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.


Time: Monday, February 25, 4:10 PM
Place: SC 1312
Speaker: Adam Prenosil
Title: Introduction to abstract algebraic logic III

I will finish this series of talks about abstract algebraic logic by introducing the Leibniz hierarchy of logics, which among other classes of logics includes algebraizable, equivalential, and protoalgebraic logics. We will focus in particular on protoalgebraic logics, providing several equivalent characterizations of this class. Some of these will be be couched in semantic terms (the monotonicity of the Leibniz operator, or the closure of the class of reduced models under subdirect products), while others will involve the existence of a suitable set of formulas (which validates the rule of Modus Ponens and the axiom of Reflexivity, or defines the Leibniz congruence). As in the case of Maltsev conditions in universal algebra, the difficult part consists in showing that an abstract semantic condition such as the monotonicity of the Leibniz operator is always witnessed syntactically.


Time: Monday, March 18, 4:10 PM
Place: SC 1312
Speaker: Hayden Jananthan
Title: Posner-Robinson for Hyperjumps of Turing Degrees, Part 1 Slides
Abstract: One version of the Posner-Robinson Theorem states that for any A Turing above 0′ and non-recursive Z Turing reducible to A, there exists B such that A is Turing equivalent to B+Z is Turing equivalent to B’. In this way, any non-recursive Z is (relative to some B) a Turing jump. Here we give an unpublished proof (due to Slaman, and also proven independently by Woodin) of the hyperjump version, namely that for any A Turing above Kleene’s O and non-hyperarithmetical Z Turing reducible to A, there exists B such that A Turing equivalent to B+Z is Turing equivalent to O^B, so that any non-hyperarithmetical Z is (relative to some B) a hyperjump. In this first talk of two, I will cover some of the foundational background to make sense of this result. In the second talk, I will prove the theorem using Kumabe-Slaman forcing.


Time: Friday, March 22, 3:10 PM
Place: SC 1312
Speaker: Matthew Moore (University of Kansas)
Title: Indecision: finitely generated, finitely related clones
Abstract: A clone is a set of operations on a set that is closed under composition
and variable manipulations. There are two common methods of finitely
specifying a clone of operations. The first is to generate the clone
from a finite set of operations via composition and variable
manipulations. The second method is to specify the clone as all
operations preserving a given finite set of relations. Clones specified
in the first way are called finitely generated, and clones specified in
the second way are called finitely related.

Since the 1970s, it has been conjectured that there is no algorithmic
procedure for deciding whether a finitely generated clone is finitely
related. We give an affirmative answer to this conjecture by associating
to each Minsky machine M a finitely generated clone C such that C is
finitely related if and only if M halts, thus proving that deciding
whether a given clone is finitely related is impossible.

The full proof is quite intricate. This talk will focus on exposition
and explanation of the result, and should be accessible to a general
audience.



Time: Monday, March 25, 4:10 PM
Place: SC 1312
Speaker: Keith Kearnes (CU Boulder)
Title: Dualizable algebras in residually small congruence modular varieties.
Abstract: Call an algebra A strongly homogeneous if for any subalgebra B it is the case that any two homomorphisms of B into A differ by an automorphism of A. I will discuss the dualizability of strongly homogeneous finite algebras in residually small congruence modular varieties.


Time: Monday, April 1, 4:10 PM
Place: SC 1312
Speaker: Hayden Jananthan
Title: Posner-Robinson for Hyperjumps of Turing Degrees, Part 2
Abstract: One version of the Posner-Robinson Theorem states that for any A Turing above 0′ and non-recursive Z Turing reducible to A, there exists B such that A is Turing equivalent to B+Z is Turing equivalent to B’. In this way, any non-recursive Z is (relative to some B) a Turing jump. Here we give an unpublished proof (due to Slaman, and also proven independently by Woodin) of the hyperjump version, namely that for any A Turing above Kleene’s O and non-hyperarithmetical Z Turing reducible to A, there exists B such that A Turing equivalent to B+Z is Turing equivalent to O^B, so that any non-hyperarithmetical Z is (relative to some B) a hyperjump. In this first talk of two, I will cover some of the foundational background to make sense of this result. In the second talk, I will prove the theorem using Kumabe-Slaman forcing.


Time: Monday, April 8, 4:10 PM
Place: SC 1312
Speaker: Nick Galatos (joint work with Gavin St. John)
Title: Decidability and undecidability for residuated lattices
Abstract: Residuated lattices generalize lattice-ordered groups, Boolean and Heyting algebras and also arise as lattices of ideals of rings and lattices of binary relations. At the same time they form algebraic semantics for various non-classical logical systems. We discuss decidability questions for various varieties of residuated lattices and present both positive and negative results. The decidability results are obtained either by a proof-theoretic analysis or by the use of well-ordered sets. For undecidability results we use encodings of counter machines and also embeddings of semigroups with undecidable word problem.


Time: Monday, April 15, 4:10 PM
Place: SC 1312
Speaker: Bogdan Chornomaz
Title: Title: SSP ?= RC Slides
Abstract: Sauer-Shelah-Perles lemma says that a set family shatters at least as many sets as it has. From a lattice standpoint, this lemma implicitly considers a boolean lattice of all subsets of a base set. It does not in general hold for arbitrary lattice. We denote the class of those lattices for which SSP lemma does hold by, well, SSP and try to characterize this class somehow. At the moment, a standing conjecture (probably wrong) is that SSP = RC, and we will keep a little intrigue about what RC stands for.

This talk stems from a preprint “A Sauer–Shelah–Perles Lemma for Lattices” by Zeev Dvir, Yuval Filmus and Shay Moran, and from a brief collaboration of the speaker with these authors.


Time: Monday, April 22, 4:10 PM
Place: SC 1312
Speaker: Ralph McKenzie
Title: SSP and RC lattices
Abstract: Last week’s seminar was about two classes of finite lattices: relatively complemented lattices and lattices that have the Sauer-Shelah-Perles property. I will prove two of the characterizations of these classes that were cited in the talk.


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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