A Note on Dilworth's Embedding Theorem

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 v L = 2_ 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 > 2, the smallest positive integer t v for which the distributive lattice L — 1 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, U C, U« • • U C where each C. is r lit i a chain of at most k — 1 points. 1. Preliminaries. A poset consists of a pair (X, P) where X is a set and P is a reflexive, antisymmetric, and transitive relation on X. The notations (x, y) £ P and x < y in P are used interchangeably. If x and y are distinct points in X and neither (x, y) nor (y, x) is in P, then we say x and y are incomparable and write xly. For convenience we will frequently use a single symbol to denote a poset. If X and Y are isomorphic posets, then we write X = Y and if X is isomorphic to a subposet of Y, then we write X C Y. The dual of a poset X, denoted X, is the poset on the same set with x < y in X iff y < x in X. If (X, P) and (Y, Q) are posets, their free sum, denoted X+Y, is the poset (X 0 Y, P 0 Q) where 0 denotes disjoint union. Their cartesian product X x Y is the poset (X x Y, S) where S = i((x, y), (z, w)): x < z in X and y < w in Yt. The cartesian product of n copies of X is denoted X". The join of (X, P) and (Y, Q), denoted X© Y, is the poset (X u Y, P u 2 UXx Y). A function /: Y —» X is order preserving iff y < w in Y implies fiy) < f(w) in X. The cardinal power of X and Y, denoted X , is the poset consisting of all ordering preserving functions from Y to X with / < g in XY iff fiy) < g(y) in X for every y £ Y. A poset C for which x, y e C imply x < y or y < x is called a chain. We denote the m element chain 0<l<2<---<M-lbyM_, A chain (X, L) is said to be linear extension of (X, P) when P CL. We also say L is a linear extension of P. By a theorem of Szpilrajn Ll2], if 7 denotes the collection of all linear extensions of P, then lie = P. Presented to the Society, November 8, 1974; received by the editors July 5, 1974. AMS (MOS) subject classifications (1970). Primary 06A10, 06A35.