"preordered" meaning in English

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  "lang_code": "en",
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        "simple past and past participle of preorder"
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  "word": "preordered"
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  "lang": "English",
  "lang_code": "en",
  "pos": "adj",
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      "examples": [
        {
          "ref": "1980, Johann Schröder, Operator Inequalities, Academic Press, page 8:",
          "text": "Most of the theory of this section on linear order spaces (and cones) can be generalized to preordered linear spaces (and wedges).",
          "type": "quote"
        },
        {
          "ref": "1998, Ghanshyam B. Mehta, “Chapter 1: Preference and Utility”, in Salvador Barbera, Peter Hammond, Christian Seidl, editors, Handbook of Utility Theory, Volume 1: Principles, Springer, page 30:",
          "text": "It can be proved [see Herden (1989a,c, 1990, 1993a,b) and Herden and Mehta (1997a) that a continuous isotone (or strictly isotone) function defined on a closed subset K of a preordered topological space X has a continuous isotone (or strictly isotone) extension to a function g on X if and only if there is a separable system on X which satisfies certain \"separation\" conditions.",
          "type": "quote"
        },
        {
          "ref": "1999, Rafael Villarroel-Flores, Equivariant Homotopy Type of Categories and Preordered Sets, University of Minnesota, page 24:",
          "text": "We remind the reader that the barycentric subdivision sd(P) of a preordered set P is the poset of chains in P, ordered by inclusion.",
          "type": "quote"
        }
      ],
      "glosses": [
        "Equipped with a preorder."
      ],
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        "(set theory, order theory, of a set) Equipped with a preorder."
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  "word": "preordered"
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        },
        {
          "ref": "1999, Rafael Villarroel-Flores, Equivariant Homotopy Type of Categories and Preordered Sets, University of Minnesota, page 24:",
          "text": "We remind the reader that the barycentric subdivision sd(P) of a preordered set P is the poset of chains in P, ordered by inclusion.",
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          "preorder",
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