# 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.

What is MCQ? The Mathematical Citation Quotient (MCQ) measures journal impact by looking at citations over a five-year period. Subscribers to MathSciNet may click through for more detailed information.

## A note on Dilworth’s embedding theoremHTML articles powered by AMS MathViewer

by William T. Trotter
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.
Similar Articles
• Retrieve articles in Proceedings of the American Mathematical Society with MSC: 06A35
• Retrieve articles in all journals with MSC: 06A35