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CLOSURE OPERATORS, FRAMES AND NEATEST REPRESENTATIONS

  • ROB EGROT (a1)

Abstract

Given a poset $P$ and a standard closure operator $\unicode[STIX]{x1D6E4}:{\wp}(P)\rightarrow {\wp}(P)$ , we give a necessary and sufficient condition for the lattice of $\unicode[STIX]{x1D6E4}$ -closed sets of ${\wp}(P)$ to be a frame in terms of the recursive construction of the $\unicode[STIX]{x1D6E4}$ -closure of sets. We use this condition to show that, given a set ${\mathcal{U}}$ of distinguished joins from $P$ , the lattice of ${\mathcal{U}}$ -ideals of $P$ fails to be a frame if and only if it fails to be $\unicode[STIX]{x1D70E}$ -distributive, with $\unicode[STIX]{x1D70E}$ depending on the cardinalities of sets in ${\mathcal{U}}$ . From this we deduce that if a poset has the property that whenever $a\wedge (b\vee c)$ is defined for $a,b,c\in P$ it is necessarily equal to $(a\wedge b)\vee (a\wedge c)$ , then it has an $(\unicode[STIX]{x1D714},3)$ -representation.

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Bulletin of the Australian Mathematical Society
  • ISSN: 0004-9727
  • EISSN: 1755-1633
  • URL: /core/journals/bulletin-of-the-australian-mathematical-society
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