The Symmetry of Lower and Upper Approximations, Determined by a Cyclic Hypergroup, Applicable in Control Theory

Abstract

In the first part of our paper, we construct a cyclic hypergroup of matrices using the Ends Lemma. Its properties are then, in the second part of the paper, used to describe the symmetry of lower and upper approximations in certain rough sets with respect to invertible subhypergroups of this cyclic hypergroup. Since our approach is widely used in autonomous robotic systems, we suggest an application of our results for the study of detection sensors, which are used especially in mobile robot mapping.

Special Issue on Symmetry in Classical and Fuzzy Algebraic Hypercompositional Structures

“…Based on the properties of a cyclic hypergroup of particular matrices, Krehlik and Vyroubalova described in [13] the symmetry of lower and upper approximations in certain rough sets connected with this hypergroup. The main tool involved in their study is the Ends lemma, in which the hypercomposition is the principal end of a partially ordered semigroup.…”

Sequences of Groups, Hypergroups and Automata of Linear Ordinary Differential Operators

“…Furthermore, for clarification and evolution of terminology, see [8]. For results obtained by means of quasi-multiautomata, see, e.g., [5][6][7][8]27]. In this section, R n [x] means, as usually, the ring of polynomials of degree at most n.…”

“…In several papers, published mainly as conference proceedings such as [2][3][4], the existing theory was extended by the use of hypercompositional structures in place of the usual algebraic structures. The use of hypercompositional generalizations has been tested in the automata theory, where it has brought several interesting results; see, e.g., [5][6][7][8]. Naturally, this approach is not the only possible one.…”

n-Ary Cartesian Composition of Multiautomata with Internal Link for Autonomous Control of Lane Shifting

“…By extensive hypergroup in the sense of hypercompositional structures we mean a hypergroup that {a, b} ⊆ a * b for all a, b ∈ H, i.e., that the elements are included in its "resalt of hyperoperation". For the same application on this theory in other science or real-life problems see, for example, [3][4][5][6][7].…”