Transitive Relations, Topologies and Partial Orders
نویسنده
چکیده
Let be a set with elements. A subset of × is a binary relation (or relation) on . The number of relations on is 22. Equivalently, there are 2 2 labeled bipartite graphs on 2 vertices, assuming the bipartition is fixed and equitable. A relation on is reflexive if for all ∈ , we have ( ) ∈ . The number of reflexive relations on is 2(−1). A relation on is antisymmetric if for all ∈ , the conditions ( ) ∈ and ( ) ∈ imply that = . The number of antisymmetric relations on is 2 · 3(−1)2. A relation on is transitive if for all ∈ , the conditions ( ) ∈ and ( ) ∈ imply that ( ) ∈ . There is no known general formula for the number of transitive relations on . It is surprising that such a simply-stated counting problem remains unsolved [1, 2, 3, 4, 5, 6]. A topology on is a collection Σ of subsets of that satisfy the following axioms:
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