Graduate Thesis Or Dissertation
 

The structure of preordered sets and their topological properties

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  • Let X be a set. Given any preorder < on the set X, there corresponds a family of subsets of X, namely, WIx E x} where L = {y y E X y <x} such that, for all elements x and y of X, x <y iff L ( L. In this thesis, it is shown that, conversely, the preorders on a set can be derived from arbitrary families of subsets of that set. Thus, an upper bound for the cardinality of all preorders on any set is obtained. Moreover, the cardinal number of the set of all partial orders on any finite set is odd. An equivalence relation is defined on the collection of all families of subsets of a set by calling two families equivalent iff they define the same preorder. All equivalence classes of this equivalence relation are closed under arbitrary nonempty unions. Therefore, every equivalence class has a greatest element with respect to set inclusion. In fact, a family of subsets of X is the greatest element of an equivalence class if and only if it is a topology for X which is also closed under arbitrary intersections. Consequently, the cardinal number of the set of all preorders on a set X is equal to the cardinal number of the set of all topologies for X which are also closed under arbitrary intersections and, in particular, if X is a finite set, then the cardinal number of the set of all preorders on the set X is equal to the cardinal number of the set of all topologies for X.
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