分野別セミナー

Interpolation is a logical property. Roughly, it says that if A implies B, then there is a C, in the vocabulary common to A and B, such that A implies C and C implies B. Coherence originated in sheaf theory, and has been studied intensively in algebra, mostly for groups, semigroups, rings and modules. A model theoretic definition of coherence is that a class K is coherent if every finitely generated substructure of a finitely presented structure in K is itself finitely presented.


Surprisingly (at least to us) it turns out that a version of interpolation is almost equivalent to coherence, and the missing bit is quite neat. We also have a general criterion for coherence, applicable to classes of ordered algebraic structures. Using it, we proved failures of coherence (and thus interpolation) for a number of such classes, in particular for Boolean algebras with operators and residuated lattices (both classes of some significance in logic), as well as lattices.


I will give some necessary background and present a proof of failure of coherence for lattices - as a toy example.
(joint work with George Metcalfe from the University of Bern)