Abstract
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 semianchoredness. 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 \({x^{m} \leq x^{n}}\) (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.
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Presented by C. Tsinakis.
Partly supported by the FWF project START Y544-N23, the project GAP202/10/1826 and JSPS KAKENHI 25330013.
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Baldi, P., Terui, K. Densification of FL chains via residuated frames. Algebra Univers. 75, 169–195 (2016). https://doi.org/10.1007/s00012-016-0372-5
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DOI: https://doi.org/10.1007/s00012-016-0372-5