Thomas Kucera : Infinitary properties of infinite sequences in modules

Posted by James Freitag , part of the Logic Seminar.

At

Sept. 6, 2022, 4 p.m.

In

636 SEO

Abstract

Joint work with Philipp Rothmaler, CUNY. Elementary duality of positive primitive formulas over left and right modules is a model- theoretic tool (introduced by Prest [1988] and developed extensively by I. Herzog [1993]) in the context of the finitary first order model theory of modules that relates the category of left R- modules to the category of right R-modules, and much more. Prest, Rothmaler, and Ziegler [1994] extended the basic ideas to certain infinitary analogues of ppfs, (with finitely many free variables). Rothmaler and I extend these results further, to positive primitive properties of infinite sequences of elements in a module. We identify the syntactic forms of the dualizable properties, how to compute the duals, and the two kinds of infinitary existential quantifier that arise. We show that a certain class of modules, the locally projective modules of Zimmermann-Huisgen [1976], has a straightforward axiomatization by implications of dualizable formulas of our kind (but not likely by the formulas of [PRZ]). The elementary dual theory is easy to describe formally; but the algebraic content of it remains stubbornly obscure. I will give a general overview of this work, while avoiding most of the very technical machinery underlying it.