Uniform Interpolation and Coherence

Kowalski, Tomasz; Metcalfe, George (2019). Uniform Interpolation and Coherence. Annals of pure and applied logic, 170(7), pp. 825-841. Elsevier 10.1016/j.apal.2019.02.004

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A variety V is said to be coherent if every finitely generated subalgebra of a finitely presented member of V is finitely presented. It is shown here that coherence corresponds to a key ingredient of uniform deductive interpolation for equational consequence in V: the property that any compact congruence on a finitely generated free algebra of V restricted to a free algebra over fewer generators is compact. A general criterion is derived for establishing failures of coherence, and hence also of uniform deductive interpolation. The criterion is then applied in conjunction with properties of canonical extensions to prove that coherence and uniform deductive interpolation fail for certain varieties of Boolean algebras with operators (including varieties for the modal logic K and KT), double-Heyting algebras, residuated lattices, and lattices.

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