On filter theory of residuated lattices
type: Item
instance of: scholarly article
title: On filter theory of residuated lattices
zbMATH Open document ID: 1228.03045
author: Yiquan Zhu
author: Yang Xu
published in: Information Sciences
publication date: 2010-09-08T00:00:00Z
review text: 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 w…
reviewed by: Afrodita Iorgulescu
Mathematics Subject Classification ID: 03G25
Mathematics Subject Classification ID: 06D35
Mathematics Subject Classification ID: 06F05
Mathematics Subject Classification ID: 06F35
zbMATH DE Number: 5781095
zbMATH Keywords: residuated lattice
zbMATH Keywords: BL-algebra
zbMATH Keywords: MTL-algebra
zbMATH Keywords: MV-algebra
zbMATH Keywords: IMTL-algebra
zbMATH Keywords: NM-algebra
zbMATH Keywords: Boolean filter
zbMATH Keywords: implicative filter
zbMATH Keywords: positive implicative filter, MV-filter
zbMATH Keywords: fantastic filter
zbMATH Keywords: fuzzy filter
MaRDI profile type: Publication
full work available at URL: j.ins.2010.05.034
OpenAlex ID: W2083614943
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DOI: 10.1016/J.INS.2010.05.034
description: scientific article; zbMATH DE number 5781095