Characterizations of Standard Elements in Posets

The fundamental characterization theorem of standard elements in lattices is extended to posets. Several other characterizations of standard elements are obtained in a sectionally semi-complemented poset and also in an atomistic, dually sectionally semi-complemented poset.     

Decompositions in Quantum Logic

We present a method of constructing an orthomodular poset from a relation algebra. This technique is used to show that the decompositions of any algebraic, topological, or relational structure naturally form an orthomodular poset, thereby explaining the source of orthomodularity in the ortholattice of closed subspaces of a Hilbert space. Several known methods of producing orthomodular posets are shown to be special cases of this result. These include the construction of an orthomodular poset from a modular lattice and the construction of an orthomodular poset from the idempotents of a ring.

Particular attention is paid to decompositions of groups and modules. We develop the notion of a norm on a group with operators and of a projection on such a normed group. We show that the projections of a normed group with operators form an orthomodular poset with a full set of states. If the group is abelian and complete under the metric induced by the norm, the projections form a -complete orthomodular poset with a full set of countably additive states.

We also describe some properties special to those orthomodular posets constructed from relation algebras. These properties are used to give an example of an orthomodular poset which cannot be embedded into such a relational orthomodular poset, or into an orthomodular lattice. It had previously been an open question whether every orthomodular poset could be embedded into an orthomodular lattice.

     

Effect algebras with the maximality property

The maximality property was introduced in orthomodular posets as a common generalization of orthomodular lattices and orthocomplete orthomodular posets. We show that various conditions used in the theory of effect algebras are stronger than the maximality property, clear up the connections between them and show some consequences of these conditions. In particular, we prove that a Jauch–Piron effect algebra with a countable unital set of states is an orthomodular lattice and that a unital set of Jauch–Piron states on an effect algebra with the maximality property is strongly order determining.     

On the structure of orthomodular posets

It is shown how all orthomodular posets (of various kinds) are constructible from families of sets satisfying various conditions, usually with the generating family emerging as identical with (or contained in) the family of frames (that is, maximal orthogonal subsets of the non-zero elements) of the constructed orthomodular poset.     

Order Dimension of Orthomodular Amalgamations Over Trees

We consider the amalgamation of bounded involution posets over a strictly directed graph as applied to orthomodular lattices, orthomodular posets or orthoalgebras. In the finite setting, we show that the order dimension of the amalgamation does not exceed that of the amalgamated structures by more than one. We also present conditions under which equality obtains.      

Riesz ideals in generalized effect algebras and in their unitizations

We show that quotients of generalized effect algebras by Riesz ideals preserve some important special properties, e.g., homogeneity and hereditary Riesz decomposition properties; moreover, quotients of generalized orthoalgebras, weak generalized orthomodular posets, generalized orthomodular lattices and generalized MV-algebras with respect to Riesz ideals belong to the same class. We give a necessary and sufficient condition under which a Riesz ideal I of a generalized effect algebra P is a Riesz ideal also in the unitization E of P. We also study relations between Riesz ideals and central elements in GEAs and in their unitizations. In the last section, we demonstrate the notion of Riesz ideals by some illustrative examples. Received June 28, 2005; accepted in final form January 23, 2007.     

Uniquely Complemented Posets

We study complementation in bounded posets. It is known and easy to see that every complemented distributive poset is uniquely complemented. The converse statement is not valid, even for lattices. In the present paper we provide conditions that force a uniquely complemented poset to be distributive. For atomistic resp. atomic posets as well as for posets satisfying the descending chain condition we find sufficient conditions in the form of so-called LU-identities. It turns out that for finite posets these conditions are necessary and sufficient.     

Modular functions: Uniform boundedness and compactness

We prove the Nikodym Boundedness, Brooks-Jewett and Vitali-Hahn-Saks theorems for modular functions on orthomodular lattices with SIP and on particular complemented or sectionally complemented lattices, and the equivalence, for any complemented or sectionally complemented lattice, between the Brooks-Jewett and Vitali-Hahn-Saks theorems for group-valued modular functions. As consequence, we obtain characterizations of relative, sequential and weak compactness in spaces of modular functions.     

Frink quasicontinuous posets

In this paper, the concept of Frink quasicontinuous posets is introduced. The main results are: (1) a poset is a Frink quasicontinuous poset if and only if its normal completion is a quasicontinuous lattice; (2) a poset is precontinuous if and only if it is Frink quasicontinuous and meet precontinuous; (3) when a Frink quasicontinuous poset satisfies certain conditions, the way below relation has the interpolation property; (4) the category of quasicontinuous lattices with complete homomorphisms is a full reflective subcategory of the category of Frink quasicontinuous posets with cut-stable maps.     

Orthocomplemented Posets with a Symmetric Difference

We endow orthocomplemented posets with a binary operation–an abstract symmetric difference of sets–and we study algebraic properties of this class, . Denoting its elements by ODP, we first investigate on the features related to compatibility in ODPs. We find, among others, that any ODP is orthomodular. This explicitly links with the theory of quantum logics. By analogy with Boolean algebras, we then ask if (when) an ODP is set representable. Though we find that general ODPs do not have to be set representable, many natural ODPs are shown to be. We characterize the set-representable ODPs in terms of two valued morphisms and prove that they form a quasivariety. This quasivariety contains the class of pseudocomplemented ODPs as we show afterwards. At the end we ask whether any orthomodular poset can be converted or, more generally, enlarged to an ODP. By countre-examples we answer these questions to the negative. The authors acknowledge the support of the research plan MSM 0021620839 that is financed by the Ministry of Education of the Czech Republic and the grant GAČR 201/07/1051 of the Czech Grant Agency.     

Irreducible orthomodular lattices which are simple

It is well known that for a chain finite orthomodular lattice, all congruences are factor congruences, so any directly irreducible chain finite orthomodular lattice is simple. In this paper it is shown that the notions of directly irreducible and simple coincide in any variety generated by a set of orthomodular lattices that has a uniform finite upper bound on the lengths of their chains. The prototypical example of such a variety is any variety generated by a set ofn dimensional orthocomplemented projective geometries.Presented by B. Jónsson.Supported by a grant from NSERC.     

A Poset Fiber Theorem for Doubly Cohen-Macaulay Posets and Its Applications

This paper studies topological properties of the lattices of non-crossing partitions of types A and B and of the poset of injective words. Specifically, it is shown that after the removal of the bottom and top elements (if existent) these posets are doubly Cohen-Macaulay. This strengthens the well-known facts that these posets are Cohen-Macaulay. Our results rely on a new poset fiber theorem which turns out to be a useful tool to prove double (homotopy) Cohen- Macaulayness of a poset. Applications to complexes of injective words are also included.     

Parity representations of posets

Parity representations, introduced in this paper, comprise a new method of representation of posets that yields insight into the combinatorics of the poset of all intervals of a poset. Results here generalize some results previously obtained for the face lattices of binary partition polytopes.     

Topologies on products of partially ordered sets II: Ideal topologies

Within the theory of ideals in partially ordered sets, several difficulties set in which do not occur in the special case of lattices (or bidirected posets). For example, a finite product of ideals in the factor posets need not be an ideal in the product poset. The notion ofstrict ideals is introduced in order to remedy some deficiencies occurring in the general case of an arbitrary product of posets. Besides other results, we show the following main theorem: The ideal topology (cf. [2]) of a product of non-trivial posets coincides with the product topology if and only if the number of factors is finite (4.19.). Presented by L. Fuchs     

Modular posets and semigroups

Posets and poset homomorphisms (preserving both order and parallelism) have been shown to form a category which is equivalent to the category of pogroupoids and their homomorphisms. Among the posets those posets whose associated pogroupoids are semigroups are identified as being precisely those posets which are (C ₂+1)-free. In the case of lattices this condition means that the lattice is alsoN ₅-free and hence modular. Using the standard connection: semigroup to poset to pogroupoid, it is observed that in many cases the image pogroupoid obtained is a semigroup even if quite different from the original one. The nature of this mapping appears intriguing in the poset setting and may well be so seen from the semigroup theory viewpoint.     

On the greedy dimension of a partial order

This paper introduces a new concept of dimension for partially ordered sets. Dushnik and Miller in 1941 introduced the concept of dimension of a partial order P, as the minimum cardinality of a realizer, (i.e., a set of linear extensions of P whose intersection is P). Every poset has a greedy realizer (i.e., a realizer consisting of greedy linear extensions). We begin the study of the notion of greedy dimension of a poset and its relationship with the usual dimension by proving that equality holds for a wide class of posets including N-free posets, two-dimensional posets and distributive lattices.     

Prime, irreducible elements and coatoms in posets

In this paper, some properties of prime elements, pseudoprime elements, irreducible elements and coatoms in posets are investigated. We show that the four kinds of elements are equivalent to each other in finite Boolean posets. Furthermore, we demonstrate that every element of a finite Boolean poset can be represented by one kind of them. The example presented in this paper indicates that this result may not hold in every finite poset, but all the irreducible elements are proved to be contained in each order generating set. Finally, the multiplicative auxiliary relation on posets and the notion of arithmetic poset are introduced, and some properties about them are generalized to posets.     

Parcours dans les graphes: Un outil pour l'algorithmique des ensembles ordonnes

A finite poset can easily be represented by a directed acyclic graph. This work intends to make use of performant graph search methods as a tool for checking order properties. Semi-modular posets and semilattices are investigated here.An underlying idea consists in turning to account structural properties of the poset, and deriving effective algorithms. This purpose leads us to ‘good’ theoretic characterizations — that is, directly available from an algorithmic point of view — and especially a new one for modular lattices.     

A Boolean topological orthomodular poset

Wilce introduced the notion of a topological orthomodular poset and proved any compact topological orthomodular poset whose underlying orthomodular poset is a Boolean algebra is a topological Boolean algebra in the usual sense. Wilce asked whether the compactness assumption was necessary for this result. We provide an example to show the compactness assumption is necessary.