Poset Structure Research Articles

M-Quasinormal f-rings

Given a completely regular topological space X, we wish to determine the poset structure of the root system of prime ideals of the ring C( X) of real-valued continuous functions on X; and vice versa. Here, we describe three measures on the poset of prime subgroups of a lattice-ordered group which determine the arithmetic of the group. Then we show that C( X) has the property that the sum of any m minimal prime ideals is a maximal ideal or the entire ring if and only if the subspace ⋂ j=1 m cl( coz( f j ))⊂ X is a P-space for every pairwise disjoint family { f j } j=1 m ⊂ C( X).

A Cyclage Poset Structure for Littlewood–Richardson Tableaux

A graded poset structure is defined for sets of Littlewood–Richardson (LR) tableaux whose cardinalities are the multiplicities of irreducible gl(n)-modules in the tensor product of several irreducible gl(n)-modules indexed by rectangular partitions. This is a generalization of the cyclage poset on tableaux defined by Lascoux and Schützenberger. This combinatorial construction is of independent interest and may be understood without reference to the underlying algebraic geometry. It is shown that the polynomials obtained by enumerating LR tableaux by shape and a generalized charge statistic, are the graded multiplicities of irreducibles in certain graded gl(n)-modules supported in the closure of a nilpotent conjugacy class. In particular, explicit tableau formulas are obtained for the special cases of the Kostka–Foulkes polynomials, the coefficient polynomials of two-column Macdonald–Kostka polynomials, and the graded multiplicities of coordinate rings of closures of conjugacy classes of nilpotent matrices. These polynomials coincide with the q -enumeration of rigged configurations and conjecturally coincide with the q -analogues of LR coefficients defined by the spin–weight generating functions of ribbon tableaux introduced by Lascoux, Leclerc, and Thibon.

Nielsen fixed point theory for partially ordered sets

In this paper, we introduce a Nielsen type number N ∗ (f, P ) for any selfmap f of a partially ordered set P of spaces. This Nielsen theory relates to various existing Nielsen type fixed point theories for different settings such as maps of pairs of spaces, maps of triads, fibre preserving maps, equivariant maps and iterates of maps, by exploring their underlying poset structures.

On Subdivision Posets of Cyclic Polytopes

There are two related poset structures, the higher Stasheff–Tamari orders, on the set of all triangulations of the cyclic d polytope with n vertices. In this paper it is shown that both of them have the homotopy type of a sphere of dimension n−d− 3. Moreover, we resolve positively a new special case of the Generalized Baues Problem: the Baues poset of all polytopal decompositions of a cyclic polytope of dimension d≤ 3 has the homotopy type of a sphere of dimensionn−d− 2.

Integral subschemes of codimension two

An even linkage class L of two-codimensional subschemes in P n has a natural partial ordering given by domination. In this paper we give a necessary condition for X ∈ L to be integral in terms of its location in the poset structure on L . The condition is almost sufficient in the sense that if a subscheme dominates an integral subscheme and satisfies the necessary conditions, then it can be deformed with constant cohomology to an integral subscheme. In particular, the necessary conditions are sufficient in the case that Lazarsfeld and Rao originally studied, since the minimal element for L was a smooth connected space curve.

Triangulations of cyclic polytopes and higher Bruhat orders

Recently Edelman and Reiner suggested two poset structures, (n, d) and (n, d) on the set of all triangulations of the cyclic d-polytope C(n, d) with n vertices. Both posets are generalizations of the well-studied Tamari lattice. While (n, d) is bounded by definition, the same is not obvious for (n, d). In the paper by Edelman and Reiner the bounds of (n, d) were also confirmed for (n, d) whenever d≤5, leaving the general case as a conjecture.

Decomposition of a bidirected graph into strongly connected components and its signed poset structure

A bidirected graph is a graph each arc of which has either two positive end-vertices (tails), two negative end-vertices (heads), or one positive end-vertex (a tail) and one negative end-vertex (a head). We define the strong connectivity of a bidirected graph as a generalization of the strong connectivity of an ordinary directed graph. We show that a bidirected graph is decomposed into strongly connected components and that a signed poset structure is naturally defined on the set of the consistent strongly connected components. We also give a linear time algorithm for decomposing a bidirected graph into strongly connected components. Furthermore, we discuss the relationship between the decomposition of a bidirected graph and the minimization of a certain bisubmodular function.

Tychonoff Poset Structures and Auxiliary Relations

Tychonoff Poset Structures and Auxiliary Relations

The poset structure of positive implicative BCK-algebras

Many algebraic structures can be characterized in terms of some order structure. Among them are lattices, semilattices, boolean algebras, and semi-boolean algebras. In this paper we show that a pBCK-algebra can be characterized in terms of two order relations, one of which also makes it an implicative BCK-algebra. One of these orders is the well-known order for the larger class of BCK-algebras; the other one is a new order we introduce here.

The local Kostant-PBW ordering

Commutation relations in Lie theory give rise to families of finite partially ordered sets connected with Kostant's partition function. Techniques originally used to obtain arithmetic properties of Kostant's function mesh well with these poset structures giving a great deal of geometric structure to the Hasse diagrams. Quite apart from any use in Lie theory these posets offer a wealth of examples of nicely behaved tractable posets which are neither rank symmetric nor satisfy LYM.

Design and administration of distributed and hierarchical information networks under partial orderings

In this paper, we extend previous results [1] in the protection and administration of information networks under partial orderings. We summarize previous results, and extend them to cover hierarchical networks. We consider distributed hierarchical administration, show a hierarchical network, and demonstrate a provably secure communications technique in which partial orderings are used to control flow, traffic analysis is minimized, local compromise does not cause global compromise, and distributed hierarchical administration works. We show means by which trusted and untrusted computing bases may be connected to form provably secure distributed information networks under partial orderings, and a risk analysis technique which takes advantage of the POset structure to reduce the complexity of analysis for these networks. We summarize results and propose further extensions of this work.

劣モジュラ・システムに関連する基多面体のすべての面の特徴づけ

For a distributive lattice D ⫋ 2^E and a submodular function f on D with φ ∈ D and f(φ) = 0, the pair (D,f) is called a submodular system and, when E ∈ D, the polyhedron given by B(f) = {x | x ∈ R^E, ∀X ∈De : x(X) ≦ f(X), x(E) =f(E)} is called the base polyhedron associated with (D, f)・ We examine the structure of the base polyhedron B(f) and give a characterization of all the faces of B(f). Faces of B(f) are made correspond one-to-one to certain sublattices of D, so that the collection D of all such sublattices of D is anti-isomorphic with the collection F of all the nonempty faces of B(f). Here, D and Fare considered as posets relative to set inclusion. The incidence relation among faces, dimensions of faces, and extreme points and extreme rays of faces are given based on the structure of the sublattices in D. These include as special cases recent results on (1) a poset structure of a polymatroid extreme point and connected components by Bixby, Cunningham and Topkis, (2) extreme rays of a cone determined by a distributive lattice by Tomizawa, and (3) adjacency for polymatroid extreme points by Topkis. Moreover, given a sublattice D_1 of D_2 on which f is modular, F(D_1) = {x | x ∈ B(f), ∀X ∈D_1 : x(X) = f(X) } is a nonempty face of B(f) and there uniquely exists a sublattice D_2 in D which corresponds to the face F(D_1). We show a theorem which characterizes the relationship between D_1 and D_2. D_2 is considered as a closure of D_1 and this closure operation is closely related to the concept of maximal skeleton recently considered by Nakamura and Iri. Algorithmic aspects of these characterizations are also discussed.

Entropy of L-fuzzy sets

The notion of “entropy” of a fuzzy set, introduced in a previous paper in the case of generalized characteristic functions whose range is the interval [0, 1] of the real line, is extended to the case of maps whose range is a poset L (or, in particular, a lattice). Some of the reasons giving rise to the non-comparability of the truth values and then the necessity of considering poset structures as range of the maps are discussed. The interpretative problems of the given mathematical definitions regarding the connections with decision theory are briefly analyzed.