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  <doc>
    <id>1627</id>
    <completedYear>2023</completedYear>
    <publishedYear>2023</publishedYear>
    <thesisYearAccepted/>
    <language>eng</language>
    <pageFirst>1</pageFirst>
    <pageLast>33</pageLast>
    <pageNumber>33</pageNumber>
    <edition/>
    <issue>2</issue>
    <volume>84</volume>
    <type>article</type>
    <publisherName>Springer Nature</publisherName>
    <publisherPlace>Berlin</publisherPlace>
    <creatingCorporation/>
    <contributingCorporation/>
    <belongsToBibliography>0</belongsToBibliography>
    <completedDate>2023-04-25</completedDate>
    <publishedDate>2023-04-25</publishedDate>
    <thesisDateAccepted>--</thesisDateAccepted>
    <title language="eng">Topology of closure systems in algebraic lattices</title>
    <abstract language="eng">Algebraic lattices are spectral spaces for the coarse lower topology. Closure systems in algebraic lattices are studied as subspaces. Connections between order theoretic properties of a closure system and topological properties of the subspace are explored. A closure system is algebraic if and only if it is a patch closed subset of the ambient algebraic lattice. Every subset X in an algebraic lattice P generates a closure system〈X〉P . The closure system〈Y 〉P generated by the patch closure Y of X is the patch closure of〈X〉P. If X is contained in the set of nontrivial prime elements of P then〈X〉P is a frame and is a coherent algebraic frame if X is patch closed in P. Conversely, if the algebraic lattice P is coherent then its set of nontrivial prime elements is patch closed.</abstract>
    <parentTitle language="eng">Algebra universalis</parentTitle>
    <identifier type="doi">10.1007/s00012-023-00815-7</identifier>
    <identifier type="urn">urn:nbn:de:101:1-2023090808111682406614</identifier>
    <enrichment key="opus.doi.autoCreate">false</enrichment>
    <enrichment key="opus.urn.autoCreate">false</enrichment>
    <enrichment key="review.accepted_by">2</enrichment>
    <licence>Creative Commons - CC BY - Namensnennung 4.0 International</licence>
    <author>Niels Schwartz</author>
    <subject>
      <language>eng</language>
      <type>uncontrolled</type>
      <value>Poset</value>
    </subject>
    <subject>
      <language>eng</language>
      <type>uncontrolled</type>
      <value>Complete lattice</value>
    </subject>
    <subject>
      <language>eng</language>
      <type>uncontrolled</type>
      <value>Algebraic lattice</value>
    </subject>
    <subject>
      <language>eng</language>
      <type>uncontrolled</type>
      <value>Frame</value>
    </subject>
    <subject>
      <language>eng</language>
      <type>uncontrolled</type>
      <value>Closure system</value>
    </subject>
    <subject>
      <language>eng</language>
      <type>uncontrolled</type>
      <value>Closure operator</value>
    </subject>
    <subject>
      <language>eng</language>
      <type>uncontrolled</type>
      <value>Spectral space</value>
    </subject>
    <subject>
      <language>eng</language>
      <type>uncontrolled</type>
      <value>Specialization</value>
    </subject>
    <subject>
      <language>eng</language>
      <type>uncontrolled</type>
      <value>Coarse lower topology</value>
    </subject>
    <subject>
      <language>eng</language>
      <type>uncontrolled</type>
      <value>Scott topology</value>
    </subject>
    <subject>
      <language>eng</language>
      <type>uncontrolled</type>
      <value>Patch topology</value>
    </subject>
    <collection role="ddc" number="510">Mathematik</collection>
    <collection role="open_access" number="">open_access</collection>
    <collection role="institutes" number="">Fakultät für Informatik und Mathematik</collection>
    <collection role="Transformationsvertrag" number="">DEAL Springer Nature</collection>
    <file>https://opus4.kobv.de/opus4-uni-passau/files/1627/Schwartz_TopologyOfClosureSystems.pdf</file>
  </doc>
</export-example>
