Basic substructural core fuzzy logics and their extensions: Mianorm-based logics

Abstract

This paper addresses standard completeness for non-associative, non-commutative, substructural fuzzy logics and their axiomatic extensions. First, fuzzy systems, which are based on mianorms (binary monotonic identity aggregation operations on the real unit interval [0,1]), their corresponding algebraic structures, and algebraic completeness results are discussed. Next, completeness with respect to algebras whose lattice reduct is [0,1], so-called standard completeness, is established for these systems using construction in the style of Jenei–Montagna. Finally, some axiomatic extensions of the non-associative, non-commutative core fuzzy logics having axioms corresponding to the structural rule(s) of exchange and/or associativity are considered.