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Using the operators of taking upper and lower cones in a poset with a unary operation, we define operators M (x , y) and R (x , y) in the sense of multiplication and residuation, respectively, and we show that by using these operators, a general modification of residuation can be introduced. A relatively pseudocomplemented poset can be considered as a prototype of such an operator residuated poset. As main results, we prove that every Boolean poset as well as every pseudo-orthomodular poset can be organized into a (left) operator residuated structure. Some results on pseudo-orthomodular posets are presented which show the analogy to orthomodular lattices and orthomodular posets. [ABSTRACT FROM AUTHOR]

Let 〈 R , ∼ 〉 be the Rado graph, Emb ( R ) the monoid of its self-embeddings, P ( R ) = { f ( R ) : f ∈ Emb ( R ) } the set of copies of R contained in R , and I R the ideal of subsets of R which do not contain a copy of R . We consider the poset 〈 P ( R ) , ⊂ 〉 , the algebra P ( R ) / I R , and the inverse of the right Green's preorder on Emb ( R ) , and show that these preorders are forcing equivalent to a two step iteration of the form P ⁎ π , where the poset P is similar to the Sacks perfect set forcing: adds a generic real, has the ℵ 0 -covering property and, hence, preserves ω 1 , has the Sacks property and does not produce splitting reals, while π codes an ω -distributive forcing. Consequently, the Boolean completions of these four posets are isomorphic and the same holds for each countable graph containing a copy of the Rado graph. [ABSTRACT FROM AUTHOR]

The Rosenbloom–Tsfasman metric has attracted the attention of many researchers as a generalization of the Hamming metric that is relevant to practical problems. Codes for this metric were considered. In particular, Reed–Solomon codes were generalized to be compatible with this metric. In this paper, a generalization of BCH codes for the Rosenbloom–Tsfasman metric is proposed. This generalization is based on considering BCH codes as subfield subcodes of Reed–Solomon codes. By characterizing these subfield subcodes, an explicit construction of BCH codes for the Rosenbloom–Tsfasman metric is provided. Two important properties of Reed–Solomon codes and BCH codes for the Rosenbloom–Tsfasman metric are studied and compared with those for the Hamming metric. These properties are cyclic structure and duality. The approach is based on Galois-Fourier transforms associated with Hasse derivatives. [ABSTRACT FROM PUBLISHER]

This study presents an application of the Hasse diagram technique (HDT) as the assessment tool to select the most appropriate analytical procedures according to their greenness or the best analytical performance. The dataset consists of analytical procedures for benzo[ a]pyrene determination in sediment samples, which were described by 11 variables concerning their greenness and analytical performance. Two analyses with the HDT were performed-the first one with metrological variables and the second one with 'green' variables as input data. Both HDT analyses ranked different analytical procedures as the most valuable, suggesting that green analytical chemistry is not in accordance with metrology when benzo[ a]pyrene in sediment samples is determined. The HDT can be used as a good decision support tool to choose the proper analytical procedure concerning green analytical chemistry principles and analytical performance merits. [ABSTRACT FROM AUTHOR]

The dimension of a poset P, denoted $$\dim (P)$$ , is the least positive integer d for which P is the intersection of d linear extensions of P. The maximum dimension of a poset P with $$|P|\le 2n+1$$ is n, provided $$n\ge 2$$ , and this inequality is tight when P contains the standard example $$S_n$$ . However, there are posets with large dimension that do not contain the standard example $$S_2$$ . Moreover, for each fixed $$d\ge 2$$ , if P is a poset with $$|P|\le 2n+1$$ and P does not contain the standard example $$S_d$$ , then $$\dim (P)=o(n)$$ . Also, for large n, there is a poset P with $$|P|=2n$$ and $$\dim (P)\ge (1-o(1))n$$ such that the largest d so that P contains the standard example $$S_d$$ is o( n). In this paper, we will show that for every integer $$c\ge 1$$ , there is an integer $$f(c)=O(c^2)$$ so that for large enough n, if P is a poset with $$|P|\le 2n+1$$ and $$\dim (P)\ge n-c$$ , then P contains a standard example $$S_d$$ with $$d\ge n-f(c)$$ . From below, we show that $$f(c)={\varOmega }(c^{4/3})$$ . On the other hand, we also prove an analogous result for fractional dimension, and in this setting f( c) is linear in c. Here the result is best possible up to the value of the multiplicative constant. [ABSTRACT FROM AUTHOR]

Abstract: The crown is a height-2 poset whose Hasse diagram is a cycle of length 2t. A family of subsets of is -free if is not a weak subposet of . Let be the largest size of -free families of subsets of . De Bonis–Katona–Swanepoel proved . Griggs and Lu proved that for all even . In this paper, we prove for all odd . [Copyright &y& Elsevier]

We describe the classical Hasse principle for the existence of nontrivial zeros for quadratic forms over number fields, namely, local zeros over all completions at places of the number field imply nontrivial zeros over the number field itself. We then go on to explain more general questions related to the Hasse principle for nontrivial zeros of quadratic forms over function fields, with reference to a set of discrete valuations of the field. This question has interesting consequences over function fields of p-adic curves. We also record some open questions related to the isotropy of quadratic forms over function fields of curves over number fields. [ABSTRACT FROM AUTHOR]

The role of indicators to measure trends in every area of interest is increasing. Especially in the field of politics and sociology, where modeling based on multiple indicators typically is difficult, multi-indicator systems call for attention. Multi-indicator systems are most often the first step to derive a ranking indicator. Correspondingly there is a high interest in how to transform multi-indicator systems into a one-dimensional metric scale. The scientific discipline of decision support systems provides many well-known techniques, classical examples are PROMETHEE, ELECTRE, DEA or the simple TOPSIS. The mathematical technique is pretty sophisticated, therefore the simplest variant, namely the weighted sum of indicators plays its role too, just because of its simplicity and transparency. Although the need of a derivation of a one-dimensional scale is evident, we argue that there is an interim step, between the multi-indicator system and the ranking index, provided by simple elements of partial order theory. This interim step bears useful information too and in this paper we show how and which useful additional information can be derived. We derive for example a bias-free sensitivity study, where the indicator 'chronic and sustained human flight' turns out to be the most important indicator within the multi-indicator system of 12 indicators, which are the basis of the Failed State Index. [ABSTRACT FROM AUTHOR]

Mazurkiewicz traces are a widely used model for describing the languages of concurrent systems computations. The causal structure of atomic actions occurring in a process modeled as a trace generates a partial order. Hasse diagrams of such order are very common structures used for presentation and investigation in the concurrency theory, especially from the behavioural perspective. We present effective algorithms for Hasse diagrams construction and transformation. Later on, we use them for enumeration of all linearisations of the partial order that represents a concurrent process. Additionally, we attach the flexible visual implementation of all considered algorithms. [ABSTRACT FROM AUTHOR]

It is known that the First-Fit algorithm for partitioning a poset P into chains uses relatively few chains when P does not have two incomparable chains each of size k. In particular, if P has width w, then Bosek, Krawczyk, and Szczypka [SIAM J. Discrete Math., 23 (2010), pp. 1992-1999], proved an upper bound of ckw² on the number of chains used by First-Fit for some constant c, while Joret and Milans [Order, 28 (2011), pp. 455-464] gave one of ck²w. In this paper we prove an upper bound of the form ckw. This is most possible up to the value of c. [ABSTRACT FROM AUTHOR]

We give an explicit nonrecursive complete matching for the Hasse diagram of the strong Bruhat order of any interval in any Coxeter group. This yields a new derivation of the Möbius function, recovering a classical result due to Verma. [ABSTRACT FROM AUTHOR]

The set of acyclic orientations of a connected graph with a given sink has a natural poset structure. We give a geometric proof of a result of Jim Propp: this poset is the disjoint union of distributive lattices. [ABSTRACT FROM AUTHOR]

Convex rank tests are partitions of the symmetric group which have desirable geometric properties. The statistical tests defined by such partitions involve counting all permutations in the equivalence classes. Each class consists of the linear extensions of a partially ordered set specified by data. Our methods refine existing rank tests of nonparametric statistics, such as the sign test and the runs test, and are useful for exploratory analysis of ordinal data. We establish a bijection between convex rank tests and probabilistic conditional independence structures known as semigraphoids. The subclass of submodular rank tests is derived from faces of the cone of submodular functions or from Minkowski summands of the permutohedron. We enumerate all small instances of such rank tests. Of particular interest are graphical tests, which correspond to both graphical models and to graph associahedra. [ABSTRACT FROM AUTHOR]

We have proved that the Frankl’s Conjecture is true for the class of finite posets satisfying the dual covering property. [ABSTRACT FROM AUTHOR]

Abstract: In this paper, a new similarity model is proposed, which based on rough set to evaluate the similarity degree of the two concepts of concept lattice. The proposed method combines featural and structural information into decision and has a higher correlation with human judgement, which can be viewed as the development of Tversky’s similarity model. Compared with other similarity models this approach is convenient to measure the similarity of the concepts of the large contexts, by which we can avoid constructing Hasse diagram and looking through all concepts of the context. [Copyright &y& Elsevier]

The concept of a semiprime ideal in a poset is introduced. The relations between the semiprime (prime) ideals of a poset and the ideals of the set of all ideals of the poset are established. A result analogous to Separation Theorem is obtained in respect of semiprime ideals. Further, a generalization of Stone's Separation Theorem for posets is obtained in respect of prime ideals. Some counterexamples are also given. [ABSTRACT FROM AUTHOR]

Abstract: We consider a problem of searching an element in a partially ordered set (poset). The goal is to find a search strategy which minimizes the number of comparisons. Ben-Asher, Farchi and Newman considered a special case where the partial order has the maximum element and the Hasse diagram is a tree (tree-like posets) and they gave an -time algorithm for finding an optimal search strategy for such a partial order. We show that this problem is equivalent to finding edge ranking of a simple tree corresponding to the Hasse diagram, which implies the existence of a linear-time algorithm for this problem. Then we study a more general problem, namely searching in any partial order with maximum element. We prove that such a generalization is hard, and we give an -approximate polynomial-time algorithm for this problem. [Copyright &y& Elsevier]

Abstract: The poset retraction problem for a poset is whether a given poset containing as a subposet admits a retraction onto , that is, whether there is a homomorphism from onto which fixes every element of . We study this problem for finite series-parallel posets . We present equivalent combinatorial, algebraic, and topological charaterisations of posets for which the problem is tractable, and, for such a poset , we describe posets admitting a retraction onto . [Copyright &y& Elsevier]

In this paper, we consider a characterization of the folding of 3D model proteins. In this characterization, we describe the different kinds of 3-step path conformation, then construct an augmented Hasse matrix which reflects some properties of the protein folding. This method can characterize the folding degree of 3D model proteins and evaluate the similarity/diversity of these five model proteins. © 2007 Wiley Periodicals, Inc. Int J Quantum Chem, 2007 [ABSTRACT FROM AUTHOR]

Abstract: Suppose is a partially ordered set with the partial order defined by divisibility, that is, for any two elements satisfying divides , we have . A table of real numbers is said to be consistent with , provided that for any two elements satisfying divides , . Given a real number , we want to determine whether , by comparing with as few entries of as possible. In this paper, we investigate the complexity , measured by the number of comparisons, of the above search problem. We present a search algorithm for and prove a lower bound on using an adversary argument. [Copyright &y& Elsevier]

This study attempts to characterize national passenger and freight transportation sustainability. Based on the indicator that measures the conformity of the growths of all sectors with transportation, the Islamic countries are comparatively studied. The proposed measure, elasticity, for each pair of variables indicates the extent to which the two variables have been changing consistently. The study database consisted of key aspects of transportation sustainability in the form of national variables including transportation, economic, social, and environmental categories for the period 1980-1995. Firstly, the elasticity of the social, environmental, and economic variables with respect to passenger and freight transportation variables was developed. Using individual elasticities, composite passenger and freight sustainability indices were suggested. Then, utilizing partial order theory and Hasse Diagram Technique (HDT), two composite indices were employed to visualize the comparative situations of countries. Based on comparative appraisal achieved by HDT, country ranking were developed. The methodology may be applied to any other time and geographic area for addressing pertinent issues for balancing and sustainable development of transportation systems. [ABSTRACT FROM AUTHOR]

This paper describes a new top-down algorithm for the stepwise generation of the different levels or Hasse diagrams of the Hasse tree associated to the fuzzy preorder closure (min-transitive closure) of a given reflexive binary fuzzy relation. The algorithm is based upon a recently established weight-driven method for computing the min-transitive closure of a reflexive binary fuzzy relation. The way in which this method gradually establishes the fuzzy preorder closure implies that for the generation of a specific level of the Hasse tree, the newly proposed algorithm does not require the complete computation of this closure. [ABSTRACT FROM AUTHOR]

Hasse constants and their basic properties are introduced to facilitate the connection between the lattice of subalgebras of an algebra $$ \mathcal{C} $$ and the natural action of the automorphism group Aut( $$ \mathcal{C} $$ ) on $$ \mathcal{C} $$ . These constants are then used to describe the lattice of subloops of the smallest nonassociative simple Moufang loop. [ABSTRACT FROM AUTHOR]

In this project, we apply the method of partial ordering on the ranking of 74 contaminated sites located in the county of West Zealand (Denmark). The method is based on the concept that the parameters are kept separated through the ranking analysis, and thus no weighing of the different parameter values is necessary. The ranking is displayed in a graphical form by the Hasse diagram technique to ease the interpretation. A critical comparison is made of the ranking of contaminated sites by the partial ordering method and an index function used by the county of West Zealand. Comparing the ranking by the partial ordering method to the index function shows that the choice of score points and index function highly influences the ranking result, as only four sites are equally ranked. The importance of the parameters used to identify the environmental hazard of the contaminated sites is analyzed in order to evaluate the influence of each parameter on the ranking. From among a total of six different parameters, two have high influence, two medium, and two low because of both the construction of the scoring system and the characteristics of the data. [ABSTRACT FROM AUTHOR]

Given a graph G and an orientation σ of some of its edges, consider the graph AOσ(G) which is defined as follows: The vertices are the acyclic orientations of G which agree with σ, and two of these are adjacent if they differ only by the reversal of a single edge. AOσ(G) is easily seen to be bipartite. The purpose of this note is to show that it need not contain a Hamilton path even if both partite sets have the same cardinality. This answers a question of C. D. Savage [SIAM Rev., 39 (1997), pp. 605–629] and sheds new light onto two well-known open questions in the field of combinatorial Gray codes. [ABSTRACT FROM AUTHOR]

The graph of linear extensions G(P) of a poset P has as vertices the linear extensions of P, and two vertices are adjacent if they differ only by an adjacent transposition. This graph has so far been investigated mainly with respect to Hamilton paths and intrinsic geodesic convexity. The work on the latter topic especially shows how the graph-theoretic properties of G(P) reflect the order-theoretic structure of P. The aim of this paper is to study the graph G(P) with respect to topics from classical graph theory, e.g., connectivity, cycle space, isometric embeddings, and to find order-theoretic interpretations of these notions. The main theorems in this paper are the equality of the connectivity of G(P) and the jump number of P, the existence of a certain generating system for the cycle space, and a relationship of P and its subposets obtained via an embedding of the graph of linear extensions. [ABSTRACT FROM AUTHOR]