Uniform interpolation and coherence

TL;DR

This paper explores the concept of coherence in algebraic varieties and establishes a connection with uniform deductive interpolation, providing criteria for failure and demonstrating these failures in several algebraic structures.

Contribution

It introduces a new equivalence between coherence and a restricted form of uniform deductive interpolation and applies this to identify failures in various algebraic varieties.

Findings

Coherence is equivalent to a restricted uniform deductive interpolation.

Criteria for failure of coherence are established.

Coherence and uniform deductive interpolation fail in several algebraic varieties.

Abstract

A variety V is said to be coherent if any finitely generated subalgebra of a finitely presented member of V is finitely presented. It is shown here that V is coherent if and only if it satisfies a restricted form of uniform deductive interpolation: that is, any compact congruence on a finitely generated free algebra of V restricted to a free algebra over a subset of the generators is again compact. A general criterion is obtained for establishing failures of coherence, and hence also of uniform deductive interpolation. This criterion is then used in conjunction with properties of canonical extensions to prove that coherence and uniform deductive interpolation fail for certain varieties of Boolean algebras with operators (in particular, algebras of modal logic K and its standard non-transitive extensions), double-Heyting algebras, residuated lattices, and lattices.