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Representing quasi-orders by embeddability ordering of families of topological spaces

Authors: A. E. McCluskey and T. B. M. McMaster
Journal: Proc. Amer. Math. Soc. 127 (1999), 1275-1279
MSC (1991): Primary 06A06, 54H10
Published electronically: January 27, 1999
MathSciNet review: 1600113
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Abstract: An elementary argument constructs, for each cardinal $\alpha$, a topological space whose subspaces, ordered by homeomorphic embeddability, can model every partial order on $\alpha$-many points. We show how to modify this procedure to deal also with quasi-orders (where the antisymmetry condition may fail), obtaining an initial estimate of the cardinality of the space then required.

References [Enhancements On Off] (What's this?)

  • 1. C. Kuratowski, Sur la puissance de l'ensemble des "nombres de dimension" au sens de M. Fréchet, Fund. Math. 8 (1926), 201-208.
  • 2. C. Kuratowski and W. Sierpi\'{n}ski, Sur un problème de M. Fréchet concernant les dimensions des ensembles linéaires, Fund. Math. 8 (1926), 193-200.
  • 3. P.T. Matthews and T.B.M. McMaster, Families of spaces having prescribed embeddability order-type, Rend. Ist. Mat. Univ. Trieste 25 (1993), 345-352. MR 96e:54015
  • 4. P.T. Matthews and T.B.M. McMaster, Quasiorder contexts for total negation, Proc. Eighth Prague Topological Symp. (1996), 283-287. CMP 98:13
  • 5. A.E. McCluskey, T.B.M. McMaster and W.S. Watson, Representing set-inclusion by embeddability among subspaces of the real line, to appear in Top. Appl.
  • 6. A.E. McCluskey and T.B.M. McMaster, Realizing quasiordered sets by subspaces of continuum-like spaces, submitted to Order.
  • 7. A. Pultr and V. Trnková, Combinatorial, Algebraic and Topological Representations of Groups, Semigroups and Categories, North-Holland, Amsterdam (1980). MR 81d:18001
  • 8. W. Sierpi\'{n}ski, Sur un problème concernant les types de dimensions, Fund. Math. 19 (1932), 65-71.

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Additional Information

A. E. McCluskey
Affiliation: Department of Mathematics, National University of Ireland, Galway, Ireland

T. B. M. McMaster
Affiliation: Department of Pure Mathematics, Queen’s University of Belfast, Belfast BT7 1NN, Northern Ireland

Keywords: Partial order, quasi-order, ordering by embeddability
Received by editor(s): August 9, 1997
Published electronically: January 27, 1999
Communicated by: Alan Dow
Article copyright: © Copyright 1999 American Mathematical Society

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