On some questions concerning the axiomatisation of WNM algebras and their subvarieties
In a seminal paper Esteva and Godo introduced monoidal t norm based logic MTL and some of its prominent extensions such as NM and WNM. We notice that NM is axiomatisable from IMTL, and hence MTL, with one variable axioms, by instantiating the WNM axiom over one variable. This observation leads us here to study the logic axiomatised by extending MTL by this one variable axiom. We shall refer to its equivalent algebraic semantics as the variety of GHP algebras, for those algebras will be shown to form the largest variety of MTL algebras such that the falsum free reducts of the positive cones of their chains are the most general totally ordered Godel hoops. Among other results we obtain a general description of GHP standard algebras, and use the latter to characterise those extensions of WNM that can be obtained from GHP via the same set of extending axioms.
Fuzzy Sets Syst.