Uniquely Complemented Posets

Chajda, Ivan; Länger, Helmut; Paseka, Jan

doi:10.1007/s11083-017-9440-5

Cited by 4 publications

(1 citation statement)

References 12 publications

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“…• One can use the machinery involved for posets used by the authors also in their previous papers [3] - [10] (partly written with further coauthors) on complemented posets, posets with an antitone involution or weakly orthomodular posets etc.…”

Section: Introductionmentioning

confidence: 99%

Sublattices and $Δ$-blocks of orthomodular posets

Chajda,

Länger

2020

Preprint

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For orthoposets we introduce a binary relation ∆ and a binary operator d(x, y) which are generalizations of the binary relation C and the commutator c(x, y), respectively, known for orthomodular lattices. We characterize orthomodular posets among orthoposets and orthogonal posets. Moreover, we describe connections between the relations ∆ and ↔ and the operator d(x, y). In details we investigate certain orthomodular posets of subsets of a finite set. In particular we describe maximal orthomodular sublattices and Boolean subalgebras of such orthomodular posets. Finally, we study properties of ∆-blocks with respect to Boolean sublattices and distributive subposets they include.

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Section: Introductionmentioning

confidence: 99%

Sublattices and $Δ$-blocks of orthomodular posets

Chajda,

Länger

2020

Preprint

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Sublattices and Δ-blocks of orthomodular posets

Chajda

Länger

2020

Journal of Logic and Computation

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States of quantum systems correspond to vectors in a Hilbert space and observations to closed subspaces. Hence, this logic corresponds to the algebra of closed subspaces of a Hilbert space. This can be considered as a complete lattice with orthocomplementation, but it is not distributive. It satisfies a weaker condition, the so-called orthomodularity. Later on, it was recognized that joins in this structure need not exist provided the subspaces are not orthogonal. Hence, the resulting structure need not be a lattice but a so-called orthomodular poset, more generally an orthoposet only. For orthoposets, we introduce a binary relation $\mathrel \Delta$ and a binary operator $d(x,y)$ that are generalizations of the binary relation $\textrm{C}$ and the commutator $c(x,y)$, respectively, known for orthomodular lattices. We characterize orthomodular posets among orthogonal posets. Moreover, we describe connections between the relations $\mathrel \Delta$ and $\leftrightarrow$ (the latter was introduced by P. Pták and S. Pulmannová) and the operator $d(x,y)$. In addition, we investigate certain orthomodular posets of subsets of a finite set. In particular, we describe maximal orthomodular sublattices and Boolean subalgebras of such orthomodular posets. Finally, we study properties of $\Delta$-blocks with respect to Boolean subalgebras and distributive subposets they include.

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A topological duality for strong Boolean posets

Yuan

2019

Mathematica Slovaca

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In this paper, we define a new class of posets which are complemented and ideal-distributive, we call these posets strong Boolean. This definition is a generalization of Boolean lattices on posets, and is different from Boolean posets. We give a topology on the set of all prime Frink ideals in order to obtain the Stone’s topological representation for strong Boolean posets. A discussion of a duality between the categories of strong Boolean posets and BP-spaces is also presented.

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Uniquely Complemented Posets

Cited by 4 publications

References 12 publications

Sublattices and $Δ$-blocks of orthomodular posets

Sublattices and $Δ$-blocks of orthomodular posets

Sublattices and Δ-blocks of orthomodular posets

A topological duality for strong Boolean posets

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