Algebraic Properties of Paraorthomodular Posets. (8th July 2021)

• Record Type: Journal Article
• Authors: Chajda, Ivan; Fazio, Davide; Länger, Helmut; Ledda, Antonio; Paseka, Jan
• Is Part Of: Logic journal of the IGPL. Volume 30:Number 5(2022)
• Page Start: 840
• Page End: 869
• Publication Date: 2021-07-08
• Subjects: poset with an antitone involution -- orthomodular lattice -- orthomodular poset -- paraorthomodular lattice -- paraorthomodular poset -- orthoalgebra -- effect algebra -- commutative directoid -- D-continuous poset -- Dedekind–MacNeille completion; Logic, Symbolic and mathematical -- Periodicals; 511.3
• DOI: 10.1093/jigpal/jzab024
• Languages: English
• ISSNs: 1367-0751

Abstract

Abstract: Paraorthomodular posets are bounded partially ordered sets with an antitone involution induced by quantum structures arising from the logico-algebraic approach to quantum mechanics. The aim of the present work is starting a systematic inquiry into paraorthomodular posets theory both from algebraic and order-theoretic perspectives. On the one hand, we show that paraorthomodular posets are amenable of an algebraic treatment by means of a smooth representation in terms of bounded directoids with antitone involution. On the other, we investigate their order-theoretical features in terms of forbidden configurations. Moreover, sufficient and necessary conditions characterizing bounded posets with an antitone involution whose Dedekind–MacNeille completion is paraorthomodular are provided.