Composition Of Binary Relations Research Articles

Asynchronous Tracking Control of Leader-Follower Multiagent Systems With Input Uncertainties Over Switching Signed Digraphs.

Signed digraphs with both positive and negative weighted edges are widely applied to explain cooperative and competitive interactions arising from various social, biological, and physical systems. This article formulates and solves the asynchronous tracking control problem of multiagent systems with input uncertainties on switching signed digraphs. In the interaction setting, we assume that the leader moves at a time-varying acceleration that cannot be measured by the followers accurately, and further suppose that each agent receives its neighbors' states information at certain instants determined by its own clock, which is not necessary to be synchronized with those of other agents. Using dynamically changing spanning subdigraphs of signed digraphs to describe graphically asynchronous interactions, the asynchronous tracking problem is equivalently transformed into a convergence problem of products of general substochastic matrices (PGSSM), in which the matrix elements are not necessarily non-negative and the row sums are less than or equal to 1. With the help of the matrix analysis technique and the composition of binary relations, we propose a new and original method to deal with the convergence problem of PGSSM, and further establish a spanning tree condition for asynchronous tracking control. Finally, the validity of the theoretical findings is verified through several numerical examples.

BUILDING A LOGICAL NETWORK FOR SOLVING THE PROBLEM OF CAR RENTAL BY MEANS ALGEBRA OF FINITE PREDICATES

The article is devoted to the research of the tools of algebra of finite predicates for the system analysis and formalization of the task of automating car rental according to the selected parameters. In particular, the process of optimal car selection depends on the following parameters: car class, brand, availability of driver, type of trip and its duration, fuel type, tariff type, as well as season and weather conditions. Each of these criteria has its own area of definition, where you need to take into account all the relationships and influences between the values of the entered variables (criteria). The aim of the work is to increase the speed of data processing in the problem of car rental by dividing the input multi-place ratio into a binary composition. The technique is based on the means and methods of algebra of finite predicates. Introduction of the predicate of object recognition in the specified subject area allowed to formally describe data of any type, and the applied method of construction of logical networks provides increase in speed of information processing due to parallelization of processing processes. Thus, a complex multi-place relation was divided into a composition of binary relations described in the language of predicate algebra, taking into account the detailed system analysis of the subject area. A scientific novelty is the constructed mathematical model of the car rental problem, which is represented by a predicate that depends on thirteen variables. This predicate is characterized by a system of twelve binary relations, which are represented in the article by dual graphs and formulas of the corresponding predicates. The model predicate is a composition of all constructed binary predicates. The practical significance is due to the logical network built on the basis of a mathematical model, which allows from the relationship "many to many" to move to the relationship "to each other" and parallelize the process of information processing. The result is a logical network of car rental problems, which works iteratively until it receives stable results in two consecutive steps and allows you to solve problems of analysis and synthesis for car rental according to selected parameters.

Containment Control of Asynchronous Discrete-Time General Linear Multiagent Systems With Arbitrary Network Topology

In this contribution, we propose and investigate the containment control issue for general linear multiagent systems (MASs) under the asynchronous setting, where the network topology is not subjected to any structural restrictions and the roles of the leaders and the followers are entirely determined by the network topology. It is assumed that the interaction time instants of each agent, at which this agent interacts with its neighbors, are independent of the other agents' and can be unevenly distributed. An asynchronous distributed algorithm is proposed to implement the control strategy of linear MASs. The non-negative matrix theory and the composition of binary relations are utilized to handle the asynchronous containment control issue. It is shown that the leaders in each closed and strongly connected component of the network topology will reach a common state and the followers will gradually enter the dynamic convex hull constructed by the leaders. Moreover, it is also proved that the system matrix can be strictly unstable, and the upper bound of the system matrix's spectral radius is explicitly stated. Finally, two simulation examples are also provided to verify the efficacy of our theoretical results.

Classification of Finite Commutative Semigroups for Which the Inverse Monoid of Local Automorphisms is a ∆-Semigroup

A semigroup S is called a ∆-semigroup if the lattice of its congruences forms a chain relative to the inclusion. A local automorphism of the semigroup S is called an isomorphism between its two subsemigroups. The set of all local automorphisms of the semigroup S relative to the ordinary operation of composition of binary relations forms an inverse monoid of local automorphisms. We present a classification of finite commutative semigroups for which the inverse monoid of local automorphisms is a ∆-semigroup.

Stable Quasiorderings on Some Permutable Inverse Monoids

Let G be an arbitrary group of bijections on a finite set. By I(G), we denote the set of all injections each of which is included in a bijection from G. The set I(G) forms an inverse monoid with respect to the ordinary operation of composition of binary relations. We study different properties of the semi-group I(G). In particular, we establish necessary and sufficient conditions for the inverse monoid I(G) to be permutable (i.e., ξ ○ φ = φ ○ ξ for any pair of congruences on I(G)). In this case, we describe the structure of each congruence on I(G). We also describe the stable orderings on I(A n ), where A n is an alternating group.

V-TRANSFORMATION OF STRONG VARIETIES OF PARTIAL ALGEBRAS

In this paper, we define V-transformation and prove that Vtransformation is a bijective if Vtransformation is a composition of binary relations with inverse of itself then equal relation delte of C-terms. AMS Subject Classification: 08A55, 08B15, 08B25

A survey on fuzzy relational equations, part I: classification and solvability

Fuzzy relational equations play an important role in fuzzy set theory and fuzzy logic systems, from both of the theoretical and practical viewpoints. The notion of fuzzy relational equations is associated with the concept of "composition of binary relations." In this survey paper, fuzzy relational equations are studied in a general lattice-theoretic framework and classified into two basic categories according to the duality between the involved composite operations. Necessary and sufficient conditions for the solvability of fuzzy relational equations are discussed and solution sets are characterized by means of a root or crown system under some specific assumptions.

On Definability of Hypergraphs by Their Semigroups of Homomorphisms

In studying mathematical objects on the basis of their mapping systems considerable attention is given to the endomorphism semigroups and the partial endomorphisms semigroups of the investigated objects. It is natural to apply the same attention to systems of partial homomorphisms of these objects. However, such systems were rarely considered, since in these situations the usual composition of binary relations cannot be used and it is necessary to consider such mapping systems together with other operations. Experience shows that it is possible to consider the operations of right restrictive multiplication and left restrictive multiplication introduced by Wagner in [14] in the setting of operations for systems of partial homomorphisms. For example, Schein [11] and Molchanov [8], [9] showed that in individual cases systems of partial homomorphisms endowed with the restrictive multiplications have simple structure and contain important information about the investigated objects. This paper continues investigations in this direction. We describe here a wide class of hypergraphs which are determined up to isomorphism by their restrictive semigroups of partial homomorphisms. We also apply this result in analysing some problems of endomorphism semigroups of hypergraphs. The main result of the paper — theorem 3.1 — was announced in [7]. Note also that results of this paper generalize and correct a series of statements formulated without proofs in the end of the survey article [10]. The author wishes to thank Professor Boris M. Schein and the referee of this paper for their numerous valuable suggestions.

A General Extension Theorem for Binary Relations

A general theorem on “extending” binary relations to linear (or strict linear) orders is proved. The well-known result that every partial order can be written as the intersection of linear orders in which it is embedded is obtained as a special case, as are several other extension theorems of this type. Using duality arguments, a number of results on the composition of binary relations are proved. For example, a complete, negatively transitive relation is the union of all linear orders embedded within it.Journal of Economic LiteratureClassification Number: C60.

On efficiently computing the product of two binary relations

We precisely analyse ideas of M. A. Kronrod for the efficient composition of binary relations on finite domains. It shows that the “Four Russians” algorithm for Boolean matrix multiplication behaves no different in average and worst cases, which therefore makes it less preferable for use than is often claimed.