Value semigroups, value quantales, and positivity domains

Ittay Weiss

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In 1981 and 1997 Kopperman and Flagg, respectively, proved that every topological space is metrisable, provided the symmetry and separation axioms are removed from the requirements on the metric, and the metric is allowed to take values in, respectively, a value semigroup or a value quantale. Seeking to construct a value quantale from a value semigroup we focus on a small portion of the structure present in a value semigroup, comprising what we call a positivity domain, and we construct its enveloping value quantale, forming part of a detailed comparison between value semigroups and value quantales. We obtain a representation theorem for value quantales in terms of positivity domains, and we outline how products of positivity domains can be used in the theory of continuity spaces instead of (the non-existent) products of value quantales.
Original languageEnglish
Pages (from-to)844-866
Number of pages23
JournalJournal of Pure and Applied Algebra
Issue number2
Early online date24 May 2018
Publication statusPublished - 1 Feb 2019


  • value quantale
  • value semigroup
  • continuity space
  • ordered semigroup
  • ordered monoid


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