The algebraic closure of a field K played an important role in classical field theory since it provided a universe for the class of finite algebraic extensions of K or, in other words, for the class of fields which are finitely generated as modules over K. The notion of algebraic closure can be generalised to arbitrary categories as follows. Let C be a category of models (of a fixed language) and let lambda be a cardinal number. We term a model H in C lambda-universal in C if every model in C generated by fewer than lambda generators is embeddable in H and conversely, every subgroup of H generated by fewer than lambda generators belongs to C. In this talk I will show that given an arbitrary model M (of a functional language) the aleph_0-universal model exists for the category of models discriminated by M answering a question raised by Baumslag, Miasnikov and Remeslennikov. We will further discuss the aleph_0-universal group for the class of fully residually free groups and give a model-theoretic description of the Lyndon’s free group.

TBA
No Meeting Friday, 26 November 2010, 3:10 PM

Thanksgiving Break

While it is always abstractly true that an algebraic variety (that is, a first-order class axiomatized by equations) possesses free algebras on any given number of generators, it is often a difficult problem to explicitly represent these algebras. I will discuss an example of a problem of this type, in which the free algebra on one generator in the variety axiomatized by the one equation

x * (y * z) == (x * y) * (x * z)

was only first represented using the help of a very strong large cardinal axiom. That construction is due to Laver. Subsequently Dehornoy achieved a representation of the same algebra in ZFC; however, we will discuss several open problems for this variety which are proved only modulo Laver’s large cardinal assumption.

In 2008, Siggers surprised the universal algebra community by showing that the well-known “Taylor term” Mal’cev condition, when restricted to locally finite varieties, is equivalent to a strong Mal’cev condition involving a single 6-ary term. Subsequent improvements have been given by Markovic and McKenzie, Kearnes, and Maroti. Since then, Valeriote et al have shown that congruence meet semi-distributivity is also characterizable at the locally finite level by a strong Mal’cev condition, but that no other “omitting TCT types” conditions can be so characterized. In this lecture we describe a construction which proves that if C is a strong Mal’cev condition satisfied by every locally finite variety which is n-permutable for some n and has a near-unanimity term, then C is also satisfied by the variety of semilattices. In particular, neither “congruence distributivity” nor “congruence distributivity + n-permutability for some n” can be characterized by a strong Mal’cev condition at the locally finite level.

A partially ordered set (poset) L is called an extension of the poset P provided P is a subset of L and the order of L restricts to that of P. In case every element of L is a join of elements of L, we say that L is a join-extension of P. We use the term join-completion for a join-extension that is a complete lattice. The concepts of a meet-extension and a meet-completion are defined dually. Join-completions were introduced by B. Banaschewski in 1956, and they are intimately related to representations of complete lattices studied systematically by J.R. Buchi in 1952. They include the well-studied Dedekind-MacNeille completion and various ideal completions. The aim of this talk is to present a systematic study of join-completions of ordered structures, including lattice ordered groups and MV-algebras.

A partially ordered set (poset) L is called an extension of the poset P provided P is a subset of L and the order of L restricts to that of P. In case every element of L is a join of elements of L, we say that L is a join-extension of P. We use the term join-completion for a join-extension that is a complete lattice. The concepts of a meet-extension and a meet-completion are defined dually. Join-completions were introduced by B. Banaschewski in 1956, and they are intimately related to representations of complete lattices studied systematically by J.R. Buchi in 1952. They include the well-studied Dedekind-MacNeille completion and various ideal completions.

The aim of this talk is to present a systematic study of join-completions of ordered structures, including lattice ordered groups and MV-algebras.

Slides

Gödel conjectured in 1933 that there exists a translation between formulas of intuitionistic logic (Int) and modal logic (S4) such that a formula is provable in Int iff its translation is provable in S4. McKinsey and Tarski proved this in 1948, using the recently developed tools of algebraic logic. This talk will go through the details of that proof, as well as discuss some extensions of that result.

Varieties of de Morgan algebras (bounded distributive lattices equipped with a dual bounded lattice endomorphism identically satisfying x“=x) have been intensively studied by several authors since the Sixties. We present recent work (in collaboration with Leonardo Cabrer) on finite projective deMorgan algebras, including a charaterization of finite projective Kleene algebras. We exploit the fact that in any (locally finite) variety, (finite) projective algebras correspond to retracts of (finitely generated) free algebras. In the first part of the talk, we preliminarily show, following work by Kalman, that there are exactly three (nontrivial) subdirectly irreducible de Morgan algebras, and they generate a chain of three (nontrivial) varieties of de Morgan algebras, namely Boolean, Kleene, and de Morgan algebras. We next characterize free Kleene and deMorgan algebras, following work by Berman and Blok.

A tame filtration of an algebra is defned by the growth of its terms, which has to be majorated by an exponential function. A particular case is the degree filtration used in the definition of the growth of finitely generated algebras. The notion of tame filtration is useful in the study of possible distortion of degrees of elements when one algebra is embedded as a subalgebra in another. A geometric analogue is the distortion of the (Riemannian) metric of a (Lie) subgroup when compared to the metric induced from the ambient (Lie) group. The distortion of a subalgebra in an algebra also reflects the degree of complexity of the membership problem for the elements of this algebra in this subalgebra. One of our goals is to investigate, mostly in the case of associative or Lie algebras, if a tame filtration of an algebra can be induced from the degree filtration of a larger algebra. (This is a joint work with Yuri Bahturin.)

Gödel conjectured in 1933 that there exists a translation between formulas of intuitionistic logic (Int) and modal logic (S4) such that a formula is provable in Int iff its translation is provable in S4. McKinsey and Tarski proved this in 1948, using the recently developed tools of algebraic logic. This talk will go through the details of that proof, as well as discuss some extensions of that result.

This talk will present classical material on two of the basic types of Large Cardinals, namely Inaccessible Cardinals and Measurable Cardinals. We will motivate the discussion from both the analytic and logical viewpoints, and particular show (using Gödel’s Second Incompleteness Theorem) that both of these types of cardinal number cannot be proved to exist from the axioms of set theory.