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