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Proceedings of the American Mathematical Society

Published by the American Mathematical Society, the Proceedings of the American Mathematical Society (PROC) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

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

The 2020 MCQ for Proceedings of the American Mathematical Society is 0.85.

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A note on Dilworth’s embedding theorem
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by William T. Trotter PDF
Proc. Amer. Math. Soc. 52 (1975), 33-39 Request permission

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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Additional Information
  • © Copyright 1975 American Mathematical Society
  • 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