Embedding Partially Ordered Sets into Chain-products

Embedding partially ordered sets into chain-products is already known to be NP-complete (see Yannakakis 30] for dimension or Stahl and Wille 26] for 2-dimension). In this paper, we introduce a new dimension parameter and show that encoding using terms (or k-dimension) is not better than bit-vector (or 2-dimension) and vice versa. A decomposition theory is introduced using coatomic lattices. An algorithm is provided to compute the associated decomposition tree. Such a tree is unique for a lattice and we show how it allows bit-vectors encoding computations. In the meantime a conjecture of Caseau for 2-dimension is discussed. In recent applications in computer science (cf. A t-Kaci et al 3] for logic programming, Caseau 5] for object programming, Agrawal et al 1] for databases, Ellis 9] for conceptual graphs management and Mattern 23] for distributed systems), the problem of partial order encoding has come into light. In those applications big hierarchies have to be eeciently stored in a computer. EEcient here means that the total storage is optimal with respect to fast answers for reachability queries (i.e. x y?). In some particular cases, the hierarchy is the directed covering graph of a lattice and some extra operations are required such as the computation of x _ y and x ^ y for any two elements x and y (see Caseau 5]). For a survey of these applications and encoding techniques, see Fall 10]. Another well studied particular case is obtained for trees in which eecient nearest common ancestor computations are needed (see Harel and Tarjan 17]).