Amalgamation in classes of involutive commutative residuated lattices

Abstract

The amalgamation property and its variants are in strong relationship with various syntactic interpolation properties of substructural logics, hence its investigation in varieties of residuated lattices is of particular interest. The amalgamation property is investigated in some classes of non-divisible, non-integral, and non-idempotent involutive commutative residuated lattices in this paper. It is proved that the classes of odd and even totally ordered, involutive, commutative residuated lattices fail the amalgamation property. It is also proved that the varieties of odd and even semilinear, idempotent symmetric, involutive, commutative residuated lattices have the amalgamation property, and hence also the transferable injections property.

Group Representation for Even and Odd Involutive Commutative Residuated Chains

“…Ultimately, this process led to a representation of odd or even involutive FL e -chains by bunches of layer groups. This representation has been lifted to a categorical equivalence in [6], and has proven to be a powerful weaponry to prove amalgamation results in classes of involutive FL e -algebras which are neither integral, nor divisible, nor idempotent [7]. In Theorem 3.1 below, we establish a more direct constructional relationship between an (odd or even involutive) FL e -chain and its corresponding layer groups, bypassing the intermediary stage of layer algebras.…”

“…. 7 Here and also in (3.20) uv stands for max ≤κ (u, v). This notation does not cause any inconsistency with the notation of Theorem 3.1/(A) since for any two positive idempotent elements u, v of an odd or even involutive FLe-chain (X, ≤, •, →, t, f ) it holds true that uv = max ≤ (u, v) which is further equal to max ≤κ (u, v) by (3.2).…”