# 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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