A generalization of Hiraguchi's: Inequality for posets |
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Authors: | Williani T Trotter |
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Institution: | Department of Mathematics and Computer Science, University of South Carolina, Columbia, South Carolina 29208, USA |
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Abstract: | For a poset X, Dim(X) is the smallest positive integer t for which X is isomorphic to a subposet of the cartesian product of t chains. Hiraguchi proved that if | X | ? 4, then Dim(X) ? | X |/2]. For each k ? 2, we define Dimk(X) as the smallest positive integer t for which X is isomorphic to a subposet of the cartesian product of t chains, each of length k. We then prove that if | X | ? 5, Dim3(X) ? {| X |/2} and if | X | ? 6, then Dim4(X) ? | X |/2]. |
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