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.     

偏序集的代数完备

引进一个偏序集的代数完备, 并且构造任意偏序集的一个代数完备.有最小元的并半格的代数完备正好是它的理想完备. 一个偏序集的代数完备同构于它的一个由下集作为元的完备格,并且这个完备格包含所有主理想. 基于代数完备的Galois联络的下扩张仍然是一个Galois联络.     

Varieties of Posets

We introduce and investigate the notion of a homomorphism, of a congruence relation, of a substructure of a poset and consequently the notion of a variety of posets. These notions are consistent with those used in lattice theory and multilattice theory. There are given some properties of the lattice of all varieties of posets.     

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.

     

Betti-linear ideals

We introduce the notion of a Betti-linear monomial ideal, which generalizes the notion of lattice-linear monomial ideal introduced by Clark. We provide a characterization of Betti-linearity in terms of Tchernev's poset construction. As an application we obtain an explicit canonical construction for the minimal free resolutions of monomial ideals having pure resolutions.     

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.     

Some linked subsets of posets

Recently Frank Tall has introduced the notion of a poset being “endowed” which is used in a forcing and reflection type proof to show that certain properties are preserved by forcing with such a poset. Subsequently Bill Fleissner introduced the property of a boolean algebra having a “lynx” and used this in an “Axiom and combinatorics” type proof of the same result. The two notions are basically the same thing. In this paper we prove that some of the usual Cohen algebras have lynxes, that it is consistent that nearly all of them have lynxes and investigate more generally the notion of lynxes. Research supported by NSERC of Canada.     

On <Emphasis Type="Italic">Q</Emphasis>-Upper Algebras

Given a poset we introduce the notion of Q-upper algebras and study the (positive) implicativity, commutativity and quasi-commutativity in Q-upper algebras.     

On thecd-variation polynomials of André and simsun permutations

We prove a conjecture of Stanley on thecd-index of the semisuspension of the face poset of a simplicial shelling component. We give a new signed generalization of André permutations, together with a new notion ofcd-variation for signed permutations. This generalization not only allows us to compute thecd-index of the face poset of a cube, but also occurs as a natural set of orbit representatives for a signed generalization of the Foata-Strehl commutative group action on the symmetric group. From the induction techniques used, it becomes clear that there is more than one way to define classes of permutations andcd-variation such that they allow us to compute thecd-index of the same poset. This research was supported by the UQAM Foundation.     

Level Eulerian Posets

The notion of level posets is introduced. This class of infinite posets has the property that between every two adjacent ranks the same bipartite graph occurs. When the adjacency matrix is indecomposable, we determine the length of the longest interval one needs to check to verify Eulerianness. Furthermore, we show that every level Eulerian poset associated to an indecomposable matrix has even order. A condition for verifying shellability is introduced and is automated using the algebra of walks. Applying the Skolem–Mahler–Lech theorem, the ab-series of a level poset is shown to be a rational generating function in the non-commutative variables a and b. In the case the poset is also Eulerian, the analogous result holds for the cd-series. Using coalgebraic techniques a method is developed to recognize the cd-series matrix of a level Eulerian poset.     

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.     

Order varieties generated by finite posets

In a 1981 paper, Duffus and Rival define an order variety as a class of posets that is closed under the formation of products and retracts. They also introduce the notion of an irreducible poset. In the present paper we define nonextendible colored posets and certain minimal nonextendible colored posets that we call zigzags. We characterize via nonextendible colored posets the order varieties generated by a set of posets. If the generating set contains only finite posets our characterization is via zigzags. By using these theorems we give a characterization of finite irreducible posets.As an application we show that two different finite irreducible posets generate two different order varieties. We also show that there is a poset which has two different representations by irreducible posets. We thereby settle two open problems listed in the Duffus and Rival paper.     

Approximation of Metric Spaces by Partial Metric Spaces

Partial metrics are generalised metrics with non-zero self-distances. We slightly generalise Matthews' original definition of partial metrics, yielding a notion of weak partial metric. After considering weak partial metric spaces in general, we introduce a weak partial metric on the poset of formal balls of a metric space. This weak partial metric can be used to construct the completion of classical metric spaces from the domain-theoretic rounded ideal completion.     

On the facets of the secondary polytope

The secondary polytope of a point configuration A is a polytope whose face poset is isomorphic to the poset of all regular subdivisions of A. While the vertices of the secondary polytope - corresponding to the triangulations of A - are very well studied, there is not much known about the facets of the secondary polytope.The splits of a polytope, subdivisions with exactly two maximal faces, are the simplest examples of such facets and the first that were systematically investigated. The present paper can be seen as a continuation of these studies and as a starting point of an examination of the subdivisions corresponding to the facets of the secondary polytope in general. As a special case, the notion of k-split is introduced as a possibility to classify polytopes in accordance to the complexity of the facets of their secondary polytopes. An application to matroid subdivisions of hypersimplices and tropical geometry is given.     

Left-exact localizations of ∞-topoi II: Grothendieck topologies

We revisit the work of Toën–Vezzosi and Lurie on Grothendieck topologies, using the new tools of acyclic classes and congruences. We introduce a notion of extended Grothendieck topology on any ∞-topos, and prove that the poset of extended Grothendieck topologies is isomorphic to that of topological localizations, hypercomplete localizations, Lawvere–Tierney topologies, and covering topologies (a variation on the notion of pretopology). It follows that these posets are small and have the structure of a frame. We revisit also the topological–cotopological factorization by introducing the notion of a cotopological morphism. And we revisit the notions of hypercompletion, hyperdescent, hypercoverings and hypersheaves associated to an extended Grothendieck topology.We also introduce the notion of forcing, which is a tool to compute with localizations of ∞-topoi. We use this in particular to show that the topological part of a left-exact localization of an ∞-topos is universally forcing the generators of this localization to be ∞-connected instead of inverting them.     

Duality respecting representations and compatible complexity measures for gammoids

We show that every gammoid has special digraph representations, such that a representation of the dual of the gammoid may be easily obtained by reversing all arcs. In an informal sense, the duality notion of a poset applied to the digraph of a special representation of a gammoid commutes with the operation of forming the dual of that gammoid. We use these special representations in order to define a complexity measure for gammoids, such that the classes of gammoids with bounded complexity are closed under duality, minors, and direct sums.     

Adjacency posets of planar graphs

In this paper, we show that the dimension of the adjacency poset of a planar graph is at most 8. From below, we show that there is a planar graph whose adjacency poset has dimension 5. We then show that the dimension of the adjacency poset of an outerplanar graph is at most 5. From below, we show that there is an outerplanar graph whose adjacency poset has dimension 4. We also show that the dimension of the adjacency poset of a planar bipartite graph is at most 4. This result is best possible. More generally, the dimension of the adjacency poset of a graph is bounded as a function of its genus and so is the dimension of the vertex-face poset of such a graph.