{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T02:01:14Z","timestamp":1760148074386,"version":"build-2065373602"},"reference-count":10,"publisher":"MDPI AG","issue":"4","license":[{"start":{"date-parts":[[2023,3,27]],"date-time":"2023-03-27T00:00:00Z","timestamp":1679875200000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"DOI":"10.13039\/501100002428","name":"Austrian Science Fund (FWF)","doi-asserted-by":"publisher","award":["I 4579-N","20-09869L"],"award-info":[{"award-number":["I 4579-N","20-09869L"]}],"id":[{"id":"10.13039\/501100002428","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/501100001824","name":"Czech Science Foundation (GA\u010cR)","doi-asserted-by":"publisher","award":["I 4579-N","20-09869L"],"award-info":[{"award-number":["I 4579-N","20-09869L"]}],"id":[{"id":"10.13039\/501100001824","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Symmetry"],"abstract":"<jats:p>We present the smallest non-lattice orthomodular poset and show that it is unique up to isomorphism. Since not every Boolean poset is orthomodular, we consider the class of skew orthomodular posets previously introduced by the first and third author under the name \u201cgeneralized orthomodular posets\u201d. We show that this class contains all Boolean posets and we study its subclass consisting of horizontal sums of Boolean posets. For this purpose, we introduce the concept of a compatibility relation and the so-called commutator of two elements. We show the relationship between these concepts and introduce a kind of ternary discriminator for horizontal sums of Boolean posets. Numerous examples illuminating these concepts and results are included in the paper.<\/jats:p>","DOI":"10.3390\/sym15040810","type":"journal-article","created":{"date-parts":[[2023,3,27]],"date-time":"2023-03-27T04:31:21Z","timestamp":1679891481000},"page":"810","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":0,"title":["Orthomodular and Skew Orthomodular Posets"],"prefix":"10.3390","volume":"15","author":[{"ORCID":"https:\/\/orcid.org\/0000-0003-3840-3879","authenticated-orcid":false,"given":"Ivan","family":"Chajda","sequence":"first","affiliation":[{"name":"Department of Algebra and Geometry, Faculty of Science, Palack\u00fd University Olomouc, 771 46 Olomouc, Czech Republic"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Miroslav","family":"Kola\u0159\u00edk","sequence":"additional","affiliation":[{"name":"Department of Computer Science, Faculty of Science, Palack\u00fd University Olomouc, 771 46 Olomouc, Czech Republic"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-7030-4080","authenticated-orcid":false,"given":"Helmut","family":"L\u00e4nger","sequence":"additional","affiliation":[{"name":"Department of Algebra and Geometry, Faculty of Science, Palack\u00fd University Olomouc, 771 46 Olomouc, Czech Republic"},{"name":"Institute of Discrete Mathematics and Geometry, Faculty of Mathematics and Geoinformation, Technische Universit\u00e4t Wien, 1040 Vienna, Austria"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2023,3,27]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"823","DOI":"10.2307\/1968621","article-title":"The logic of quantum mechanics","volume":"37","author":"Birkhoff","year":"1936","journal-title":"Ann. Math."},{"key":"ref_2","first-page":"766","article-title":"Studies on the foundation of quantum mechanics. I","volume":"19","author":"Husimi","year":"1937","journal-title":"Proc. Phys.-Math. Soc. Jpn."},{"key":"ref_3","doi-asserted-by":"crossref","first-page":"57","DOI":"10.1017\/S1446788700005978","article-title":"On orthomodular posets","volume":"11","author":"Finch","year":"1970","journal-title":"J. Austral. Math. Soc."},{"key":"ref_4","doi-asserted-by":"crossref","first-page":"275","DOI":"10.1515\/ms-2022-0018","article-title":"Logical and algebraic properties of generalized orthomodular posets","volume":"72","author":"Chajda","year":"2022","journal-title":"Math. Slovaca"},{"key":"ref_5","unstructured":"Pt\u00e1k, P., and Pulmannov\u00e1, S. (1991). Orthomodular Structures as Quantum Logics, Kluwer."},{"key":"ref_6","first-page":"13","article-title":"Translations of distributive and modular ordered sets","volume":"27","year":"1988","journal-title":"Acta Univ. Palack. Olomuc. Fac. 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Symmetry, 14.","DOI":"10.3390\/sym14020323"}],"container-title":["Symmetry"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/2073-8994\/15\/4\/810\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,10]],"date-time":"2025-10-10T19:03:52Z","timestamp":1760123032000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/2073-8994\/15\/4\/810"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2023,3,27]]},"references-count":10,"journal-issue":{"issue":"4","published-online":{"date-parts":[[2023,4]]}},"alternative-id":["sym15040810"],"URL":"https:\/\/doi.org\/10.3390\/sym15040810","relation":{},"ISSN":["2073-8994"],"issn-type":[{"type":"electronic","value":"2073-8994"}],"subject":[],"published":{"date-parts":[[2023,3,27]]}}}