Thomas Kucera (University of Manitoba, Canada), Saturated free algebras and almost indiscernible theories: an overview

Tue, 12 April 2022, 1 pm MDT

This is work motivated by questions at the intersection of algebra and model theory, and using advanced techniques of model theory.

Baldwin and Shelah (Algebra Universalis, 1983) studied saturated free algebras. Pillay and Sklinos (Bull. Symb. Logic 2015), following the lead of this paper, studied 'almost indiscernible theories', taking the opportunity to refine the statements of the major results and improve the proofs. We extend these results to large infinite contexts, both in the size of the language and the kinds of tuples allowed in a 'basis'; and return to examples and applications in algebra, in particular in the theory of modules.

The theory develops by noting various analogies. The model-theoretic concept 'indiscernible sequence' generalizes 'linearly independent set' in a vector space, 'free (generating) set' of an algebra, 'algebraic independence' in an algebraically closed field, and similar concepts. 'Saturated model' generalizes concepts such as 'injective envelope of a module', 'algebraic closure of a field', and similar constructions. A complete first-order theory is "almost indiscernible" if it has a (sufficiently large) saturated model which lies in the algebraic closure of an indiscernible set (of sequences). Requiring that a saturated model be generated by an indiscernible set imposes strong structural constraints, but nonetheless there are natural motivating examples.

I start with some history and motivation from algebra, then I will give an overview of the main model theoretic concepts and techniques, motivating them as much as possible by examples from algebra. I'll state the new technical structural results for almost indiscernible theories in our more general context, with no more than informal 'hand-waving' about the proof techniques. Then I will present some consequences for free algebras and for theories of modules, including structure theorems and some examples. I conclude with a list of open questions.

This is joint work with Anand Pillay.

[slides] [paper] [video]