{
  "abstract": "Pseudocomplemented lattices satisfy a lot of nice properties and were studied by a number of algebraists. However, even when a poset P with 0 is not pseudocomplemented, we can assign to every element a of P the set $a^p$ of all maximal elements being orthogonal to a. This assignment $^p$ is a non-trivial operator provided P satisfies the so-called Ascending Chain Condition (ACC). In particular, we investigate this operator when the poset P in question is a semilattice or a lattice. We characterize this operator and describe its properties that turn out to be similar to the corresponding properties of pseudocomplementation. Moreover, we introduce an equivalence relation $Θ_p$ on P induced by $^p$ similarly as it was done by V. I. Glivenko for distributive pseudocomplemented lattices. Finally, we investigate for which lattices L with 0 satisfying the ACC this equivalence relation turns out to be a congruence and when the quotient lattice $L/Θ_p$ is complemented respectively modular.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.35988",
  "author_total": 2,
  "authors": [
    {
      "name": "Ivan Chajda"
    },
    {
      "name": "Helmut Länger"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/b2f1mvymk0xg1.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.35988",
  "fulltext_url": null,
  "id": "b2f1mvymk0xg1",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.35988",
  "language": null,
  "licence": "CC-BY-4.0",
  "licence_label": "CC BY 4.0",
  "licence_source": "article",
  "licence_url": "http://creativecommons.org/licenses/by/4.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "math.CO"
    ]
  ],
  "openalex": null,
  "pdf_url": "https://arxiv.org/pdf/2610.35988",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-26",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/b2f1mvymk0xg1.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:35:53Z",
      "native_id": "2610.35988",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.35988"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.CO",
      "subfield": "combinatorics"
    }
  ],
  "title": "A generalization of pseudocomplementation in semilattices, lattices and posets with 0",
  "title_text": "A generalization of pseudocomplementation in semilattices, lattices and posets with 0",
  "type": "preprint",
  "updated_at": "2026-10-30T12:35:53Z",
  "url": "https://research.draconitic.com/paper/b2f1mvymk0xg1",
  "venue": "arXiv",
  "versions": []
}