem can have only two complexities, polynomial and NP-complete. We will discuss an approach using the techniques of universal algebra, which has proved most successful so far. We will review the latest known results, including the one by the speaker (in collaboration with C. Carvalho and V. Dalmau) which was proved in August 2007. |
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| Time: Tuesday, September 11. (SC 1312, 4:10 – 5:30 p.m.) Speaker: Franco Montagna, University of Siena, Italy Title: Representations of GBL-algebras Abstract: My objective is to develop different types of representations for BL and GBL-algebras. The presentation will be self-contained and include definitions and detailed discussion of the main mathematical structures arising in this talk. These representations will be used to establish various decidability results. |
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