Coherence and deductive interpolation

Tomasz Kowalski (La Trobe University)

On 24 May, Tomasz Kowalski (La Trobe) will present "Coherence and deductive interpolation" in Old Arts 143 at 11. 

Abstract: Coherence is an algebraic property, important and studied mostly in classical algebra. A class V of algebras closed under subalgebras (typically a class of rings or modules) is coherent, if every finitely generated subalgebra of a finitely presented algebra in V is itself finitely presented. 

Deductive interpolation is a logical property. For equational languages, it can be formulated as follows. For any finite sets of variables X,Y,Z and any finite set S(X,Y) or equations over X, Y, there exists a set of equations I(Y) such that for any equation e(Y,Z) we have S(X,Y) |= e(Y,Z) iff I(Y) |= e(Y,Z). If furthermore I(Y) can be chosen finite, the property is called uniform. A variety V has uniform deductive interpolation, if the above property holds with turnstile relativised to V. 

A rather natural restriction of uniform deductive interpolation occurs when we require Z to be empty. It turns out that this property, for a variety V, is equivalent to coherence. 

This leads to a criterion of coherence, which essentially amounts to closure under a form of completions and presence of a certain gadget. We use the criterion to show failures of coherence for a wide range of varieties of ordered algebras. A number of previous results are subsumed, notably failure of uniform interpolation for S4 (Ghilardi, Zawadowski), and failure of coherence for lattices (McKenzie, Schmidt). This is joint work with George Metcalfe (Bern).