On filter theory of residuated lattices
The authors study filters, Boolean filters (implicative filters), G-filters (positive implicative filters), MV-filters (fantastic filters) and also the corresponding fuzzy filters, fuzzy Boolean filters (fuzzy implicative filters), fuzzy G-filters (fuzzy positive implicative filters), fuzzy MV-filters (fuzzy fantastic filters) of a residuated lattice. They also introduce and study, in a similar manner as in the above cases, the notions of regular filter and fuzzy regular filter of a residuated lattice. Characterizations of some particular residuated lattices in terms of (fuzzy) filters are presented. Finally, relations among special types of (fuzzy) filters of residuated lattices are established. Reviewer's remarks: (0) The authors write ``fuzzy Boolean filter'' instead of ``Boolean fuzzy filter'', etc. (1) The authors and the referees of the paper have ignored that: {\parindent=4mm \begin{itemize}\item[--] residuated lattices are categorically equivalent to bounded reversed left-BCK(P) lattices (see [\textit{A. Iorgulescu}, ``Some direct ascendents of Wajsberg and MV algebras'', Sci. Math. Jpn. 57, No. 3, 583--647 (2003; Zbl 1050.06006)]), hence residuated lattices are particular cases of (reversed left-) BCK-algebras; \item[--] filter theory on reversed left-BCK algebras (ideal theory on BCK-algebras) was studied a long time ago (see for example [\textit{K. Iséki} and \textit{S. Tanaka}, ``An introduction to the theory of BCK-algebras'', Math. Jap. 23, 1--26 (1978; Zbl 0385.03051)]); \item[--] Boolean algebras are bounded implicative BCK-algebras (cf. the cited paper by Iséki and Tanaka, Theorem 12); \item[--] any implicative BCK-algebra is commutative and positive implicative (cf. the cited paper by Iséki and Tanaka, Theorem 10); \item[--] the so-called regular residuated lattices are in fact the Girard monoids, or involutive residuated lattices or residuated lattices with condition (DN) (Double Negation), hence they are particular cases of bounded BCK-algebras with condition (DN); \item[--] MV algebras are categorically equivalent to bounded commutative BCK-algebras (see [\textit{D. Mundici}, Math. Jap. 31, 889--894 (1986; Zbl 0633.03066)]). \end{itemize}} Consequently, Lemma 2.6 and Propositions 2.8, 2.10 contain known results. (2) The residuated lattice from Example 5.2 is regular and one can easily prove that any filter of a regular residuated lattice is a regular one; that is why, in Example 5.2, the considered filter \(F=(c,1]\) is a regular one. (3) Concerning the authors' statement on page 3631 that ``fuzzy filters are a useful tool to obtain results on classical filters'', our opinion is contrary: all the relations among fuzzy filters reflect the relations among the classical filters; hence, the fuzzy theory is only a reflection of the basic ``crisp'' (classical) theory, which remains the most important one. There are already many other levels of fuzzifications in the literature, but all are based on the ``crisp'' level.
scientific article; zbMATH DE number 5781095
scientific article