Left residuated lattices induced by lattices with a unary operation
type: Item
instance of: scholarly article
title: Left residuated lattices induced by lattices with a unary operation
zbMATH Open document ID: 1457.03075
author: Ivan Chajda
author: Helmut Länger
published in: Soft Computing
publication date: 2020-07-15T00:00:00Z
review text: The authors recently have shown that every orthomodular lattice \(L\) can be converted into a left residuated lattice with operations \(\otimes\) and \(\to\) defined as term operations \((x \vee y') \wedge y\), resp., \((x \wedge y) \vee x'\) in \(L\); see [the authors, Miskolc Math. Notes 18, No. 2, 685--689 (2017; Zbl 1399.06020)]. This result is now generalized. Let now \(L\) be a bounded lattice with an unary operation \('\) and binary operations \(\otimes\) and \(\to\) defined as above; then this lattice is left residuated and satisfies the equation \(x \to 0 = x\) if and only if the identities \(x \wedge (((x \vee y') \wedge y) \vee y') = x\) and \(x \vee (((x \wedge y) \vee y') \wedge y) = x\) are valid in \(L\). In the rest of the paper, the connections between these two identities and weak and dually weak orthomodularity in the sense of [the authors, Asian-Eur. J. Math. 11, No. 2, Article ID 1850093, 18 p. (2018; Zbl 1386.06009)] are investigated. Finally, the mentioned result is improved: with the simple definitions \(x \otimes y = x \wedge y\) and \(x \to y = x' \wedge y\), the two identities may be replaced by \(x \wedge ((x \wedge y \vee y') = x\) and \(x \vee ((x \vee y') \wedge y) = x\).
reviewed by: Jānis Cīrulis
Mathematics Subject Classification ID: 03G25
Mathematics Subject Classification ID: 06B20
Mathematics Subject Classification ID: 06C15
Mathematics Subject Classification ID: 06F05
zbMATH DE Number: 7220820
zbMATH Keywords: left adjointness
zbMATH Keywords: left residuated lattice
zbMATH Keywords: variety of lattices
zbMATH Keywords: weakly orthomodular lattice
zbMATH Keywords: dually weakly orthomodular lattice
zbMATH Keywords: complementation
MaRDI profile type: Publication
full work available at URL: s00500-019-04461-x
OpenAlex ID: W2988141478
Wikidata QID: Q126842585
cites work: Q3222894
cites work: A note on orthomodular lattices
cites work: Residuation in orthomodular lattices
cites work: Orthomodular lattices can be converted into left residuated l-groupoids
cites work: Weakly orthomodular and dually weakly orthomodular lattices
cites work: Orthomodular lattices that are horizontal sums of Boolean algebras
cites work: Lattices with unique complementation
DOI: 10.1007/S00500-019-04461-X
description: scientific article; zbMATH DE number 7220820