scholarly journals Various Topologies Generated from E j -Neighbourhoods via Ideals

Complexity ◽  
2021 ◽  
Vol 2021 ◽  
pp. 1-11
Author(s):  
Rodyna A. Hosny ◽  
Baravan A. Asaad ◽  
A. A. Azzam ◽  
Tareq M. Al-Shami
Keyword(s):  
Binary Relation ◽  
Binary Relations ◽  
Accuracy Measures ◽  
Better Than

One of the considerable subjects in mathematics is the study of topology. Deducing topology from arbitrary binary relations has enticed the attention of many researchers. So, we devote this article to generate some kinds of topologies from ideals and E j -neighborhoods which are induced from any binary relation. We define new types of approximations and accuracy measures from these topologies and then compare them with their counterparts induced directly from E j -neighborhoods and ideals. Also, we show that the approximations and accuracy measures given, herein, are better than those introduced in some previous studies under any arbitrary relation.

Variable Neighborhood Model for Agent Control Introducing Accessibility Relations Between Agents with Linear Temporal Logic

2014 ◽  
Vol 18 (6) ◽  
pp. 937-945
Author(s):  
Seiki Ubukata ◽  
◽  
Tetsuya Murai ◽  
Yasuo Kudo ◽  
Seiki Akama ◽  
...  
Keyword(s):  
Topological Space ◽  
Temporal Logic ◽  
Binary Relation ◽  
Linear Temporal Logic ◽  
Binary Relations ◽  
Cellular Space ◽  
Current Time ◽  
Low Level ◽  
Local Properties ◽  
Agent Control

In general, there are two types of agents, reflex and deliberative. The former does not have the ability for deep planning that produces higher-level actions to attain goals cooperatively, which is the ability of the latter. Can we cause reflex agents to act as though they could plan their actions? In this paper, we propose a variable neighborhood model for reflex agent control, that allows such agents to create plans in order to attain their goals. The model consists of three layers: (1) topological space, (2) agent space, and (3) linear temporal logic. Agents with their neighborhoods move in a topological space, such as a plane, and in a cellular space. Then, a binary relation between agents is generated each time from the agents’ position and neighborhood. We call the pair composed of a set of agents and binary relations the agent space. In order to cause reflex agents to have the ability to attain goals superficially, we consider the local properties of the binary relation between agents. For example, if two agents have a symmetrical relation at the current time, they can struggle to maintain symmetry or they could abandon symmetry at the next time, depending on the context. Then, low-level behavior, that is, the maintenance or abandonment of the local properties of binary relations, grant reflex agents a method for selecting neighborhoods for the next time. As a result, such a sequence of low-level behavior generates seemingly higher-level actions, as though reflex agents could attain a goal with such actions. This low-level behavior is shown through simulation to generate the achievement of a given goal, such as cooperation and target pursuing.

On transitive subrelations of binary relations

2011 ◽  
Vol 76 (4) ◽  
pp. 1429-1440 ◽  
Author(s):  
Christopher S. Hardin
Keyword(s):  
Binary Relation ◽  
Transitive Closure ◽  
Infinite Chain ◽  
Binary Relations ◽  
Transitive Relation

AbstractThe transitive closure of a binary relation R can be thought of as the best possible approximation of R “from above” by a transitive relation. We consider the question of approximating a relation from below by transitive relations. Our main result is that every thick relation (a relation whose complement contains no infinite chain) on a countable set has a transitive thick subrelation. This allows for a solution to a problem arising from previous work by the author and Alan Taylor. We also exhibit a thick relation on an uncountable set with no transitive thick subrelation.

An algebraic approach to binary relations

2015 ◽  
Vol 08 (02) ◽  
pp. 1550017 ◽  
Author(s):  
Ivan Chajda ◽  
Helmut Länger ◽  
Petr Ševčik
Keyword(s):  
Binary Relation ◽  
Binary Operation ◽  
Algebraic Approach ◽  
Binary Relations ◽  
Point Extension

Several attempts were made to assign to a given binary relation a certain binary operation in order to allow an algebraic approach for investigating binary relations. However, the previous attempts by the first two authors were restricted to the case of so-called directed binary relations. In this paper, this restriction is omitted and a general approach is developed. We assign to every binary relation a partial binary operation in such a way that the properties of the relation can be described by properties of the assigned operation. These properties are expressed mostly in terms of existential or strong identities. The partial binary operation can be extended to an everywhere defined binary operation by means of the so-called one-point extension. This enables us to get a genuine algebraic approach similar to that for directed binary relations.

Approximations in Rough Sets vs Granular Computing for Coverings

2012 ◽  
pp. 152-163
Author(s):  
Guilong Liu ◽  
William Zhu
Keyword(s):  
Knowledge Discovery ◽  
Binary Relation ◽  
Set Theory ◽  
Rough Set ◽  
Rough Sets ◽  
Rough Set Theory ◽  
Knowledge Discovery In Databases ◽  
Equivalence Relations ◽  
Binary Relations ◽  
Covering Rough Set

Rough set theory is an important technique in knowledge discovery in databases. Classical rough set theory proposed by Pawlak is based on equivalence relations, but many interesting and meaningful extensions have been made based on binary relations and coverings, respectively. This paper makes a comparison between covering rough sets and rough sets based on binary relations. This paper also focuses on the authors’ study of the condition under which the covering rough set can be generated by a binary relation and the binary relation based rough set can be generated by a covering.

scholarly journals On the Möbius function of a non-singular binary relation

1970 ◽  
Vol 3 (2) ◽  
pp. 155-162 ◽  
Author(s):  
P. D. Finch
Keyword(s):  
Binary Relation ◽  
Partially Ordered Sets ◽  
Möbius Function ◽  
Binary Relations ◽  
Ordered Sets ◽  
Finite Sets ◽  
Partially Ordered ◽  
Mobius Function

Some of the results in the theory of Möbius functions of finite partially ordered sets are extended to arbitrary non-singular binary relations on finite sets.

scholarly journals Divisibility of binary relations

1971 ◽  
Vol 5 (1) ◽  
pp. 75-86 ◽  
Author(s):  
D.G. Fitz-Gerald ◽  
G.B. Preston
Keyword(s):  
Binary Relation ◽  
Alternative Proof ◽  
Binary Relations

In his paper in Mat. Sb. (N.S.) 61 (103) (1963), Zareckiĭ associated with any binary relation α an ordered pair, (Lα Mα), say, of lattices and showed that α is a left [right] divisor of β if and only if We provide an alternative proof of this result by embedding the category of relations in the category of sets. Our approach provides a unified treatment of several hitherto independent results, and gives new results for the category of partial transformations.

Fully admissible binary relations in topology

1977 ◽  
Vol 82 (2) ◽  
pp. 259-264
Author(s):  
D. M. G. McSherry
Keyword(s):  
Binary Relation ◽  
Continuous Mapping ◽  
Binary Relations ◽  
Separation Axiom

A fully admissible binary relation (3) is an operator , other than the equality operator and universal operator , which assigns to each space |S, τ|, a reflexive, symmetric, binary relation , and which is such that for any continuous mapping implies . With each such relation , we associate a ‘separation axiom’ , as well as ‘-regularity’ and ‘-connectedness’, where ≡ -regularity + T0, and -regularity + -connectedness ≡ indiscreteness.

Approximations in Rough Sets vs Granular Computing for Coverings

2010 ◽  
Vol 4 (2) ◽  
pp. 63-76 ◽  
Author(s):  
Guilong Liu ◽  
William Zhu
Keyword(s):  
Binary Relation ◽  
Set Theory ◽  
Rough Set ◽  
Rough Sets ◽  
Granular Computing ◽  
Rough Set Theory ◽  
Knowledge Discovery In Databases ◽  
Equivalence Relations ◽  
Binary Relations ◽  
Covering Rough Set

Rough set theory is an important technique in knowledge discovery in databases. Classical rough set theory proposed by Pawlak is based on equivalence relations, but many interesting and meaningful extensions have been made based on binary relations and coverings, respectively. This paper makes a comparison between covering rough sets and rough sets based on binary relations. This paper also focuses on the authors’ study of the condition under which the covering rough set can be generated by a binary relation and the binary relation based rough set can be generated by a covering.

Relation algebras of every dimension

1992 ◽  
Vol 57 (4) ◽  
pp. 1213-1229 ◽  
Author(s):  
Roger D. Maddux
Keyword(s):  
Binary Relation ◽  
Sequent Calculus ◽  
Logical Consequence ◽  
Relation Algebra ◽  
Binary Relations ◽  
Relation Algebras

AbstractConjecture (1) of [Ma83] is confirmed here by the following result: if 3 ≤ α < ω, then there is a finite relation algebra of dimension α, which is not a relation algebra of dimension α + 1. A logical consequence of this theorem is that for every finite α ≥ 3 there is a formula of the form S ⊆ T (asserting that one binary relation is included in another), which is provable with α + 1 variables, but not provable with only α variables (using a special sequent calculus designed for deducing properties of binary relations).

scholarly journals Minimizing Cost Travel in Multimodal Transport Using Advanced Relation Transitive Closure

2018 ◽  
Vol 2018 ◽  
pp. 1-7
Author(s):  
Rachid Oucheikh ◽  
Ismail Berrada ◽  
Lahcen Omari
Keyword(s):  
Binary Relation ◽  
Constraint Satisfaction ◽  
Operations Research ◽  
Transitive Closure ◽  
Constraint Satisfaction Problem ◽  
Cost Optimization ◽  
Binary Relations ◽  
Mathematical Background ◽  
Multimodal Transport ◽  
Binary Constraint

The optimization computation is an essential transversal branch of operations research which is primordial in many technical fields: transport, finance, networks, energy, learning, etc. In fact, it aims to minimize the resource consumption and maximize the generated profits. This work provides a new method for cost optimization which can be applied either on path optimization for graphs or on binary constraint reduction for Constraint Satisfaction Problem (CSP). It is about the computing of the “transitive closure of a given binary relation with respect to a property.” Thus, this paper introduces the mathematical background for the transitive closure of binary relations. Then, it gives the algorithms for computing the closure of a binary relation according to another one. The elaborated algorithms are shown to be polynomial. Since this technique is of great interest, we show its applications in some important industrial fields.

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