Universal Algebra and Logic

Fall 2020 Seminars

Time: Monday, December 7, 2–3 PM (CST, UTC -6)
Place: Zoom
Speaker: Jason Parker (Brandon University)
Title: Isotropy Groups of Quasi-Equational Theories
Slides: here
Abstract: In [2], my PhD supervisors (Pieter Hofstra and Philip Scott) and I studied the new topos-theoretic phenomenon of isotropy (as introduced in [1]) in the context of single-sorted algebraic theories, and we gave a logical/syntactic characterization of the isotropy group of any such theory, thereby showing that it encodes a notion of inner automorphism or conjugation for the theory. In the present talk, I will summarize the results of my recent PhD thesis, in which I build on this earlier work by studying the isotropy groups of (multi-sorted) quasi-equational theories (also known as essentially algebraic, cartesian, or finite limit theories). In particular, I will show how to give a logical/syntactic characterization of the isotropy group of any such theory, and that it encodes a notion of inner automorphism or conjugation for the theory. I will also describe how I have used this characterization to exactly characterize the ‘inner automorphisms’ for several different examples of quasi-equational theories, most notably the theory of strict monoidal categories and the theory of presheaves valued in a category of models. In particular, the latter example provides a characterization of the (covariant) isotropy group of a category of set-valued presheaves, which had been an open question in the theory of categorical isotropy.
[1] J. Funk, P. Hofstra, B. Steinberg. Isotropy and crossed toposes. Theory and Applications of Categories 26, 660–709, 2012.
[2] P. Hofstra, J. Parker, P.J. Scott. Isotropy of algebraic theories. Electronic Notes in Theoretical Computer Science 341, 201–217, 2018.

Time: Monday, September 28, 2–3 PM (CDT, UTC -5)
Place: Zoom
Speaker: Bogdan Chornomaz
Title: Introduction to Möbius functions
Slides: here
Abstract: We will give a brief introduction to Möbius functions on lattices and discuss their basic properties. We will also prove that lattices with nonvanishing Möbius function are SSP and that geometric lattices have a nonvanishing (moreover, alternating) Möbius function.

Time: Monday, September 21, 2–3 PM (CDT, UTC -5)
Place: Zoom
Speaker: Bogdan Chornomaz
Title: SSP ?= RC
Slides: here
Preprint: here
Abstract: We will talk about SSP ?= RC conjecture, which states that a finite lattice satisfies an analogue of Sauer-Shelah-Perles lemma iff it is relatively complemented. In particular, we discuss one strategy of attacking the conjecture using certain colored bipartite graphs. We will hit the wall by constructing a “counterexample” in the language of those graphs. Then we will try to lift it to the counterexample to the general conjecture using some “pumping” constructions and will hit the wall again.

Time: Monday, September 14, 2–3 PM (CDT, UTC -5)
Place: Zoom
Speaker: Adam Prenosil
Title: Four-valued logics of truth, non-falsity, and material equivalence
Slides: here
Abstract: the purpose of this talk is to demonstrate how to work with quasivarieties (universal Horn classes) in a relational signature which consists of more than one predicate symbol. While universal algebraists work with the binary equality predicate, and algebraic logicians work with the unary truth predicate, the four-valued Belnap–Dunn logic provides a natural setting where two unary predicates arise (namely the truth and non-falsity predicates), as well as an equality predicate. We show how to axiomatize the logic of truth and non-falsity, as well as the logic of truth and equality, determined by the four-valued algebraic semantics of Belnap and Dunn.

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

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