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Orthogonality Spaces Associated with Posets

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Abstract

An orthogonality space is a set equipped with a symmetric, irreflexive relation called orthogonality. Every orthogonality space has an associated complete ortholattice, called the logic of the orthogonality space. To every poset, we associate an orthogonality space consisting of proper quotients (that means, nonsingleton closed intervals), equipped with a certain orthogonality relation. We prove that a finite bounded poset is a lattice if and only if the logic of its orthogonality space is an orthomodular lattice. We prove that that a bounded poset is a chain if and only if the logic of the associated orthogonality space is a Boolean algebra.

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Funding

This research is supported by grants VEGA 2/0142/20 and 1/0006/19, Slovakia and by the Slovak Research and Development Agency under the contracts APVV-18-0052 and APVV-20-0069.

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  1. Department of Mathematics and Descriptive Geometry, Faculty of Civil Engineering, Slovak University of Technology, Radlinského 11, Bratislava, 810 05, Slovakia

    Gejza Jenča

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  1. Gejza Jenča
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Correspondence to Gejza Jenča.

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The author expresses his gratitude for the possibility to present this material on the conference SSAOS 2022 in Tatranská Lomnica.

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Jenča, G. Orthogonality Spaces Associated with Posets. Order 40, 575–588 (2023). https://doi.org/10.1007/s11083-022-09623-5

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  • DOI: https://doi.org/10.1007/s11083-022-09623-5

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