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Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)

 

 

A note on Dilworth's embedding theorem


Author: William T. Trotter
Journal: Proc. Amer. Math. Soc. 52 (1975), 33-39
MSC: Primary 06A35
DOI: https://doi.org/10.1090/S0002-9939-1975-0373988-0
MathSciNet review: 0373988
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Abstract: The dimension of a poset $ X$ is the smallest positive integer $ t$ for which there exists an embedding of $ X$ in the cartesian product of $ t$ chains. R. P. Dilworth proved that the dimension of a distributive lattice $ L = {\underline 2 ^X}$ is the width of $ X$. In this paper we derive an analogous result for embedding distributive lattices in the cartesian product of chains of bounded length. We prove that for each $ k \geqslant 2$, the smallest positive integer $ t$ for which the distributive lattice $ L = {\underline 2 ^X}$ can be embedded in the cartesian product of $ t$ chains each of length $ k$ equals the smallest positive integer $ t$ for which there exists a partition $ X = {C_1} \cup {C_2} \cup \cdots \cup {C_t}$ where each $ {C_i}$ is a i a chain of at most $ k - 1$ points.


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DOI: https://doi.org/10.1090/S0002-9939-1975-0373988-0
Keywords: Distributive lattice, dimension of a partitially ordered set, matching
Article copyright: © Copyright 1975 American Mathematical Society