Semiantichains and Unichain Coverings in Direct Products of Partial Orders

We conjecture a generalization of Dilworth’s theorem to direct products of partial orders. In particular, we conjecture that the largest “semiantichain” and the smallest “unichain covering” have the same size. We consider a special c lass of semiantichains and unichain coverings and determine when equality holds for them. This conjecture implies the existence of k-saturated partitions. A stronger conjecture, for which we also prove a special case, implies the Greene-Kleitman result on simultaneous k and (k + I)-saturated partitions., ’ Whiie at Stanford this research was supported in part by National Science Foundation grant MCS-77-23738 and by Office of Naval Research contract N00014-76-C-0688. Reproduction in whole or in part is permitted for any purpose of the United States government. 1 1. Duality between semiantichains and unichain coverings. In this paper we study the relationship between semiantichains and unichain coverings in direct products of partial orders. Scmiantichains are more general objects than antichains, and unichains are a restricted class of chains. The study of antichains (colicctions of pairwise unrelated elements) in partially ordered sets admits two approaches. The earlier arises from Sperner’s theorem 1321, w h i c h . characterizes the maximum-sized antichains of a Boolean algebra. In g e n e r a l , Sperner theory obtains explicit values for the maximum size of antichains in partially ordered sets having special properties, and explicit descriptions of their composition. When the poset is ranked and the maximum-sized antichain consists of the I ank with most elements, the poset has the’sperner property.’ Generalizations of Sperner’s theorem have mostly consisted of showing that various posets h a v e the Sperner property or stronger versions of the Sperner property. Greene and Kleitman 1131 have given an excellent survey of results of this type. Diiworth’s theorem [4] bounds the size of the largest antichain by another invariant of the partial order. In particular, covering the partial order by chains is a “dual” minimization problem. No chain hits two elements of an antichain, so a .covcring always requires more items than any antichain has. Diiworth’s theorem asserts that in fact the optimum sizes are always equal. The result does not give the extremal value or extremal collections, but it applies to ail partially ordered sets. Generalizations of Dilworth’s theorem have flowed lessfreely. A number o f alternate proofs have been given, e.g. 131, [lo), but the only broad extension we have is Greene and Kleitman’s result 1121 on k-families and k-saturated partitions. e The study of k-families began with Erdos. A k-family in a partially ordered set is a collection of elements which contains no chain of size k + 1. An antichain is a l-family. Erdiis (61 generalized Sperner’s theorem by showing that the largest k-family in a Boolean algebra consists (uniquely) of the k largest ranks. A (ranked) partial order satisfying this for all k is said to have the *strong Sperner property”. Again, further Sperncr-type results on k-families can be found in 1131. Clearly any chain contains at most k elements of a k-family, so any partition C o f a ” partial order into chains {Ci} gives an upper bound of m&) = Ci min {k, ICil} on the size of the largest k-family. If the largest k-family has this size, t h e partition is called k-saturated. Greene and Kleitman proved there always exists a k-saturated partition, which for k = 1 reduces to Diiworth’s theorem. T h e y 2 showed further that for any k there exists a partition which is simultaneously k. and k + l-saturated. They applied lattice methods generalizing Dilworth’s less well-known result [5] about the lattice behavior of antichains. Saks [30] gave a shorter proof of the existence of k-saturated partitions of P by examining t h e direct product of P with a k-element chain. We consider a generalization of the Diiworth-type idea of saturated partitions to the direct product of any two partial orders. Sperner theory h a s a l s o discussed direct products. A scmiantichaia in a direct product is a collection of elements no two of which are related if they are identical in either component. The class of semiantichains includes the class of antichains. If the largest semiantichain still consists of a single rank, then the direct product has the two-part Sperner property. Results of this nature have been proved by Katona (211, 1231, . Kieitman 1241, and Griggs 1151, (171, with extensions to k-families by Katona (221, Schonheim 1311, and recently by Proctor, Saks, and Sturtevant 1271. Examples where maximum-sized semiantichain’s are not antichains were examined by West a n d Kieitman [33] and G. W. Peck 1261. To generalize Diiworth’s theorem to semiantichains we need a dual covering problem. Semiantichains are more general objects than antichains, so we need more restricted objects than chains. We define a unichajn (one-dimensional chaiq) in a direct product to be a chainin which one component remains fixed. Alternatively it is the product of an element from one order with a chain from the other. Two elements on a unichain are called unicornparable. Clearly no semiantichain can contain two elements of a unichain, so the largest semiantichain is bounded by the smallest covering by unichains. After 1331, West and Saks conjece turcd that equality always holds. We have not proved equality for general direct products, but we prove a special case here. Also, we make a stronger conjecture analogous to Green and Kieitman’s simultaneous k and k + l-saturation. If one *of the partial orders is a chain of k + 1 elements, the conjecture reduces to their result. Note that maximizing semiantichains and minimizing unichain c o v e r i n g s are dual integer programs. One such formulation has as constraint matrix t h e incidence matrix between elements and unichains. Showing that the underlying linear program has an integral optimal solution would prove the conjecture, b y guaranteeing that the integer program has no “duality gap”. 3 These dual programs form an example of the frequent duality between *packing” problems and Ucovering” problems (see [l], 121, (71, (81, [II], 1191, 1251, 1291. Diiworth’s theorem is another example; Dantzig and Hoffman [3] deduced it from duality principles. Hoffman and Schwartz (201 also used integer programming ideas to prove a slight generalization of Greene and Kieitman’s k-saturation result by transforming the problem into a transportation problem. These methods work partly because any subset of a partial order is still a partial order. However, a subset of a direct product need not be a direct product. Indeed, subs.ets of direct product orders frequently have duality gaps between their largest semiantichains and smallest unichain coverings. (The smallest example is a particular ‘T-element subset of the product of a &element chain with a 3-element chain.) Diiworth’s theorem can also be proved by transforming it to a bipartite matching problem or a network flow problem (see 191, [lo]). The difficulty in applying these latter methods to direct products is that unicomparability, unlike comparability, is not transitive. Much is known about the integrality of optima when the constraint matrix is totally unimoduiar, balanced, etc., as summarized by Hoffman [18]. Unfortunately, none of the several integer programming formulations we know of for this direct product problem have any of those properties. ’ Finally, Greene and Kieitman use lattice theoretic methods because the set of ,kfamilies and maximum k-families form well behaved lattices. We have found no reasonable partial order on semiantichains or maximum semiantichains. In the case where the largest semiantichain is also an antichain, network flow methods can be used to prove the conjecture. This result will appear in a subsequent paper. In Section 2 we find necessary and sufficient conditions for equality to hold when semiantichains and unichain coverings are required to have a particularly nice property called “decomposability”. When this happens, the size of the optimum is determined by the sizes of the largest k-families in the two + components. In Section 3 we develop the stronger form of the conjecture and show it holds in this case. We note with boundless ambition that if the first conjecture is true we can begin to ask about the existence of “k-saturated partitions” of direct products into unichains, analogously to k-saturated partitions of posets. Before embarking on the subject of decomposability, we note that this duality question can be phrased as a problem in graph theory. The “comparability graph” of a partially ordered set is formed by letting (z,~) be an edge in G(P) if z