Cardinality-restricted chains and antichains in partially ordered sets

AbstractFor a given poset and positive integer κ, four problems are considered. Covering: Determine a minimum cardinality cover of the poset elements by chains (antichains), each of length (width) at most κ. Optimization: Given also weights on the poset elements, find a chain (antichain) of maximum total weight among those of length (width) at most κ. It is shown that the chain covering problem is NP-complete, while chain optimization is polynomial-time solvable. Several classes of facets are derived for the polytope generated by incidence vectors of antichains of width at most κ. Certain of these facets are then used to develop a polyhedral combinatorial algorithm for the antichain optimization problem. Computational results are given for the algorithm on randomly generated posets with up to 1005 elements and 4 ⩽ κ ⩽ 30

Partially Ordered Sets and Their Invariants

We investigate how much information cardinal invariants can give on the structure of the ordered set on which they are de�ned. We consider the basic de�nitions of an ordered set and see how they are related to one another. We generalize some results on cardinal invariants for ordered sets and state some useful characterizations. We investigate how cardinal invariants can in uence the existence of some special suborderings. We generalize some results on the Dilworth and Sierpinski theorems and explore the conjecture of Miller and Sauer. We address some open problems on dominating numbers. We investigate Model Games to �nd some characterizations on the cardinality of a set.

Extreme k-families

AbstractLet P be a poset. A subset A of P is a k-family iff A contains no (k + 1)-element chain. For i ⩾ 1, let Ai be the set of elements of A at depth i − 1 in A. The k-families of P can be ordered by defining A ⩽ B iff, for all i, Ai is included in the order ideal generated by Bi. This paper examines minimal r-element k-families, defined as k-families A such that |A| = r and for every B < A, |B| < r. Minimal k-families are related to maximal r-antichains and an operation called Sperner closure, which have been used to obtain extremal results for families of sets with width restrictions. Let Mk,r be the set of minimal r-element k-families and let Mk = ∪r ≥ 0 Mk,r. It is shown that Mk is a join-subsemilattice by the lattice Ak of k-families. Mk is a lower semimodular lattice, where the rth rank is given by Mk,r. If wk is the maximum size of a k-family, then |Mk,r| ⩽ (wrk)and |∪Mk| ⩽ Σi = 1wk ⌈i/k⌉. Let D(A) = max{|B| − |A| | B is a k-family and B ⩽ A}. For k-families A and B, D(A v B) ⩽ D(A) + D(B). This result shows that {A | D(A) = 0} is also a join-subsemilattice of Ak

Diversity as Width

It is argued that if the population of options is a finite poset, diversity comparisons may be conveniently based on widths i.e. on the respective maximum numbers of pairwise incomparable options included in the relevant subposets. The width-ranking and the undominated width-ranking are introduced and characterized

Irreflexive and reflexive dimension

AbstractThere are three equivalent definitions of dimension for partially ordered sets. When generating these three definitions to C-dimension over an arbitrary class of orders C, the three definitions diverge. We compare these three definitions and determine certain requirements under which they are equivalent