Set Of Partial Orders Research Articles

Partition complexes, duality and integral tree representations

We show that the poset of non-trivial partitions of 1,2,...,n has a fundamental homology class with coefficients in a Lie superalgebra. Homological duality then rapidly yields a range of known results concerning the integral representations of the symmetric groups S_n and S_{n+1} on the homology and cohomology of this partially-ordered set.

Crisp dimension theory and valued preference relations

Representation of binary preference relations in a real space where each coordinate suggests the existence of underlying criteria is a standard and indeed suggestive approach. Classical dimension theory addresses this problem, showing that whenever crisp preferences define a partial order set, it can be represented in a real space, and then we can search for a minimal representation. Valued preference relation being a much more complex structure, there is an absolute need for meaningful representations, being manageable by decision-makers. In this paper, we continue analyzing the concept of a generalized dimension function of valued preference relations, i.e. a mapping assigning a generalized dimension value to every α-cut of any given valued preference relation, as introduced in a previous paper. We should of course be expecting deep computational problems within this generalized dimension context, since they are already present in crisp dimension theory. In this paper, we present some properties of such a generalized dimension function, pointing out that our approach allows alternative representations depending on some underlying rationality core the decision-maker may change.

Density and unique decomposition theorems for the lattice of cellular classes

A class C of pointed spaces is called a cellular class if it is closed under weak equivalences, arbitrary wedges and pointed homotopy pushouts. The smallest cellular class containing X is denoted by C(X), and a partial order relation much less than is defined by: X much less than Y if Y is an element of C(X). In this text we investigate the sub partial order sets generated respectively by simply connected finite CW-complexes and by rational spaces. For rational spaces we prove a unique decomposition theorem, a density theorem and the existence of infinitely many non-comparable elements. We then prove the density theorem for a generic class of finite CW-complexes.

Relating axiomatic and operational semantics of place/transition nets: From process terms to partial orders

In this paper we define partial orders associated to process terms of place/transition nets. Further, we construct a set of partial orders from a given set of process terms. For place/transition nets without loops we prove the following result:The set of minimal partial orders constructed from process terms is in a one-to-one correspondence with the set of minimal partial orders defined by processes of the given net.

Least Squares Isotonic Regression in Two Dimensions

Given a finite partially-ordered set with a positive weighting function defined on its points, it is well known that any real-valued function defined on the set has a unique best order-preserving approximation in the weighted least squares sense. Many algorithms have been given for the solution of this isotonic regression problem. Most such algorithms either are not polynomial or they are of unknown time complexity. Recently, it has become clear that the general isotonic regression problem can be solved as a network flow problem in time O(n4) with a space requirement of O(n2), where n is the number of points in the set. Building on the insights at the basis of this improvement, we show here that, in the case of a general two-dimensional partial ordering, the problem can be solved in O(n3) time and, when the two-dimensional set is restricted to a grid, the time can be further improved to O(n2).

A novel consistency control algorithm in Cooperative Graphics Editor (CGE)

The research work presented in this paper is based on the concrete background of the Cooperative Graphics Editor (CGE), allowing two or more persons to remotely edit a graphic document simultaneously. A new concurrency control algorithm based on partial order set is presented, which has fast response and less undo-redo operations as there are no lock mechanisms. It is used to solve inconsistency caused by operation on intersecting graphics concurrently. CGE also possesses a mask strategy to solve inconsistency caused by operation on the same graphic concurrently.

Jump number maximization for proper interval graphs and series—parallel graphs

In this paper, the order-theoretic problem of finding linear extensions of a partially ordered set (POSet) with certain properties is mapped to the corresponding graphtheorectic problem. Given a POSet, a jump exists between two consecutive symbols in a topological sequence if there is no relation between these two symbols in the POSet. We want to find a topological sequence that minimizes or maximizes the total number of jumps. In graph-theoretic terms, define the jump number optimization problem as finding a linear ordering of vertex sequence of a graph such that the number of pairs of consecutive vertices that are adjacent is maximized (jump number minimization problem) or minimized (jump number maximization problem). The complexities of both problems with respect to some special graphs are investigated first in this paper. We then study the jump maximization problem for proper interval graphs and series—parallel graphs. From this study, we hope that more researches can be derived from these interdisciplinary knowledge domains.

A Poset Dimension Algorithm

This article presents an algorithm which computes the dimension of an arbitrary finite poset (partial order set). This algorithm is based on the chromatic number of a graph instead of the classical approach based on the chromatic number of some hypergraph. The relation between both approaches is analyzed. With this algorithm, the dimension of many modest size posets can be computed. Otherwise, an upper bound for the poset dimension is obtained. Some computational results are included.

The common order-theoretic structure of version spaces and ATMSs

We demonstrate how order-theoretic abstractions can be useful in identifying, formalizing, and exploiting relationships between seemingly dissimilar AI algorithms that perform computations on partially-ordered sets. In particular, we show how the order-theoretic concept of an anti-chain can be used to provide an efficient representation for such sets when they satisfy certain special properties. We use anti-chains to identify and analyze the basic operations and representation optimizations in the version space learning algorithm and the assumption-based truth maintenance system (ATMS). Our analysis allows us to (1) extend the known theory of admissibility of concept spaces for incremental version space merging, and (2) develop new, simpler label-update algorithms for ATMSs with DNF assumption formulas.

Trees and semantics

Several types of trees are proposed in the literature to model the behaviour of nondeterministic processes. The purpose of this paper is to clarify the distinction between the various types of trees and to discuss their use as semantic domains. Trees (in the sense of Milner's derivation trees) and bisimulation equivalence classes of trees are considered in three different settings: in the category of sets, in the metric setting and in the partial order setting. Each type of tree is characterized by a domain equation. Special attention is given to the use of trees as models for processes which allow for unbounded nondeterminism as for instance Milners CCS processes. In this case one has to deal with infinitely branching trees which cause problems to give denotational semantics that are fully abstract w.r.t. a transition system based operational semantics.

Extensions of G-posets and Quillen's complex

AbstractWe develop techniques to compute the homology of Quillen's complex of elementary abelian p-subgroups of a finite group in the case where the group has a normal subgroup of order divisible by p. The main result is a long exact sequence relating the homologies of these complexes for the whole group, the normal subgroup, and certain centralizer subgroups. The proof takes place at the level of partially-ordered sets. Notions of suspension and wedge product are considered in this context, which are analogous to the corresponding notions for topological spaces. We conclude with a formula for the generalized Steinberg module of a group with a normal subgroup, and give some examples.

Uppers to zero in intersections of prime ideals

Let R be a Noetherian integral domain. The structure of the partially-ordered set of prime ideals of R[z], the polynomial ring in one indeterminate over R, is not fully understood. I demonstrate that if p1,…,pn are prime ideals in R[x] with ht(pi) > 2 and either n = 1 or R is not a Henselian local domain of dimension 2 each of which contains a monic polynomial, then their intersection contains [R] many prime ideals meeting R at (0), each containing a monic polynomial.

Forcing isomorphism

If two models of a first-order theory are isomorphic, then they remain isomorphic in any forcing extension of the universe of sets. In general however, such a forcing extension may create new isomorphisms. For example, any forcing that collapses cardinals may easily make formerly nonisomorphic models isomorphic. However, if we place restrictions on the partially-ordered set to ensure that the forcing extension preserves certain invariants, then the ability to force nonisomorphic models of some theory T to be isomorphic implies that the invariants are not sufficient to characterize the models of T.A countable first-order theory is said to be classifiable if it is superstable and does not have either the dimensional order property (DOP) or the omitting types order property (OTOP). If T is not classifiable, Shelah has shown in [5] that sentences in L∞,λ do not characterize models of T of power λ. By contrast, in [8] Shelah showed that if a theory T is classifiable, then each model of cardinality λ is described by a sentence of L∞,λ. In fact, this sentence can be chosen in the . ( is the result of enriching the language by adding for each μ < λ a quantifier saying the dimension of a dependence structure is greater than μ) Further work ([3], [2]) shows that ⊐+ can be replaced by ℵ1.

The poset of infinitary traces

Partially commutative monoids, also called trace monoids, are among the most-studied formalisms to describe the behaviour of distributed systems. In order to model systems which never stop, we have to consider an extension of traces, namely infinite traces. Finite-trace monoids are strongly related to partial-order sets ( PoSets), domains and event structures, which are other models to describe the behaviour of distributed systems. The aim of this paper is to establish similar connexions between infinite-trace monoids, PoSets and event structures. We prove that the set of finite and infinite traces with the prefix order is a Scott domain and a coherently complete prime algebraic PoSet. Moreover, we establish a representation theorem between the class of finite- and infinite-trace PoSets and a subclass of labelled prime event structures.

On chains and posets within the power set of a continuum

Transfinite induction is employed to construct a copy of an arbitrary partially‐ordered set of cardinality at most c within the power set (quasi‐ordered by sub‐chain embeddability) of the real line.

Mutually complementary partial orders

Two partial orders P=( X,⩽) and Q=( X, ⩽′) are complementary if P ∩ Q={( x, x): x ε x} and the transitive closure of P ∩ Q is {( x, y : x, y ε x}. We investigate here the size ω( n) of the largest set of pairwise complementary partial orders on a set of size n. In particular, for large n we constructπ( n/log n) mutually complementary partial orders of order n, and show on the other hand that ω( n)<0.486 n for all sufficiently large n. This provides an estimate of the maximum number of mutually complementary T 0 topologies on a set of size n.

Models and logics for true concurrency

A distributed computer system consists of different processes oragents that function largely autonomously and coordinate their actions by communicating with each other. In such a situation, actions may be performed by different agents of the system locally, in a concurrent manner.In this paper, we first discuss formal models of distributed systems in which concurrency is specifiedexplicitly, in contrast to more traditional approaches where concurrency is representedimplicitly as a nondeterministic choice between all possible sequentializations of concurrent actions. This naturally leads to models based on partially-ordered sets of actions rather than sequences of actions and is often called thetrue concurrency approach. The models we focus on are distributed transition systems, elementary net systems and event structures.In the second half of the paper, we develop a family of logics to specify and reason about the behavioural properties of the models we have described. The logics we define are extensions of temporal logic with new modalities to directly describe concurrency.This paper is essentially a survey of work done by the authors during the last few years on modelling distributed systems with true concurrency and using logic to reason about these models. The emphasis is on motivating definitions through examples and on presenting major results, without going into too many formal details. We provide pointers to the literature where these details can be found.

Connectedness and synchronization

Given a description language for concurrent systems, the specification of its semantics amounts usually to assigning to each description an object from a hopefully well understood domain D whose elements are called semantical processes. Here is a general accepted “taxonomy” of event oriented processes: (1) Linear-process: a set of runs, where a run is a sequence of events (actions) [3]. (2) Pomset process: a set of node-labelled partial orders, henceforth referred to as pomsets [lo, 12, 51. (3) Automaton process: a rooted transition diagram whose edges are labelled by actions [7]. (4) Event structure [9]; also behavior structure [13]. This list reflects the alternatives: l “Linear” time (1,2) vs branching time (3,4). l Interleaving (1,3) vs causality (“true concurrency”) (2,4). The four domains support different levels of abstraction in semantical specifications: linear processes are the coarsest, whereas event structures (behavior structures), which integrate branching and causality are the most discriminating. On each of these levels synchronization is a basic operation of composing event oriented concurrent systems. In general, by synchronization we mean a composition

The Euclidean distance construction of order homomorphisms

The Euclidean distance technique of Arrow and Hahn is used to construct an upper semicontinuous order homomorphism (partial utility function) from ( X, ≻) to ( R , >), where X is a closed, convex subset of R N and ≻ is a continuous strict partial order on X. It is also shown that the order homomorphism is upper semicontinuous as a function on X× P(X) , where P(X) is the set of continuous strict partial orders on X, taken with the topology of closed convergence.

Bound Sets in Partial Orders and the Fixed Point Property

AbstractIn this paper we introduce several properties closely related to the fixed point property of a partially ordered set P: the comparability property, the fixed point property for cones, and the fixed point extension property. We apply these properties to the sets of common bounds of the minimal (maximal) elements of the partially ordered set P in order to derive fixed point theorems for P.