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2 results

  • Independence in Combinatorial Geometries of Rank Three

  • Japheth Hall, Jr.
  • Journal:
    Canadian Mathematical Bulletin / Volume 18 / Issue 2 / June 1975
    Published online by Cambridge University Press:
    20 November 2018, pp. 217-221
    Print publication:
    June 1975
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  • The class of all combinatorial geometries of rank three shall coincide with the class of all pairs (V, S) such that V is a set and S is a collection of non-empty subsets of V such that each pair of distinct elements of V belong to exactly one member of S. (See [3].)

    Consider a combinatorial geometry (V, S) of rank three.

  • A Condition for Equality of Cardinals of Minimal Generators Under Closure Operators

  • Japheth Hall, Jr
  • Journal:
    Canadian Mathematical Bulletin / Volume 14 / Issue 4 / December 1971
    Published online by Cambridge University Press:
    20 November 2018, pp. 569-570
    Print publication:
    December 1971
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  • Let C be an operator on the subsets of a set X with values among the subsets of X. We assume that C is a closure operator in X, i.e. a monotone, idempotent and extensive operator in X (cf., e.g., Birkhoff [3, p. 39], Schmidt [1], [2]). If A ⊆ X and B ⊆ X, we say that A and B are C-equivalent if C(A) = C(B) (Bleicher- Marczewski [4, p. 210]).