A Characterisation of Posets that Are nearly Antichains

We present a characterisation of posets of size n with linear discrepancy n − 2. These are the posets that are “furthest” from a linear order without being an antichain. This revised version was published online in September 2006 with corrections to the Cover Date.     

Tensor Products of Quantum Structures and Their Applications in Quantum Measurements

Tensor products of quantum logics and effect algebras with some known problems are reviewed. It is noticed that although tensor products of effect algebras having at least one state exist, in the category of complex Hilbert space effect algebras similar problems as with tensor products of projection lattices occur. Nevertheless, if one of the two coupled physical systems is classical, tensor product exists and can be considered as a Boolean power. Some applications of tensor products (in the form of Boolean powers) to quantum measurements are reviewed.     

Fock Theories and Quantum Logics

The notion of Fock theory is introduced in the framework of quantum logics, which are here orthomodular atomic lattices satisfying the covering property. It is shown that there are some fundamental facts concerning particles, which may be successfully discussed in this general context. One of these facts is to establish the theoretical conditions for considering particles as sharply defined entities. The other refers to the theoretical circumstances, which almost impose to consider that some particles have a structure, meaning they are composed from other particles. This last problem is strongly related with the conservative time evolutions.     

Maximal σ‐polynomials of connected 3‐chromatic graphs

In the set of graphs of order n and chromatic number k the following partial order relation is defined. One says that a graph G is less than a graph H if c_i(G) ≤ c_i(H) holds for every i, k ≤ i ≤ n and at least one inequality is strict, where c_i(G) denotes the number of i‐color partitions of G. In this paper the first ? n/2 ? levels of the diagram of the partially ordered set of connected 3‐chromatic graphs of order n are described. © 2003 Wiley Periodicals, Inc. J Graph Theory 43: 210–222, 2003     

Large-Scale Regularities of Lattice Embeddings of Posets

Let N denote the set of natural numbers and let P =(N ^k , ) be a countably infinite poset on the k-dimensional lattice N ^k . Given x N ^k , we write max(x) (min(x)) for the maximum (minimum) coordinate of x. Let be the directed-incomparability graph of P which is defined to be the graph with vertex set equal to N ^k and edge set equal to the set of all (x, y) such that max(x) max(y) and x and y not comparable in P. For any subset D N ^k , we let P _D and _D denote the restrictions of P and to D. Points x N ^k with min(x) = 0 will be called boundary points. We define a geometrically natural notion of when a point is interior to P or relative to the lattice N ^k , and an analogous notion of monotone interior with respect to or _D . We wish to identify situations where most of these interior points are exposed to the boundary of the lattice or, in the case of monotone interior points, not concealed very much from the boundary. All of these ideas restrict to finite sublattices F ^k and/or infinite sublattices E ^k of N ^k . Our main result shows that for any poset P and any arbitarily large integer M > 0, there is an F E with F = M where, relative to the sublattices F ^k E ^k , the ideal situation of total exposure of interior points and very little concealment of monotone interior points must occur. Precisely, we prove that for any P =(N ^k , ) and any integer M > 0, there is an infinite E N and a finite D F ^k with F E and F = M such that (1) every interior vertex of P _{E k} or _{E k} is exposed and (2) there is a fixed set C E, C k ^k , such that every monotone-interior point of _D belonging to F ^k has its monotone concealment in the set C. In addition, we show that if P ¹ =(N ^k , ₁),..., P ^r =(N ^k , _r ) is any sequence of posets, then we can find E,D, and F so that the properties (1) and (2) described above hold simultaneously for each P ⁱ . We note that the main point of (2) is that the bound k ^k depends only on the dimension of the lattice and not on the poset P. Statement (1) is derived from classical Ramsey theory while (2) is derived from a recent powerful extension of Ramsey theory due to H. Friedman and shown by Friedman to be independent of ZFC, the usual axioms of set theory. The fact that our result is proved as a corollary to a combinatorial theorem that is known to be independent of the usual axioms of mathematics does not, of course, mean that it cannot be proved using ZFC (we just couldn"t find such a proof). This puts our geometrically natural combinatorial result in a somewhat unusual position with regard to the axioms of mathematics.     

Homotopy of Non-Modular Partitions and the Whitehouse Module

We present a class of subposets of the partition lattice _n with the following property: The order complex is homotopy equivalent to the order complex of _n – 1, and the S _n -module structure of the homology coincides with a recently discovered lifting of the S _n – 1-action on the homology of _n – 1. This is the Whitehouse representation on Robinson's space of fully-grown trees, and has also appeared in work of Getzler and Kapranov, Mathieu, Hanlon and Stanley, and Babson et al.One example is the subposet P _n ⁿ – 1 of the lattice of set partitions _n , obtained by removing all elements with a unique nontrivial block. More generally, for 2 k n – 1, let Q _n ^k denote the subposet of the partition lattice _n obtained by removing all elements with a unique nontrivial block of size equal to k, and let P _n ^k = _{i = 2} ^k Q _n ⁱ . We show that P _n ^k is Cohen-Macaulay, and that P _n ^k and Q _n ^k are both homotopy equivalent to a wedge of spheres of dimension (n – 4), with Betti number . The posets Q _n ^k are neither shellable nor Cohen-Macaulay. We show that the S _n -module structure of the homology generalises the Whitehouse module in a simple way.We also present a short proof of the well-known result that rank-selection in a poset preserves the Cohen-Macaulay property.     

Lattice Approach to Wigner-Type Theorems

The Wigner's Theorem states that a bijective transformation of the set of all one-dimensional linear subspaces of a complex Hilbert space which preserves orthogonality is induced by either a unitary or an anti-unitary operator. There exist many Wigner-type theorems, in particular in indefinite metric spaces, von Neumanns algebras and Banach spaces and we try to find a common origin of all these results by using properties of the lattice subspaces of certain topological vector spaces. We prove a Wigner-type theorem for a pair of dual spaces which allows us to obtain, as particular cases, the usual Wigner's Theorem and some of its generalizations. PACS: 02.40.Dr, 03.65.Fd,03.65.Ta AMS Subject Classification (1991): 06C15, 46A20, 81P10.     

The Orthomodular Poset of Projections of A Symmetric Lattices

In a Hilbert space, there exists a natural correspondence between continuous projections and particular pairs of closed subspaces. In this paper, we generalize this situation and associate to a symmetric lattice L a subset P(L) of L× L, called its projection poset. If L is the lattice of closed subspaces of a topological vector space then elements of P(L) correspond to continuous projections and we prove that automorphisms of P(L) are determined by automorphisms of the lattice L when this lattice satisfies some basic properties of lattices of closed subspaces. Primary: 06C15, Secondary: 03G12 81P10.