An algebraic axiomatization of orthogonal posets.
type: Item
instance of: scholarly article
title: An algebraic axiomatization of orthogonal posets.
zbMATH Open document ID: 1329.06003
published in: Soft Computing
publication date: 2015-11-09T00:00:00Z
review text: An orthogonal poset is a bounded poset \(\mathcal P=(P,\leq,{'},0,1)\) with an antitone involution such that \(x\vee y\) exists whenever \(x\leq y'\) and satisfying the identity \(x\vee x'\approx 1\). An orthocomplemented directoid is an algebra \(\mathcal D=(D,+,{'},0,1)\) of type \((2,1,0,0)\) satisfying the identities \(x+x\approx x\), \(x+y\approx y+x\), \(x+((x+y)+z)\approx(x+y)+z\), \((x')'\approx x\), \(x+0\approx x\), \(x+x'\approx 1\) and \((((x+z)+(y+z)')'+(y+z)')+z'\approx z'\). It is proved that if \(\mathcal D\) is an orthocomplemented directoid and one defines \(x\leq y\) if \(x+y=y\) (\(x,y\in D\)) then \(\mathbb P(\mathcal D):=(D,\leq,{'},0,1)\) is an orthogonal poset. Conversely, if \(\mathcal P\) is an orthogonal poset and one defines \(x+y:=x\vee y\) if \(x\vee y\) exists and \(x+y\) to be an arbitrary upper bound of \(x\) and \(y\) otherwise (\(x,y\in P\)) then \(\mathbb D(\mathcal P):=(P,+,{'},0,1)\) is an orthocomplemented directoid. We have \(\mathbb P(\mathbb D(\mathcal P))=\mathcal P\). Moreover, the variety of orthocomplemented directoids is congruence distributive.
Mathematics Subject Classification ID: 06C15
zbMATH DE Number: 6506243
zbMATH Keywords: orthogonal posets
zbMATH Keywords: bounded posets
zbMATH Keywords: antitone involutions
zbMATH Keywords: orthocomplemented directoids
zbMATH Keywords: congruence distributive varieties
zbMATH Keywords: axioms
author: Ivan Chajda
reviewed by: Helmut Länger
MaRDI profile type: Publication
full work available at URL: s00500-013-1047-1
OpenAlex ID: W2094518669
cites work: On a characterization of classical and nonclassical probabilities
cites work: Q3222894
cites work: The logic of quantum mechanics
cites work: Q5250893
cites work: Q3102713
cites work: Spaces of abstract events
cites work: Testing for classicality of a physical system
cites work: Q2762622
cites work: The center of an effect algebra
cites work: Q4291523
DOI: 10.1007/S00500-013-1047-1
description: scientific article; zbMATH DE number 6506243