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We study some connections between hypergroups and n-hypergroups. For each of them we give examples which involve binary relations, lattices and hypergraphs. Finally, we study when two finite join spaces associated with lattices, d...
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We study some connections between hypergroups and n-hypergroups. For each of them we give examples which involve binary relations, lattices and hypergraphs. Finally, we study when two finite join spaces associated with lattices, defined on the same set, are isomorphic.
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In this paper, we introduce six basic types of composition of ternary relations, four of which are associative. These compositions are based on two types of composition of a ternary relation with a binary relation recently introdu...
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In this paper, we introduce six basic types of composition of ternary relations, four of which are associative. These compositions are based on two types of composition of a ternary relation with a binary relation recently introduced by Zedam et al. We study the properties of these compositions, in particular the link with the usual composition of binary relations through the use of the operations of projection and cylindrical extension.
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Szpilrajn's extension theorem on binary relations and its strengthening by Dushnik and Miller are fundamental in economic and game theories. Szpilrajn's result entails that each partial order extends to a linear order. Dushnik and...
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Szpilrajn's extension theorem on binary relations and its strengthening by Dushnik and Miller are fundamental in economic and game theories. Szpilrajn's result entails that each partial order extends to a linear order. Dushnik and Miller use Szpilrajn's theorem to show that each partial order has a realizer. Since then, many authors utilize Szpilrajn's theorem and the well-ordering principle to prove more general theorems on extending binary relations. The original extension theorems of Szpilrajn, Dushnik-Miller and Moulin-Weymark are called: Szpilrajn extension theorem, Dushnik-Miller extension theorem and Moulin-Weymark's Pareto extension theorem respectively. The generalizations of these theorems are called: Szpilrajn-type extension theorem, Dushnik-Miller-type extension theorem and Moulin-Weymark's Pareto-type extension theorem respectively. The presented results generalize well-known extension theorems in the literature.
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This paper deals with Nonconical Nonsmooth Multiobjective Programming problem. The generalized directional derivate of a mapping w. r. t a binary relation and the derivate of a binary relation are introduced and used as basic tool...
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This paper deals with Nonconical Nonsmooth Multiobjective Programming problem. The generalized directional derivate of a mapping w. r. t a binary relation and the derivate of a binary relation are introduced and used as basic tools. The optimality conditions and duality theorems are proved.
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In this paper, we generalize the notion of traces of a binary relation to the setting of ternary relations. With a given ternary relation, we associate three binary relations: its left, middle and right trace. As in the binary cas...
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In this paper, we generalize the notion of traces of a binary relation to the setting of ternary relations. With a given ternary relation, we associate three binary relations: its left, middle and right trace. As in the binary case, these traces facilitate the study and characterization of properties of a ternary relation. Interestingly, the traces themselves turn out to be the greatest solutions of relational inequalities associated with newly introduced compositions of a ternary relation with a binary relation (and vice versa).
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Relational mathematics, as it is studied in fields like mathematical economics and social choice theory for some time, provides a rich and general framework and appears to be a natural and direct way to paraphrase optimization goa...
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Relational mathematics, as it is studied in fields like mathematical economics and social choice theory for some time, provides a rich and general framework and appears to be a natural and direct way to paraphrase optimization goals, to represent user preferences, to justify fairness criterions, to cope with QoS or to valuate utility. Here, we will focus on the specific application aspects of formal relations in network design and control problems and provide the general concept of relational optimization. In relational optimization, we represent the optimization problem by a formal relation, and the solution by the set of maximal (or non-dominated) elements of this relation. This appears to be a natural extension of standard optimization, and covers other notions of optimality as well. Along with this, we will provide a set of fairness relations that can serve as maximizing relations in relational optimization according to various application needs, and we specify a meta-heuristic approach derived from evolutionary multi-objective optimization algorithms to approximate their maximum sets.
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A relation R is (≤k)-reconstructible (k a positive integer) if it is isomorphic with any relation S on the same vertex set with the property that the relations induced byRand S on any set of at most k vertices are isomorphic; it ...
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A relation R is (≤k)-reconstructible (k a positive integer) if it is isomorphic with any relation S on the same vertex set with the property that the relations induced byRand S on any set of at most k vertices are isomorphic; it is (≤k)-self dual if every restriction to at most k vertices is self dual, i.e. isomorphic to its dual relation (the relation obtained by reversing its arcs). In particular, relying on the description of (≤k)-self dual binary relations, we characterize, for each k ≥ 4, all (≤k)-reconstructible binary relations: A binary relation is (≤k)-reconstructible if and only if its modules that are chains are finite and its (≤k)-self dual modules are self dual.
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A simple head-to-head voting scheme in which voters hold complete and transitive preferences over alternatives generates all binary relations on finite sets. The minimal number of voters required to generate a binary relation prov...
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A simple head-to-head voting scheme in which voters hold complete and transitive preferences over alternatives generates all binary relations on finite sets. The minimal number of voters required to generate a binary relation provides a measure of complexity for binary relations. Complexity so defined tells us, by how much a given binary relation fails to qualify as a total preorder.
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The purpose of this paper is the study of hypergroups associated with hypergraphs. In this regards, we construct a ρ-hypergroup by means a given hypergraph by defining a special relation ρ, and then we investigate some related p...
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The purpose of this paper is the study of hypergroups associated with hypergraphs. In this regards, we construct a ρ-hypergroup by means a given hypergraph by defining a special relation ρ, and then we investigate some related properties. Further, we introduce a special product of ρ-hypergroups. Also, we bridge between subhypergraphs and subhypergroups. Finally, the fundamental relation of a ρ-hypergroup is studied.
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The language of relation algebras is expanded with variables denoting individual elements in the domain and with the binder form hybrid logic. Every elementary property of binary relations is expressible in the resulting language,...
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The language of relation algebras is expanded with variables denoting individual elements in the domain and with the binder form hybrid logic. Every elementary property of binary relations is expressible in the resulting language, something which fails for the relation algebraic language. That the new language is natural for speaking about binary relations is indicated by the fact that both Craig's Interpolation, and Beth's Definability theorems hold for its set of validities. The paper contains a number of worked examples.
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