Craig interpolation for semilinear substructural logics

Abstract

The Craig interpolation property is investigated for substructural logics whose algebraic semantics are varieties of semilinear (subdirect products of linearly ordered) pointed commutative residuated lattices. It is shown that Craig interpolation fails for certain classes of these logics with weakening if the corresponding algebras are not idempotent. A complete characterization is then given of axiomatic extensions of the “R‐mingle with unit” logic (corresponding to varieties of Sugihara monoids) that have the Craig interpolation property. This latter characterization is obtained using a model‐theoretic quantifier elimination strategy to determine the varieties of Sugihara monoids admitting the amalgamation property.

Uniform interpolation and compact congruences

“…Example 5.4 (Sugihara monoids). The variety of Sugihara monoids is locally finite, congruence distributive (as it has a lattice reduct), and has the amalgamation property [22]. Moreover, for any finite sets x, y, the compact lifting of the inclusion homomorphism i : F(y) ֒→ F(x, y) preserves intersections.…”

“…Building on their work, we relate the algebraic uniform interpolation properties introduced in this paper to the existence of a model completion. This approach is also related to the use in [21,22] of model-theoretic methods (in particular, quantifier elimination) to establish the amalgamation property for certain varieties of semilinear commutative residuated lattices.…”

Amalgamation and interpolation in ordered algebras

“…Amalgamation and interpolation properties for varieties of pointed residuated lattices (and their associated substructural logics) have been investigated by a number of authors (see, e.g., [24], [53], [44], [49], [50]), very often making use of various relationships between these properties. Let us begin here by recalling some useful facts for commutative varieties.…”

“…In the presence of semilinearity, Craig interpolation typically fails (see, e.g., [50]), but the deductive interpolation property and amalgamation property may still hold for the variety; this is the case, for example, for the varieties of MV-algebras ( [55]) and BL-algebras ( [53]). Model-theoretic proofs (based on quantifier-elimination) of the deductive interpolation property and amalgamation property for semilinear varieties of commutative residuated lattices can be found in [13], [49], and [50]. 2 In the remainder of this section, we describe some general conditions for the amalgamation property in varieties of pointed residuated lattices.…”

“…In Section 7, we introduce the class of residuated lattices and show that a variety of semilinear residuated lattices satisfying the congruence extension property has the amalgamation property iff the class of its totally ordered members has the amalgamation property (Theorem 49). We also investigate the connection between the amalgamation property for a class of bounded residuated lattices and the class of its residuated lattice reducts (Theorem 50), and amalgamation in the join of two independent varieties of residuated lattices (Theorem 52). Finally, in Section 8, we consider amalgamation in classes of GBL-algebras.…”