Densification of FL Chains via Residuated Frames

0438994 - ÚI 2017 RIV CH eng J - Journal Article
Baldi, Paolo - Terui, K.
Densification of FL Chains via Residuated Frames.
Algebra Universalis. Roč. 75, č. 2 (2016), s. 169-195. ISSN 0002-5240. E-ISSN 1420-8911
R&D Projects: GA ČR GAP202/10/1826
Keywords : densifiability * standard completeness * residuated lattices * residuated frames * fuzzy logic
Subject RIV: BA - General Mathematics
Impact factor: 0.625, year: 2016

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. It provides a purely algebraic proof to the standard completeness of uninorm logic as well as its extensions by knotted axioms.
Permanent Link: http://hdl.handle.net/11104/0242323