From lattices to Hv-matrices

Křehlík, Štěpán; Novák, Michal

doi:10.1515/auom-2016-0055

Cited by 5 publications

(6 citation statements)

References 15 publications

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“…Further research can also focus on finding the approximation set. For this we can look for inspiration in the intersection of the principal end and beginning of the hyperoperation in [10]. As a result we can form an approximation set which need not include the origin of the universe, i.e., R [0,0] .…”

Section: Discussionmentioning

confidence: 99%

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

Křehlík

Vyroubalová

2019

Symmetry

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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.

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Section: Discussionmentioning

confidence: 99%

“…In our paper we develop an idea similar to [9][10][11]. With the help of matrix calculus, which we believe is a suitable tool, we construct cyclic hypergroups and their invertible subhypergroups.…”

Section: Introductionmentioning

confidence: 99%

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

Křehlík

Vyroubalová

2019

Symmetry

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“…Now we can use Lemma 1 to construct a (noncommutative) hypergroup. In order to do this, we will need the following lemma, known as Ends lemma; for details see, e.g., [18][19][20]. Notice that a join space is a special case of a hypergroup-in this paper we speak of hypergroups because we want to stress the parallel with groups.…”

Section: Notation and Preliminariesmentioning

confidence: 99%

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

et al. 2021

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The main objective of our paper is to focus on the study of sequences (finite or countable) of groups and hypergroups of linear differential operators of decreasing orders. By using a suitable ordering or preordering of groups linear differential operators we construct hypercompositional structures of linear differential operators. Moreover, we construct actions of groups of differential operators on rings of polynomials of one real variable including diagrams of actions–considered as special automata. Finally, we obtain sequences of hypergroups and automata. The examples, we choose to explain our theoretical results with, fall within the theory of artificial neurons and infinite cyclic groups.

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“…The idea of EL-hyperstructures has been implicitely present in a number of works since at least the 1960s, for example Pickett [16]. The definition and first results were given by Chvalina [17] and the theory has been elaborated by Novák (later jointly with Chvalina, Křehlík, and Cristea) in a series of papers including [18][19][20][21][22]. It is to be noted that, since the class of EL-hyperstructures is rather broad, the aim of many theorems included in some of those papers was to establish a common ground for some already existing ad hoc derived results.…”

Section: Mathematical Background Of the Modelmentioning

confidence: 99%

Elements of Hyperstructure Theory in UWSN Design and Data Aggregation

2019

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In our paper we discuss how elements of algebraic hyperstructure theory can be used in the context of underwater wireless sensor networks (UWSN). We present a mathematical model which makes use of the fact that when deploying nodes or operating the network we, from the mathematical point of view, regard an operation (or a hyperoperation) and a binary relation. In this part of the paper we relate our context to already existing topics of the algebraic hyperstructure theory such as quasi-order hypergroups, E L -hyperstructures, or ordered hyperstructures. Furthermore, we make use of the theory of quasi-automata (or rather, semiautomata) to relate the process of UWSN data aggregation to the existing algebraic theory of quasi-automata and their hyperstructure generalization. We show that the process of data aggregation can be seen as an automaton, or rather its hyperstructure generalization, with states representing stages of the data aggregation process of cluster protocols and describing available/used memory capacity of the network.

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From lattices to Hv-matrices

Cited by 5 publications

References 15 publications

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

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

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

Elements of Hyperstructure Theory in UWSN Design and Data Aggregation

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