Time: Monday, 17 February 2014, 4:10 PM
Place: SC 1432
Speaker: Matthew Moore
Title: Idempotent Congruence Modular Algebras that admit a Natural Duality have Cube Terms.

Abstract: One of the earliest results in the theory of Natural Dualities was that an algebra with a near unanimity (NU) term is dualisable. A converse to this is also true: if A is congruence distributive, then A is dualisable iff A has an NU term. In some ways, the analogue of an NU term in congruence distributive algebras is the cube term in congruence modular algebras. For this reason, it has been conjectured that a similar characterization of dualisability for congruence modular algebras should also hold. In this talk, we will discuss this conjecture, review the requisite background from Universal Algebra, and then prove the result claimed in the title.

Time: Monday, 10 February 2014, 4:10 PM
Place: SC 1432
Speaker: Antonio Ledda (University of Caligari)
Title: The Failure of Amalgamation Property for Semilinear Varieties of Residuated Lattices

Abstract: The word ”amalgamation” refers to the process of combining a pair of algebras in such a way as to preserve a common subalgebra. There are no results to date of non-commutative varieties of residuated lattices enjoying the Amalgamation Property (AP). The variety SemRL of semilinear (representable) residuated lattices, i.e., the variety generated by all totally ordered residuated lattices, seems like a natural candidate for enjoying this property, since most varieties that have a manageable representation theory and satisfy the AP are semilinear. An indication that this may not be the case comes from the fact that the variety RepLG of representable lattice-ordered groups fails the AP. Indeed, we will show that both SemRL and the variety SemCanRL of semilinear cancellative residuated lattices fail the AP. In addition, we prove that the much larger variety U of all residuated lattices with distributive lattice reduct and satisfying a particular identity also fails the AP. In fact, we will show that any subvariety of this variety fails the AP, as long as its intersection with the variety of lattice-ordered groups fails the AP.

Time: Monday, 27 January 2014, 4:10 PM
Place: SC 1308
Speaker: Alexandr Kazda
Title: Easy constraint satisfaction problems

Abstract: The talk will be a flythrough of various subclasses of polynomially solvable constraint satisfaction problems, in particular those solvable by variants of the Datalog programming language. Although the classification of these “easy” CSPs is far from complete, universal algebra has made a significant progress sorting out which CSPs lie in classes below P such as logspace and nondeterministic logspace.

Time: Wednesday, 22 January 2014, 4:10 PM
Place: SC 1308
Speaker: Matthew Moore
Title: Natural Dualities

Abstract: We begin by defining a notion of algebraic duality that is “natural” and which encompasses (amongst others) the Pontryagin duality for abelian groups and the Priestley duality for distributive lattices. Results (some recent) establishing conditions for algebras such as groups, rings and modules to be dualizable will be summarized, and theorems and general techniques used to prove that an algebra has a Natural Duality will be discussed.