Universal Algebra and Logic
Fall 2018 Seminars
| Time: Monday, August 27th 2018, 4:10 PM HigherCong Place: SC 1312 Speaker: Andrew Moorhead Title: The relationship of supernilpotence to nilpotence (1) Abstract: Supernilpotence is a condition on an algebra that is definable with a higher arity commutator that generalizes the classical binary commutator for general algebras. Supernilpotent algebras have received attention lately because of theorems of the flavor ‘nice property true satisfied by finite nilpotent groups’ is satisfied by finite supernilpotent Mal’cev algebras of finite type. For example, it is now know that there is a polynomial time algorithm to solve the equation equation satisfiability problem for such algebras. The exact relationship between supernilpotence and nilpotence had been unclear. We will discuss how supernilpotence implies nilpotence for algebras with a Taylor term, but that in general the two notions are independent. |
| Time: Monday, September 10th 2018, 4:10 PM Place: SC 1312 Speaker: Andrew Moorhead Title: The relationship of supernilpotence to nilpotence (2) Abstract: Supernilpotence is a condition on an algebra that is definable with a higher arity commutator that generalizes the classical binary commutator for general algebras. Supernilpotent algebras have received attention lately because of theorems of the flavor ‘nice property true satisfied by finite nilpotent groups’ is satisfied by finite supernilpotent Mal’cev algebras of finite type. For example, it is now know that there is a polynomial time algorithm to solve the equation equation satisfiability problem for such algebras. The exact relationship between supernilpotence and nilpotence had been unclear. We will discuss how supernilpotence implies nilpotence for algebras with a Taylor term, but that in general the two notions are independent. |
| Time: Monday, September 17th 2018, 4:10 PM Place: SC 1312 Speaker: Andrew Moorhead Title: The relationship of supernilpotence to nilpotence (3) Abstract: Supernilpotence is a condition on an algebra that is definable with a higher arity commutator that generalizes the classical binary commutator for general algebras. Supernilpotent algebras have received attention lately because of theorems of the flavor ‘nice property true satisfied by finite nilpotent groups’ is satisfied by finite supernilpotent Mal’cev algebras of finite type. For example, it is now know that there is a polynomial time algorithm to solve the equation equation satisfiability problem for such algebras. The exact relationship between supernilpotence and nilpotence had been unclear. We will discuss how supernilpotence implies nilpotence for algebras with a Taylor term, but that in general the two notions are independent. |
| Time: Monday, September 24th 2018, 4:10 PM Place: SC 1312 Speaker: Andrew Moorhead Title: The relationship of supernilpotence to nilpotence (4) Abstract: Supernilpotence is a condition on an algebra that is definable with a higher arity commutator that generalizes the classical binary commutator for general algebras. Supernilpotent algebras have received attention lately because of theorems of the flavor ‘nice property true satisfied by finite nilpotent groups’ is satisfied by finite supernilpotent Mal’cev algebras of finite type. For example, it is now know that there is a polynomial time algorithm to solve the equation equation satisfiability problem for such algebras. The exact relationship between supernilpotence and nilpotence had been unclear. We will discuss how supernilpotence implies nilpotence for algebras with a Taylor term, but that in general the two notions are independent. |
| Time: Monday, October 1st 2018, 4:10 PM Place: SC 1312 Speaker: Adam Prenosil Title: Complemented envelopes of commutative bimonoids (Part I) Abstract: A recurrent theme in (especially ordered) algebra is embedding algebraic structures in which certain elements are “missing” into richer structures: completions and densifications of ordered structures are examples of this phenomenon. In this talk we consider embeddings into complemented and into complete complemented structures. Classical examples of such extensions are the group of differences of a cancellative commutative monoid and the Boolean envelope of a distributive lattice. We show how such examples fit into a common framework of complemented extensions of what we call bimonoids. It turns out that each commutative bimonoid embeds in a canonical doubly dense way into what we call its complemented Dedekind–MacNeille completion. If the bimonoid is already complemented (e.g. a Boolean algebra), this construction coincides with the ordinary Dedekind–MacNeille completion. |
| Time: Monday, October 8th 2018, 4:10 PM Place: SC 1312 Speaker: Adam Prenosil Title: Complemented envelopes of commutative bimonoids (Part II) Abstract: Having introduced the complemented Dedekind–MacNeille completion of a commutative bimonoid in the previous talk, we now look inside this completion for some tighter complemented envelopes. In particular, it is natural to ask if a commutative bimonoid has an envelope akin to the group of differences, where each element has the form a – b. We provide some sufficient conditions for the existence of such complemented envelopes and use them to obtain categorical equivalences between varieties of integral and involutive residuated structures, unifying and extending some existing equivalences. |
| Time: Monday, October 15th 2018, 4:10 PM Place: SC 1312 Speaker: Adam Prenosil Title: Complemented envelopes of commutative bimonoids (Part III) Abstract: Having introduced the complemented Dedekind–MacNeille completion of a commutative bimonoid in the previous talk, we now look inside this completion for some tighter complemented envelopes. In particular, it is natural to ask if a commutative bimonoid has an envelope akin to the group of differences, where each element has the form a – b. We provide some sufficient conditions for the existence of such complemented envelopes and use them to obtain categorical equivalences between varieties of integral and involutive residuated structures, unifying and extending some existing equivalences. |
| Time: Monday, October 22 2018, 4:10 PM VC-Dim-2018 Place: SC 1312 Speaker: Bogdan Chornomaz Title: Extremal problems for lattices with bounded VC dimension. Abstract: A natural question about finite lattices is how large they can be with respect to the size of the underlying set of join-irreducible elements. An obvious exponential bound is reachable simply by considering a boolean lattice. Yet, the question remains about what “causes” a lattice to be exponentially big. In this talk we will argue that the only reason for that is the growth of its Vapnik-Chervonekis (VC) dimention. Having restricted this dimension, we will prove a polynomial bound on size, show that this bound is tight and characterize the class of “extremal” lattices, reaching this bound. Given time, we will also show that generalized extremality yield a combinatorial characterization of convex geometries. |
| Time: Monday, October 29 2018, 4:10 PM Place: SC 1312 Speaker: Bogdan Chornomaz Title: Meet-irreducible elements in extremal lattices. Abstract: When estimating the size of a lattice, it is natural to consider not only the number of its join-irreducible, but also of its meet-irreducible elements. This generalization turns out to be much harder, so in this talk we will point out a bunch of unsolved problems in this direction, accompanied by the small number of positive results. Those are, to put it lightly, very modest comparing to the general scope. Still, to the speaker’s opinion they are neat and worth talking about. |
| Time: Monday,November 5th 2018, 4:10 PM Place: SC 1312 Speaker: Bogdan Chornomaz Title: Meet-irreducible elements in extremal lattices. (Part 2) Abstract: When estimating the size of a lattice, it is natural to consider not only the number of its join-irreducible, but also of its meet-irreducible elements. This generalization turns out to be much harder, so in this talk we will point out a bunch of unsolved problems in this direction, accompanied by the small number of positive results. Those are, to put it lightly, very modest comparing to the general scope. Still, to the speaker’s opinion they are neat and worth talking about. |
| Time: Monday,November 12th 2018, 4:10 PM Place: SC 1312 Speaker: Bogdan Chornomaz Title: Meet-irreducible elements in extremal lattices. (Part 3) Abstract: When estimating the size of a lattice, it is natural to consider not only the number of its join-irreducible, but also of its meet-irreducible elements. This generalization turns out to be much harder, so in this talk we will point out a bunch of unsolved problems in this direction, accompanied by the small number of positive results. Those are, to put it lightly, very modest comparing to the general scope. Still, to the speaker’s opinion they are neat and worth talking about. |
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