Variety of orthomodular posets

Chajda, Ivan; Kolařík, Miroslav

doi:10.18514/mmn.2014.771

Cited by 9 publications

(9 citation statements)

References 7 publications

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“…Furthermore, we show that sharp quantum structures, in particular orthoalgebras, can be represented as a proper subvariety of paraorthomodular directoids. This result extends the application of techniques developed in [6]. Finally, in Section 4 we investigate bounded posets with an antitone involution whose Dedekind-MacNeille completion are paraorthomodular lattices.…”

Section: Introductionsupporting

confidence: 60%

Algebraic properties of paraorthomodular posets

Chajda¹,

Fazio²,

Länger³

et al. 2020

Preprint

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Paraorthomodular posets are bounded partially ordered set 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 an algebraic and ordertheoretic perspective. 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.

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Section: Introductionsupporting

confidence: 60%

Algebraic properties of paraorthomodular posets

Chajda¹,

Fazio²,

Länger³

et al. 2020

Preprint

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“…The concept of a commutative directoid was introduced by J. Ježek and R. Quackenbush [1], see also [3] for details and elementary theory. A commutative meet-directoid is a groupoid (D, ) satisfying the following identities: The concept of a commutative directoid is very useful because it enables to translate ideas and problems in directed posets into groupoids where one can use standard algebraic tools for solutions, see e.g., [5]. The important concept of a commutative directoid is potentially applicable in the study of hierarchical lattices (see [7]).…”

Section: Preliminariesmentioning

confidence: 99%

“…This is of great advantage since there exist many methods and results for studying varieties in general algebra. The same method was used in [5], where so-called orthoposets and orthomodular posets (used in the formalization of the logic of quantum mechanics) were converted into algebras forming a variety.…”

Section: Introductionmentioning

confidence: 99%

Algebras Describing Pseudocomplemented, Relatively Pseudocomplemented and Sectionally Pseudocomplemented Posets

Chajda

Länger

2021

Symmetry

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In order to be able to use methods of universal algebra for investigating posets, we assigned to every pseudocomplemented poset, to every relatively pseudocomplemented poset and to every sectionally pseudocomplemented poset, a certain algebra (based on a commutative directoid or on a λ-lattice) which satisfies certain identities and implications. We show that the assigned algebras fully characterize the given corresponding posets. A certain kind of symmetry can be seen in the relationship between the classes of mentioned posets and the classes of directoids and λ-lattices representing these relational structures. As we show in the paper, this relationship is fully symmetric. Our results show that the assigned algebras satisfy strong congruence properties which can be transferred back to the posets. We also mention applications of such posets in certain non-classical logics.

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“…This is of great advantage since there exist many methods and results for studying varieties in General Algebra. The same machinery was used in [3] where so-called orthoposets and orthomodular posets (used in the formalization of the logic of quantum mechanics) were converted into algebras forming a variety.…”

Section: Introductionmentioning

confidence: 99%

Algebras describing pseudocomplemented, relatively pseudocomplemented and sectionally pseudocomplemented posets

Chajda¹,

Länger²

2021

Preprint

Self Cite

View full text Add to dashboard Cite

In order to be able to use methods of Universal Algebra for investigating posets, we assign to every pseudocomplemented poset, to every relatively pseudocomplemented poset and to every sectionally pseudocomplemented poset a certain algebra (based on a commutative directoid or on a λ-lattice) which satisfies certain identities and implications. We show that the assigned algebras fully characterize the given corresponding posets. It turns out that the assigned algebras satisfy strong congruence properties which can be transferred back to the posets. We also mention applications of such posets in certain non-classical logics.

show abstract

Variety of orthomodular posets

Cited by 9 publications

References 7 publications

Algebraic properties of paraorthomodular posets

Algebraic properties of paraorthomodular posets

Algebras Describing Pseudocomplemented, Relatively Pseudocomplemented and Sectionally Pseudocomplemented Posets

Algebras describing pseudocomplemented, relatively pseudocomplemented and sectionally pseudocomplemented posets

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