On the structure of linearly ordered pseudo-BCK-algebras
type: Item
instance of: scholarly article
title: On the structure of linearly ordered pseudo-BCK-algebras
zbMATH Open document ID: 1182.06009
author: Anatolij Dvurečenskij
author: Jan Kühr
published in: Archive for Mathematical Logic
publication date: 2009-12-14T00:00:00Z
review text: It is a nice, very interesting paper. First, the authors recall the notions of (bounded) porim, integral residuated lattice, pseudo-BCK algebra, pseudo-BCK(pP) algebra, pseudo-hoop, pseudo-BL algebra, basic pseudo-hoop, Wajsberg pseudo-hoop, pseudo-MV algebra, cone algebra, pseudo-ŁBCK algebra and their connections, in order to establish the auxiliary result that for every (linearly ordered) cone algebra \textbf{A} there exists a bounded (linearly ordered) Wajsberg pseudo-hoop \textbf{M} such that \textbf{A} can be embedded into \((M, \rightarrow, \rightsquigarrow,1)\). Then, the authors introduce the concept of \textit{cut} of a linearly ordered pseudo-BCK algebra and the identities (H) and (J), that play a central role in the paper, \[ \begin{aligned} (x \rightarrow y) \rightarrow (x \rightarrow z)&= (y \rightarrow x) \rightarrow (y \rightarrow z), \tag{H}\\ (((x \rightarrow y) \rightsquigarrow y) \rightarrow x) \rightsquigarrow x&= (((y \rightsquigarrow x) \rightarrow x) \rightsquigarrow y) \rightarrow y, \tag{J} \end{aligned} \] in order to characterize those linearly ordered pseudo-BCK algebras that arise as ordinal sums of linearly ordered cone algebras. As applications, the authors study the \(\{\rightarrow, \rightsquigarrow, 1\}\)-subreducts of representable pseudo-hoops and pseudo-BL algebras, and they study the Bosbach states and the Riečan states on bounded linearly ordered pseudo-BCK algebras. Reviewer's remarks: Pseudo-BCK algebras are more naturally connected to pseudo-Wajsberg algebras rather than to pseudo-MV algebras. Identity (H) appears in the definition of \textit{L-algebras} given in [\textit{W. Rump}, ``\(L\)-algebras, self-similarity, and \(l\)-groups'', J. Algebra 320, No. 6, 2328--2348 (2008; Zbl 1158.06009)].
reviewed by: Afrodita Iorgulescu
Mathematics Subject Classification ID: 06F35
Mathematics Subject Classification ID: 03G12
Mathematics Subject Classification ID: 06D35
zbMATH DE Number: 5646252
zbMATH Keywords: pseudo-BCK-algebra
zbMATH Keywords: pseudo-MV-algebra
zbMATH Keywords: ordinal sum
zbMATH Keywords: cone algebra
describes a project that uses: Pseudo Hoops
MaRDI profile type: Publication
full work available at URL: s00153-009-0151-5
OpenAlex ID: W2009985887
cites work: Varieties of BL-algebras. I: General properties.
cites work: On the structure of hoops
cites work: Komplementäre Halbgruppen. Axiomatik und Arithmetik
cites work: Komplementäre Halbgruppen Kongruenzen und Quotienten
cites work: Concerning cone algebras
cites work: Decomposition of BL-chains
cites work: Q4804772
cites work: States on pseudo MV-algebras
cites work: Pseudo MV-algebras are intervals in ℓ-groups
cites work: Aglianò-Montagna type decomposition of linear pseudo hoops and its applications
cites work: Every linear pseudo BL-algebra admits a state
cites work: Algebras in the positive cone of po-groups
cites work: Generalized MV-algebras
cites work: Bosbach states on fuzzy structures
cites work: Q2752146
cites work: Pseudo MV-algebras: A noncommutative extension of MV algebras
cites work: Q4542634
cites work: Q5692888
cites work: Q4654987
cites work: Q5190972
cites work: On the structure of generalized BL-algebras
cites work: Q5701829
cites work: Representable pseudo-BCK-algebras and integral residuated lattices
cites work: A non-commutative generalization of MV-algebras
cites work: On varieties of biresiduation algebras
DOI: 10.1007/S00153-009-0151-5
description: scientific article; zbMATH DE number 5646252