Department of Mathematics
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This talk will be an introduction to and a very preliminary report on joint work being carried out with Philipp Rothmaler of the City University of New York.

If \(I\) is a non-empty set an \(I\)-property of modules is some operation \(P\) on the class of modules, which for each module \(M\) selects some subset \(P(M)\) of \(M^{I}\). Similarly, an \( (I) \)-property \(Q\) selects some subset of \(M^{(I)}\).

Imposing natural algebraic constraints on \(I\)-properties force them to be representable by various kinds of infinitary pp formulas. As yet we have no natural semantics for \( (I) \)-properties.

Both kinds of properties naturally organize into lattices. For a fixed ring \(R\) there is a duality reflecting finitary first order properties (for finite \(I\)) between the lattice of properties of right \(R\)-modules and the lattice of properties of left \(R\)-modules. We attempt to generalize this "elementary duality" in various ways to infinitary properties of modules.

These matters will be discussed in more detail in subsequent talks this fall and winter.