Amalgamation in classes of involutive commutative residuated lattices

TL;DR

This paper investigates the amalgamation property in various classes of involutive commutative residuated lattices, identifying which subclasses satisfy or fail the property and exploring related properties like TIP.

Contribution

It provides a detailed analysis of amalgamation properties in specific subclasses, revealing conditions under which they hold or fail, and introduces the Transferable Injections Property for certain varieties.

Findings

Several subclasses of totally ordered involutive lattices fail AP.

Idempotent-symmetric subclasses possess AP but not SAP.

Certain varieties satisfy the Transferable Injections Property.

Abstract

Amalgamation is investigated in classes of involutive commutative residuated lattices that are neither divisible, nor integral, nor idempotent. We demonstrate that several subclasses of totally ordered involutive commutative residuated lattices fail the Amalgamation Property (AP). These include the classes of odd and even involutive lattices, whose failure of the AP stems from the same fundamental cause observed in the class of discrete linearly ordered abelian groups with positive normal homomorphisms. Conversely, we prove that three natural subclasses, consisting of idempotent-symmetric, totally ordered, involutive commutative residuated lattices, possess the AP, although they fail the Strong Amalgamation Property (SAP). This failure is attributable to the same underlying reason identified in the class of linearly ordered abelian groups. Furthermore, we show that the variety of semilinear, idempotent-symmetric, odd involutive commutative residuated lattices, as well as the variety generated by idempotent-symmetric, even involutive commutative residuated chains, satisfy the Transferable Injections Property (TIP), a strengthening of the AP. Finally, it is established that any variety of semilinear involutive commutative residuated lattices containing the variety of odd semilinear commutative residuated lattices fails the AP.