Department of Mathematics
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This talk continues the intermittent series of lectures I have given since 2011 on joint work being carried out with Philipp Rothmaler of City University of New York, January 25, 2011, February 02, 2011, February 08, 2011, October 22, 2013, January 14, 2014, January 28, 2014.

In these talks we investigate certain properties of infinite sequences in a module induced by satisfiability properties of sets of pp formulas. For a simple but useful example, consider a countable ring \(R\) and sets of formulas of the form \(r_{i}v=x_{i}\), \(i\in\mathbb{N}\). Given a left \( R \)-module \(M\) and a sequence \( (a_{i})_{i\in\mathbb{N}} \), there are several natural questions that we can ask about the set of formulas over \(M\) formed by replacing each variable \(x_i\) with the corresponding value \(a_i\). Is it satisfied in \(M\)? Is it finitely satisfiable in \(M\)? Is it consistent with the (positive) theory of \(M\)? Is it consistent with the algebraic theory of \(M\)?

We show that some properties of left \(R\)-modules \(M\) which cannot be axiomatized even by sentences of the infinitary logic \(\mathcal{L}_{\infty,\omega}\) can be characterized in terms of the kinds of properties mentioned, and prove a duality between 'injective left modules' and 'flat right modules' which is different from the usual finitary duality.

This work continues.