The Neggers-Stanley conjecture asserts that the polynomial counting the linear extensions of a labeled finite partially ordered set by the number of descents has real zeros only. We provide counterexamples to this conjecture. A finite partially ordered set (poset) P of cardinality p is said to be labeled if its elements are identified with the integers 1, 2, . . . , p. We will use the symbol ≺ ...

In this note I examine conditions under which a probability relation on a set of events is lexicographic. Chipman (1971) discussed the question of the minimal ordinal CT such that there exists an order-preserving function from an ordered set A (in his; paper A is fR with any order on it) to lRa with the lexicographic order on it. Since there always exists an ordinal /? such that there is an ord...

2. Cycles and their terminology. Let L be an n Xn Latin square or, alternatively, let Q be a quasigroup of order n. Let M be the set of n2 ordered triplets ijk, where k is the entry in the ith row and jth column of L. If 5 is the set of n2 ordered pairs ij, and if 7r,:Af—>S is the projection parallel to the ith coordinate (for example, ir2(ijk) = ik), then for each i, l^i^3, t< is onto S (or eq...

A partially ordered set (X, is called sharply transitive if its automorphism group is sharply transitive on X, that is, it is transitive and the stabilizer of every element is triviaL It is shown that every free group is the automorphism group of a sharply transitive partially ordered set. It is also shown that there exists a sharply transitive partially ordered set (-,Y, :::;) having some maxi...

Let T be an n-node tree of maximum degree 4, and let P be a set of n points in the plane with no two points on the same horizontal or vertical line. It is an open question whether T always has a planar drawing on P such that each edge is drawn as an orthogonal path with one bend (an “L-shaped” edge). By giving new methods for drawing trees, we improve the bounds on the size of the point set P f...

Given a finite partially ordered set P , for subsets or, in other words, coalitions X, Y of P let X ≤ Y mean that there exists an injection φ: X → Y such that x ≤ φ(x) for all x ∈ X. The set L(P ) of all subsets of P equipped with this relation is a partially ordered set. All partially ordered sets P such that L(P ) is a lattice are determined, and this result is extended to quasiordered set P ...

Based on a complete Heyting algebra, we modify the definition of lattice-valued fuzzifying convergence space using fuzzy inclusion order and construct in this way a Cartesian-closed category, called the category of L−ordered fuzzifying convergence spaces, in which the category of L−fuzzifying topological spaces can be embedded. In addition, two new categories are introduced, which are called th...