# Fall 2018 Seminars

 Time: Monday, August 27th 2018, 4:10 PM 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.

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Department of Mathematics
Vanderbilt University
Stevenson Center 1326
Nashville, TN 37240
U.S.A.

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