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Proceedings of the American Mathematical Society

Published by the American Mathematical Society since 1950, Proceedings of the American Mathematical Society is devoted to shorter research articles in all areas of pure and applied mathematics.

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

The 2024 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
Proc. Amer. Math. Soc. 52 (1975), 33-39
DOI: https://doi.org/10.1090/S0002-9939-1975-0373988-0

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.
References
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Bibliographic 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