Decomposing partial orderings into chains

Abstract

Dilworth's theorem gives the number of chains whose union is a given partially ordered set in terms of intrinsic properties of the partial ordering. This paper uses Dilworth's theorem to find the number of chains needed to uniquely determine a given partially ordered set in terms of intrinsic properties of the partial ordering. If a covers b and c covers d, then (a, b) and (c, d) are incomparable covers if either a or b is incomparable with either c or d. We prove that the number of chains whose transtitive closure is a given partial ordering is the largest number of elements in any set of incomparable covers plus the number of isolated elements of the partially ordered set.