Archimedean classes in integral, commutative residuated lattices

This paper investigates a quasi-variety of representable integral commutative residuated lattices axiomatized by the quasi-identity resulting from the well-known Wajsberg identity (p → q) → q ≤ (q → p) → p if it is written as a quasi-identity, i. e., (p → q) → q ≈ 1 ⇒ (q → p) → p ≈ 1. We prove that this quasi-identity is strictly weaker than the corresponding identity. On the other hand, we show that the resulting quasi-variety is in fact a variety and provide an axiomatization. The obtained results shed some light on the structure of Archimedean integral commutative residuated chains. Further, they can be applied to various subvarieties of MTL-algebras, for instance we answer negatively Hájek's question asking whether the variety of ΠMTL-algebras is generated by its Archimedean members. © 2009 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim.