Set Of Partial Orders Research Articles

Aggregation of L-Probabilistic Quasi-Uniformities

The problem of aggregating fuzzy structures, mainly fuzzy binary relations, has deserved a lot of attention in the last years due to its application in several fields. Here, we face the problem of studying which properties must satisfy a function in order to merge an arbitrary family of (bases of) L-probabilistic quasi-uniformities into a single one. These fuzzy structures are special filters of fuzzy binary relations. Hence we first make a complete study of functions between partially-ordered sets that preserve some special sets, such as filters. Afterwards, a complete characterization of those functions aggregating bases of L-probabilistic quasi-uniformities is obtained. In particular, attention is paid to the case L={0,1}, which allows one to obtain results for functions which aggregate crisp quasi-uniformities. Moreover, we provide some examples of our results including one showing that Lowen’s functor ι which transforms a probabilistic quasi-uniformity into a crisp quasi-uniformity can be constructed using this aggregation procedure.

Construction of an Energy Saving Model Based on FA and Its Applications for Detection of Battery Power Consumption

Although some algebraic theories such as partial order set, lattice and their properties on fuzzy set had been studied, these algebraic theories on fuzzy automata (FA) have not been discussed at present. For saving the energy of the battery so as to make it play the best performance, this paper proposes first an algebraic system on a finite-state deterministic fuzzy automaton (FDFA), including the partial order set, lattice and its properties. Second, this paper constructs a detection model system to monitor the consumption of battery power, and the detection model consists of four parts, as well as including supervision function network and decision-making process. Third, this paper gives the whole dynamic process of consumption and consumption rate of battery power, as well as discusses the comparison of the detection accuracy of between the detection model and other detection methods in the battery power consumption by simulation. Through the homomorphism and the isomorphic mapping of the lattice, the partial state of the FDFA is studied to realize the function of the whole state. These research results will provide a theoretical reference, system modeling and application background for many departments such as control, detection, tracking, identification and so on.

Continuity of Solution Mappings for Parametric Set Optimization Problems under Partial Order Relations

This paper mainly investigates the semicontinuity of solution mappings for set optimization problems under a partial order set relation instead of upper and lower set less order relations. To this end, we propose two types of monotonicity definition for the set-valued mapping introduced by two nonlinear scalarization functions which are presented by these partial order relations. Then, we give some sufficient conditions for the semicontinuity and closedness of solution mappings for parametric set optimization problems. The results presented in this paper are new and extend the main results given by some authors in the literature.

On difference graphs and the local dimension of posets

The dimension of a partially-ordered set (poset), introduced by Dushnik and Miller (1941), has been studied extensively in the literature. Recently, Ueckerdt (2016) proposed a variation called local dimension which makes use of partial linear extensions. While local dimension is bounded above by dimension, they can be arbitrarily far apart as the dimension of the standard example is n while its local dimension is only 3. Hiraguchi (1955) proved that the maximum dimension of a poset of order n is n∕2. However, we find a very different result for local dimension, proving a bound of Θ(n∕logn). This follows from connections with covering graphs using difference graphs which are bipartite graphs whose vertices in a single class have nested neighborhoods. We also prove that the local dimension of the n-dimensional Boolean lattice is Ω(n∕logn) and make progress toward resolving an analogue of the removable pair conjecture for local dimension.

A new technique for coincidence point theory in metric spaces endowed with graph

Coincidence theory is a generalization of fixed point theory. There are many researchs combining fixed point theory and graph theory. In this paper, a new type of multi-valued mapping and g-l-graph preserving is proposed to prove a coincidence point theorem on complete metric spaces endowed with a directed graph. Supported examples of there main theorems are also introduced. Main results are sufficiently conditions for finding a vertex in the directed graph such that itself image of the defined surjective mapping is contained in the defined multivalued mapping. The proposed theorem can be apply use to obtain the similar result in a matrix space endowed with a partial order set.

Identifying individual differences in adolescent neuropsychological function using the NIH Toolbox: An application of partially ordered classification modeling.

The Cognition Battery of the National Institutes of Heath Toolbox is a commonly utilized set of assessments of neuropsychological abilities, evaluating executive function, attention, working memory, processing speed, and episodic memory. We highlight the utility of an advanced statistical model in providing nuanced characterization of neurocognition in an adolescent population. We propose that partially ordered set (POSET) models are well suited to analyze polyfactorial tasks and identify distinct profiles of cognitive functioning. Two models were considered using POSET classification. The first modeled 5 distinct cognitive functions and allowed for multiple functions to contribute to task performance. The second simpler model involved only 2 broader-based functions without polyfactorial task specifications. Existing performance data from 745 adolescents aged 14-17 years were analyzed. Posterior probabilities of classification performance and the discriminatory properties of the estimated response distributions indicated how well the modeling approaches fit the data. The larger first model resulted in 8 profiles or states characterized by combinations of high or low functioning in 5 distinct functions. The simpler second model involved 2 broader-based functions that resulted in 4 states. Comparing model fit criteria, we believe that the finer-grained first model may better reflect the cognitive constructs associated with the tasks. Notably, POSET modeling did not always provide adequate classification of working memory because of the limited design of the Cognition Battery. We demonstrate that the use of POSET models is a feasible approach for detailed analysis of neurocognitive data that can extract information on cognitive functions, even when provided with limited task batteries. (PsycINFO Database Record (c) 2019 APA, all rights reserved).

Paths to Understanding Birational Rowmotion on Products of Two Chains

Birational rowmotion is an action on the space of assignments of rational functions to the elements of a finite partially-ordered set (poset). It is lifted from the well-studied rowmotion map on order ideals (equivariantly on antichains) of a poset $P$, which when iterated on special posets, has unexpectedly nice properties in terms of periodicity, cyclic sieving, and homomesy (statistics whose averages over each orbit are constant) [AST11, BW74, CF95, Pan09, PR13, RuSh12,RuWa15+,SW12, ThWi17, Yil17. In this context, rowmotion appears to be related to Auslander-Reiten translation on certain quivers, and birational rowmotion to $Y$-systems of type $A_m \times A_n$ described in Zamolodchikov periodicity. We give a formula in terms of families of non-intersecting lattice paths for iterated actions of the birational rowmotion map on a product of two chains. This allows us to give a much simpler direct proof of the key fact that the period of this map on a product of chains of lengths $r$ and $s$ is $r+s+2$ (first proved by D.~Grinberg and the second author), as well as the first proof of the birational analogue of homomesy along files for such posets.

Existence of Generalized Nash Equilibrium in n-Person Noncooperative Games under Incomplete Preference

To prove the existence of Nash equilibrium by traditional ways, a common condition that the preference of players must be complete has to be considered. This paper presents a new method to improve it. Based on the incomplete preference corresponding to equivalence class set being a partial order set, we translate the incomplete preference problems into the partial order problems. Using the famous Zorn lemma, we get the existence theorems of fixed point for noncontinuous operators in incomplete preference sets. These new fixed point theorems provide a new way to break through the limitation. Finally, the existence of generalized Nash equilibrium is strictly proved in the n-person noncooperative games under incomplete preference.

Compact and efficiently verifiable models for concurrent systems

Partial orders are a fundamental mathematical structure capable of representing concurrency and causality on a set of atomic events. In many applications it is essential to consider multiple partial orders, each representing a particular behavioral scenario or an operating mode of a system. With the exploding growth of the complexity of systems that software and hardware engineers design today, it is no longer feasible to represent each partial order of a large system explicitly, therefore compressed representations of sets of partial orders become essential for improving the scalability of design automation tools. In this paper we study two well known mathematical formalisms capable of the compressed representation of sets of partial orders: Labeled Event Structures and Conditional Partial Order Graphs. We discuss their advantages and disadvantages and propose efficient algorithms for transforming a set of partial orders from a given compressed representation in one formalism into an equivalent representation in another formalism without explicitly enumerating every partial order. The proposed algorithms make use of an intermediate mathematical formalism which we call Conditional Labeled Event Structures that combines the advantages of both structures. Finally, we compare these structures on a number of benchmarks coming from concurrent software and hardware domains.

Ramsey Numbers for Partially-Ordered Sets

We present a refinement of Ramsey numbers by considering graphs with a partial ordering on their vertices. This is a natural extension of the ordered Ramsey numbers. We formalize situations in which we can use arbitrary families of partially-ordered sets to form host graphs for Ramsey problems. We explore connections to well studied Turan-type problems in partially-ordered sets, particularly those in the Boolean lattice. We find a strong difference between Ramsey numbers on the Boolean lattice and ordered Ramsey numbers when the partial ordering on the graphs have large antichains.

Summarizing data using partially ordered set theory: An application to fiscal frameworks in 97 countries

The widespread use of composite indices has often been motivated by their practicality to quantify qualitative data in an easy and intuitive way. At the same time, this approach has been challenged due to the subjective and partly ad hoc nature of computation, aggregation and weighting techniques as well as the handling of missing data. Partially ordered set (POSET) theory offers an alternative approach for summarizing qualitative data in terms of quantitative indices, which relies on a computation scheme that fully exploits the available information and does not require the subjective assignment of weights. The present paper makes the case for an increased use of POSET theory in the social sciences and provides a comparison of POSET indices and composite indices (from previous studies) measuring the 'stringency' of fiscal frameworks using data from the OECD Budget Practices and Procedures survey (2007/08). (authors' abstract)

Effects of extreme prematurity on numerical skills and executive function in kindergarten children: An application of partially ordered classification modeling

Although mathematics disabilities (MD) are common in extremely preterm/extremely low birth weight (EPT/ELBW) children, little is known about the nature of these problems. In this study partially ordered set (POSET) models were applied to classify 140 EPT/ELBW kindergarten children (gestational age<28weeks and/or birth weight<1000g) and 110 normal birth weight (NBW) controls into profiles of numerical and cognitive skills. Models based on five numerical skills and five executive function and processing speed skills provided a good fit to performance data. The EPT/ELBW group had poorer skills in all areas than NBW controls but the models also revealed substantial individual variability in skill profiles. Weaknesses in executive function were associated with poorer mastery of numerical skills. The findings illustrate the applicability of POSET models to research on MD and suggest distinct types of early numerical deficits in EPT/ELBW children that are related to their impairments in executive function.

On structural decompositions of finite frames

A frame in an $n$-dimensional Hilbert space $H_n$ is a possibly redundant collection of vectors $\{f_i\}_{i\in I}$ that span the space. A tight frame is a generalization of an orthonormal basis. A frame $\{f_i\}_{i\in I}$ is said to be scalable if there exist nonnegative scalars $\{c_i\}_{i\in I}$ such that $\{c_if_i\}_{i\in I}$ is a tight frame. In this paper we study the combinatorial structure of frames and their decomposition into tight or scalable subsets by using partially-ordered sets (posets). We define the factor poset of a frame $\{f_i\}_{i\in I}$ to be a collection of subsets of $I$ ordered by inclusion so that nonempty $J\subseteq I$ is in the factor poset if and only if $\{f_j\}_{j\in J}$ is a tight frame for $H_n$. A similar definition is given for the scalability poset of a frame. We prove conditions which factor posets satisfy and use these to study the inverse factor poset problem, which inquires when there exists a frame whose factor poset is some given poset $P$. We determine a necessary condition for solving the inverse factor poset problem in $H_n$ which is also sufficient for $H_2$. We describe how factor poset structure of frames is preserved under orthogonal projections. We also consider the enumeration of the number of possible factor posets and bounds on the size of factors posets. We then turn our attention to scalable frames and present partial results regarding when a frame can be scaled to have a given factor poset.

Choice Overload and Height Ranking of Menus in Partially-Ordered Sets

When agents face incomplete information and their knowledge about the objects of choice is vague and imprecise, they tend to consider fewer choices and to process a smaller portion of the available information regarding their choices. This phenomenon is well-known as choice overload and is strictly related to the existence of a considerable amount of option-pairs that are not easily comparable. Thus, we use a finite partially-ordered set (poset) to model the subset of easily-comparable pairs within a set of options/items. The height ranking, a new ranking rule for menus, namely subposets of a finite poset, is then introduced and characterized. The height ranking rule ranks subsets of options in terms of the size of the longest chain that they include and is therefore meant to assess menus of available options in terms of the maximum number of distinct and easily-comparable alternative options that they offer.

Classifying Finite-Dimensional C*-Algebras by Posets of Their Commutative C*-Subalgebras

We consider the functor C that to a unital C*-algebra A assigns the partial order set C(A) of its commutative C*-subalgebras ordered by inclusion. We investigate how some C*-algebraic properties translate under the action of C to order-theoretical properties. In particular, we show that A is finite dimensional if and only C(A) satisfies certain chain conditions. We eventually show that if A and B are C*-algebras such that A is finite dimensional and C(A) and C(B) are order isomorphic, then A and B must be *-isomorphic.

Creation of partial orders of variants for selection of optimal alternatives in homogeneous sets

Subject of consideration. The paper deals with the method for structure creation of the initial set of homogeneous variants in the database adapted to selection task solution according to their target (functional) assignment. This task is actual for the electronic quick reference guides: on materials, components, parts, medicines, etc. which special-purpose designation in homogeneous group is more time-proof, than requirements on admissibility in each new selection task. Method. The offered approach is based on data structure creation representing the partial order of alternatives, designed by unconditional Pareto criterion, from a set of linear or partial orders of variants with smaller dimension. The dimension of quality measures taken into account specifies the partial order dimension, and its structure is predetermined by a special-purpose designation of variants in homogeneous set through taken into account quality measures. The final elements in the partial order graph represent Pareto optimal variants, which need to be tested only on admissibility in each new selection task. The resulting partial order is formed by means of factor sets. Main results. We have worked out the method of data adaptive adjustment on selection task which gives the possibility to start solution at once with Pareto optimal variants only by their admissibility check. The offered approach is much more effective than the traditional one, when in the beginning selection of valid variants is done, and then of the optimal ones, owing to reduction the number of multiple combinatorial criteria comparison in each new selection task. The method efficiency is confirmed by the fact that the power of Pareto optimal variants is always less or equal to the power of initial variants, that is theoretically proved in [1]. It means, that the search of valid variants under the same requirements on smaller power set is always less labour-intensive, than the search of valid variants on the greater power set, owing to the less number of operations of binary comparisons of variants. Practical significance. The results of work are usable at creation of the electronic quick reference guides for standard and generic items, both as part of automated design engineering systems, and at creation of the Helps for managers and suppliers of components and materials. In the first case they are satellite systems, in the second case the off-line ones.

TENSOR PRODUCT OF INCIDENCE ALGEBRAS

The aim of this work is to study the incidence functions and thetensor product of two incidence algebras. We show that the tensorproduct of two incidence algebras is an incidence algebra. We believethat our result is true for uncountable locally partial order sets. Wepresent some examples of incidence functions. We study the Jacobsonradical of the tensor product of the incidence algebras as well as whena tensor incidence algebra is an algebraic algebra over a eld.

Measurement of fiscal rules: Introducing the application of partially ordered set (POSET) theory

Data on (economic) institutions are often available only as observations on ordinal, inherently incomparable properties, which are then typically aggregated to a composite index in the empirical social science literature. From a methodological perspective, the present paper advocates the application of partially ordered set (POSET) theory as an alternative approach. Its main virtue is that it takes the ordinal nature of the data seriously and dispenses with the unavoidably subjective assignment of weights to incomparable properties, maintains a high standard of objectivity, and can be applied in various fields of economics. As an application, the POSET approach is then used to calculate new indices on the stringency of fiscal rules for 81 countries over the period 1985–2012 based on recent data by IMF (2012). The derived measures of fiscal rules are used to test their significance for public finances in a fiscal reaction function and compare the POSET with the composite index approach.

Partial Order Infinitary Term Rewriting

We study an alternative model of infinitary term rewriting. Instead of a metric on terms, a partial order on partial terms is employed to formalise convergence of reductions. We consider both a weak and a strong notion of convergence and show that the metric model of convergence coincides with the partial order model restricted to total terms. Hence, partial order convergence constitutes a conservative extension of metric convergence, which additionally offers a fine-grained distinction between different levels of divergence. In the second part, we focus our investigation on strong convergence of orthogonal systems. The main result is that the gap between the metric model and the partial order model can be bridged by extending the term rewriting system by additional rules. These extensions are the well-known B\"ohm extensions. Based on this result, we are able to establish that -- contrary to the metric setting -- orthogonal systems are both infinitarily confluent and infinitarily normalising in the partial order setting. The unique infinitary normal forms that the partial order model admits are B\"ohm trees.

On the homeomorphism of continuous mappings

Denote by C(X) the partially ordered (PO) set of all continuous epimorphisms of a space X under the natural identification of homeomorphic epimorphisms. The following homeomorphism theorem for bicompacta is implicitly contained in Magill’s 1968 paper: two bicompacta X and Y are homeomorphic if and only if the PO sets C(X) and C(Y) are isomorphic. In the present paper, Magill’s theorem is extended to the category of mappings in which the role of bicompacta is played by perfect mappings. The results are obtained in two versions, namely, in the category TOPZ (of triangular commutative diagrams) and in the category MAP (of quadrangular commutative diagrams).