Densification of FL chains via residuated frames

作者：Baldi Paolo^*; Terui Kazushige^*

来源：ALGEBRA UNIVERSALIS, 2016, 75(2): 169-195.

DOI：10.1007/s00012-016-0372-5

• substructural logic
• fuzzy logic
• Gentzen systems
• residuated frames
• residuated lattices
• standard completeness

We introduce a systematic method for densification, i.e., embedding a given chain into a dense one preserving certain identities, in the framework of FL algebras (pointed residuated lattices). Our method, based on residuated frames, offers a uniform proof for many of the known densification and standard completeness results in the literature. We propose a syntactic criterion for densification, called semi-anchoredness. We then prove that the semilinear varieties of integral FL algebras defined by semi-anchored equations admit densification, so that the corresponding fuzzy logics are standard complete. Our method also applies to (possibly non-integral) commutative FL chains. We prove that the semilinear varieties of commutative FL algebras defined by knotted axioms x(m) <= x(n) (with m, n > 1) admit densification. This provides a purely algebraic proof to the standard completeness of uninorm logic as well as its extensions by knotted axioms.

• 出版日期2016-4