Amalgamation in classes of involutive commutative residuated lattices (2012.14181v9)

Abstract: Amalgamation is investigated in classes of non-divisible, non-in-teg-ral, and non-idempotent involutive commutative residuated lattices. We demonstrate that several subclasses of totally-ordered, involutive, commutative residuated lattices fail the Amalgamation Property. These include the classes of odd and even ones, sharing the same underlying reason for their failure as observed in the class of discrete linearly ordered abelian groups with positive normal homomorphisms. Conversely, it is proven that three subclasses formed by idempotent-symmetric, totally-ordered, involutive, commutative residuated lattices possess the Amalgamation Property, albeit fail the Strong Amalgamation Property. This failure shares the same underlying reason as observed in the class of linearly ordered abelian groups. Additionally, it is demonstrated that the varieties of semilinear, idempotent-symmetric, odd, involutive, commutative residuated lattices and semilinear, idempotent-symmetric, odd or even, involutive, commutative residuated lattices have the Transferable Injections Property. Finally, it is shown that any variety of semilinear involutive commutative residuated lattices that includes the variety of odd semilinear commutative residuated lattices fails the Amalgamation Property.