EL-hyperstructures revisited

Novák, Michal; Křehlík, Štěpán

doi:10.1007/s00500-017-2728-y

Cited by 17 publications

(12 citation statements)

References 9 publications

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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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Section: Mathematical Background Of the Modelmentioning

confidence: 99%

Elements of Hyperstructure Theory in UWSN Design and Data Aggregation

2019

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“…are based upon what is known as "Ends lemma", which was proved by Chvalina [25] and studied by Novák [27,28], later with Cristea [29] and Křehlík [30,31] and Chvalina [32]. Semihypergroups of the below type are called EL-semihypergroups.…”

Section: Most Of the Examples Included Belowmentioning

confidence: 99%

Cyclicity in EL–Hypergroups

2018

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In the algebra of single-valued structures, cyclicity is one of the fundamental properties of groups. Therefore, it is natural to study it also in the algebra of multivalued structures (algebraic hyperstructure theory). However, when one considers the nature of generalizing this property, at least two (or rather three) approaches seem natural. Historically, all of these had been introduced and studied by 1990. However, since most of the results had originally been published in journals without proper international impact and later—without the possibility to include proper background and context-synthetized in books, the current way of treating the concept of cyclicity in the algebraic hyperstructure theory is often rather confusing. Therefore, we start our paper with a rather long introduction giving an overview and motivation of existing approaches to the cyclicity in algebraic hyperstructures. In the second part of our paper, we relate these to E L -hyperstructures, a broad class of algebraic hyperstructures constructed from (pre)ordered (semi)groups, which were defined and started to be studied much later than sources discussed in the introduction were published.

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“…One of these is the application of the Ends lemma [5][6][7] in which the hyperoperation is the principal end of a partially ordered semigroup. For some theoretical results regarding the construction, see for example Novák et al [7][8][9]. In [9] the authors modify the construction in order to increase its applicability.…”

Section: Introductionmentioning

confidence: 99%

“…For some theoretical results regarding the construction, see for example Novák et al [7][8][9]. In [9] the authors modify the construction in order to increase its applicability.…”

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

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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EL-hyperstructures revisited

Cited by 17 publications

References 9 publications

Elements of Hyperstructure Theory in UWSN Design and Data Aggregation

Elements of Hyperstructure Theory in UWSN Design and Data Aggregation

Cyclicity in EL–Hypergroups

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

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