Archimedean classes in integral commutative residuated chains

作者：Horcik Rostislav^*; Montagna Franco

来源：Mathematical Logic Quarterly, 2009, 55(3): 320-336.

DOI：10.1002/malq.200710091

• Residuated lattice
• representable residuated lattice
• MTL-algebra
• Pi MTL-algebra
• Wajsberg hoop
• Archimedean class
• the lattice of subvarieties
• t-norm based fuzzy logic

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 approximate to 1 double right arrow (q -> p) -> p approximate to 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 Hajek's question asking whether the variety of Pi MTL-algebras is generated by its Archimedean members.

• 出版日期2009-6