Logic Seminar
Thomas Kucera
Manitoba
Infinitary properties of infinite sequences in modules
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.
Tuesday September 6, 2022 at 4:00 PM in 636 SEO