Simplification and independence of axioms of fuzzy logic systems IMTL and NM

Monoidal t-norm-based logic MTL is the common formalization of all fuzzy logic of left continuous t-norms and their residua. This system has two important schematic extensions: involutive monoidal t-norm-based logic IMTL and nilpotent minimum logic NM. In this paper, we discuss simplification and independence of axioms of IMTL and NM, the shorter axioms of both logics, and an independent axioms of the system IMTL are given.