Universal Algebra and Logic

Spring 2019

Time: Monday, January 14, 4:10 PM
Place: SC 1312
Speaker: Ralph McKenzie
Title: Directly representable varieties, decidable varieties, varieties with few models;
characterizations
Abstract: I will present the characterization of locally finite varieties with only a
finite number of directly indecomposable members, and take a walk around
the other topics.
Time: Monday, January 28, 4:10 PM
Place: SC 1312
Speaker: Ralph McKenzie
Title: Directly representable varieties, decidable varieties, varieties with few models;
characterizations
Abstract: I will complete the characterization of locally finite varieties with only a finite number of directly indecomposable members, introduce varietal products and matrix powers (two operations on varieties derived from natural operations on their clones), and discuss some old results which naturally involve these notions.
Time: Monday, February 4, 4:10 PM
Place: SC 1312
Speaker: Adam Prenosil
Title: Introduction to abstract algebraic logic
Abstract: In this talk I will present some of the basic notions and theorems of abstract algebraic logic (AAL), a field which grew out of the investigations of mathematicians like Rasiowa, Sikorski, Blok, and Pigozzi into the relationship between logic and algebra. AAL can be thought of as a toolbox for studying non-classical logics using algebraic methods. It can also be thought of as the study of universal Horn classes in a relational signature which need not contain the equality symbol. In this talk I will try to present AAL from this perspective as a generalization of the theory of quasivarieties, focusing in particular on the role of the Leibniz operator and the so-called Leibniz hierarchy of logics.
Time: Monday, February 11, 4:10 PM
Place: SC 1312
Speaker: Adam Prenosil
Title: Introduction to abstract algebraic logic II
Abstract: Having introduced the matrix semantics of abstract algebraic logic (AAL) in the previous talk, we will now introduce the Leibniz operator and discuss the so-called Leibniz hierarchy of logics, starting with the classical results of Blok and Pigozzi describing algebraizable and protoalgebraic logics in terms of the behavior of the Leibniz operator.
Time: Monday, February 18, 4:10 PM
Place: SC 1312
Speaker: Davide Fazio
Title: Generalizing orthomodularity
Abstract: Quantum logics represent a long standing research tradition, and a vaste literature on algebraic structures related to quantum logics — sharp quantum structures (Orthomodular Posets, Orthomodular Lattices, Orthoalgebras) and unsharp quantum structures (Effect Algebras) — has appeared over the last fifty years. In this talk, we consider a generalization of the notion of orthomodularity for posets to the concept of the generalized orthomodularity property (GO-property) by considering the LU-operators. This seemingly mild generalization of orthomodular posets and its order theoretical analysis yield rather strong application to effect algebras, orthomodular posets (lattices), and Boolean posets (algebras). In presence of the GO-property, it will turn out that pastings of Boolean algebras are in fact orthomodular posets, thus establishing a novel connection between Greechie’s Theorems, orthomodular posets and the coherence law for effect algebras. It will turn out that an order theoretical notion of orthomodularity make sense, also in a weaker form, only in the framework of orthomodular posets. It would suggest that sharp and unsharp quantum logics are capable of a neat separation in terms of their order structure.

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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