Logic Seminar Abstracts
Title: Abelian algebras in relatively modular quasivarieties, 1
Speaker: Keith Kearnes
Affiliation: CU
Time: 3pm, Monday, February 4
Location: Math 220
Abstract:
I will talk about the structure of abelian algebras in relatively congruence modular (RCM) quasivarieties, and discuss the question of whether every finitely generated abelian RCM quasivariety is finitely axiomatizable.
Title: Abelian algebras in relatively modular quasivarieties, 2
Speaker: Keith Kearnes
Affiliation: CU
Time: 3pm, Monday, February 11
Location: Math 220
Abstract:
Continuation.
Title: Idempotent equational theories and their orderable models
Speaker: Ross Willard
Affiliation: U Waterloo
Time: 3pm, Monday, February 18
Location: Math 220
Abstract:
Let
be an equational theory.
A model of
is said to orderable if there
exists a nontrivial partial ordering of its universe with respect to which
every operation is monotone. We prove that if
is idempotent (meaning
that for every operation symbol f in its language,
implies the
idempotent law
)
and has an orderable model, then
has a
2-element orderable model. This solves a problem about the lattice of
interpretability types of varieties posed in a recent monograph of Kearnes
and Kiss.
Title: Abelian algebras in relatively modular quasivarieties, 3
Speaker: Keith Kearnes
Affiliation: CU
Time: 3pm, Monday, February 25
Location: Math 220
Abstract:
Continuation.
Title: Abelian algebras in relatively modular quasivarieties, 4
Speaker: Keith Kearnes
Affiliation: CU
Time: 3pm, Monday, March 4
Location: Math 220
Abstract:
Continuation.
Title: A consistent atomless Boolean algebra
Speaker: Kevin Selker
Affiliation: CU
Time: 3pm, Monday, March 11
Location: Math 220
Abstract:
The small ideal-independence of A, denoted smm(A), is the least size of a maximal ideal-independent subset of A. It is closely related to the invariant spread, which is the size of the largest discrete subspace of Spec(A). The ultrafilter number of A, denoted u(A), is the minimal size of an ultrafilter generating set. Assuming CH we construct an atomless Boolean algebra A with smm(A)<u(A). Time permitting, we will discuss how the construction generalizes to another cardinal function, f(A).
Title: A consistent atomless Boolean algebra, 2
Speaker: Kevin Selker
Affiliation: CU
Time: 3pm, Monday, March 18
Location: Math 220
Abstract:
Continuation.
Title: Conformal algebras, vertex algebras, and multi-valued logic
Speaker: Jonathan Smith
Affiliation: Iowa State University
Time: 3pm, Monday, April 1
Location: Math 220
Abstract:
Vertex (operator) algebras, and their conformal algebra reducts, have emerged as important structures in conformal field theory, moonshine, combinatorics, and elsewhere. Their specifications are not strictly algebraic, since an existential quantifier is involved in the property known as locality.
In a new algebraic approach, conformal algebras and vertex algebras are extended to two-sorted structures, with an additional component encoding the logical properties of locality. Within these algebras, locality is expressed as an identity, avoiding the existential quantifier used in the classical axiomatizations.
Two-sorted conformal algebras form a variety of two-sorted algebras, equivalent to a Mal'tsev variety of single-sorted algebras. Motivated by a question of Griess, subalgebras of reducts of conformal algebras are shown to satisfy a set of quasi-identities.
The formulation of vertex algebras involves multi-valued logic.
The class of two-sorted vertex algebras does not form a
variety, so open problems concerning the nature of that class
are posed.
Title: On constructions of modes
Speaker: Anna Romanowska
Affiliation: Warsaw Technical University
Time: 3pm, Monday, April 8
Location: Math 220
Abstract:
Modes are idempotent and entropic algebras. Typical examples come from geometry (affine spaces, convex sets) and combinatorics (quasigroup modes, many types of groupoids). I will first recall the basic definitions, examples and properties, and then focus on describing certain characteristic constructions of modes and (older and newer) results concerning them. I will also mention some further developments and open problems.
Title: Chain conditions determined by base properties
Speaker: Santi Spadaro
Affiliation: University of Opava, Czech Republic
Time: 3pm, Monday, April 15
Location: Math 220
Abstract:
We will review joint work with Kojman and Milovich about some topological cardinal invariants of an order-theoretic flavor whose behavior is similar to that of the cellularity (the supremum of the cardinalities of pairwise disjoint families of non-empty open sets), especially in the class of homogeneous compacta.
Title: A small game on Cantor space
Speaker: Charlie Scherer
Affiliation: CU
Time: 3pm, Monday, April 22
Location: Math 220
Abstract:
In 1981, Baumgartner and Komjáth published the result that if a Boolean algebra has only countable incomparable sets then it has a countable dense set. The converse is clearly false (consider the real interval algebra). However, I will argue that if an atomless Boolean algebra has a countable dense set then it has a countable incomparable set that is maximal with respect to inclusion.