Blockwise Gluings And Amalgamation Failures in Integral Residuated Lattices

TL;DR

This paper introduces a new blockwise gluing construction for integral residuated lattices, providing finite axiomatizations, proving decidability of their universal theories, and demonstrating the failure of the amalgamation property in many varieties.

Contribution

It generalizes the concept of ordinal sums, offers finite axiomatizations for certain varieties, and shows these varieties lack the amalgamation property, solving longstanding open problems.

Findings

Finite axiomatizations for varieties of glued chains.

Decidability of the universal theory for these varieties.

Many varieties lack the amalgamation property.

Abstract

We introduce the blockwise gluing construction. This describes residuated integral chains which can be decomposed into (possibly) partial algebras, stacked one on top of the other, and such that elements in a certain component multiply in blocks, i.e., in the same way, with respect to lower components. This construction generalizes that of 1-sums (or ordinal sums). As first main results, we provide finite axiomatizations for varieties generated by particular chains that are gluings of their archimedean components. For such varieties we also prove the finite embeddability property, and as a consequence, the decidability of their universal theory. Moreover, we solve in the negative several longstanding open problems in the literature about the amalgamation property (AP). Indeed, we provide denumerably many new examples of varieties lacking the AP, including: semilinear (commutative) integral residuated lattices, semilinear FL_w-algebras, MTL-algebras, involutive and pseudocomplemented MTL-algebras, and all their n-potent subvarieties for n > 1. For the commutative varieties, this also entails that the associated substructural logics do not have the deductive interpolation property.