Cyclicity in <i>EL</i>–Hypergroups
In the algebra of single-valued structures, <i>cyclicity</i> 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 p...
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| Format: | Article |
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MDPI AG
2018-11-01
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| Series: | Symmetry |
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| Online Access: | https://www.mdpi.com/2073-8994/10/11/611 |
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| author | Michal Novák Štepán Křehlík Irina Cristea |
| author_facet | Michal Novák Štepán Křehlík Irina Cristea |
| author_sort | Michal Novák |
| collection | DOAJ |
| description | In the algebra of single-valued structures, <i>cyclicity</i> 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 <inline-formula> <math display="inline"> <semantics> <mrow> <mi>E</mi> <mi>L</mi> </mrow> </semantics> </math> </inline-formula>-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. |
| first_indexed | 2024-04-13T06:18:40Z |
| format | Article |
| id | doaj.art-0cca77067580488ea603a5b2c8392c11 |
| institution | Directory Open Access Journal |
| issn | 2073-8994 |
| language | English |
| last_indexed | 2024-04-13T06:18:40Z |
| publishDate | 2018-11-01 |
| publisher | MDPI AG |
| record_format | Article |
| series | Symmetry |
| spelling | doaj.art-0cca77067580488ea603a5b2c8392c112022-12-22T02:58:43ZengMDPI AGSymmetry2073-89942018-11-01101161110.3390/sym10110611sym10110611Cyclicity in <i>EL</i>–HypergroupsMichal Novák0Štepán Křehlík1Irina Cristea2Faculty of Electrical Engineering and Communication, Brno University of Technology, Technická 8, 616 00 Brno, Czech RepublicDepartment of Applied Mathematics and Computer Science, Masaryk University, Lipová 41a, 602 00 Brno, Czech RepublicCentre for Information Technologies and Applied Mathematics, University of Nova Gorica, Vipavska cesta 13, 5000 Nova Gorica, SloveniaIn the algebra of single-valued structures, <i>cyclicity</i> 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 <inline-formula> <math display="inline"> <semantics> <mrow> <mi>E</mi> <mi>L</mi> </mrow> </semantics> </math> </inline-formula>-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.https://www.mdpi.com/2073-8994/10/11/611cyclic groupcyclic hypergroupEL-hyperstructurepreorder |
| spellingShingle | Michal Novák Štepán Křehlík Irina Cristea Cyclicity in <i>EL</i>–Hypergroups Symmetry cyclic group cyclic hypergroup EL-hyperstructure preorder |
| title | Cyclicity in <i>EL</i>–Hypergroups |
| title_full | Cyclicity in <i>EL</i>–Hypergroups |
| title_fullStr | Cyclicity in <i>EL</i>–Hypergroups |
| title_full_unstemmed | Cyclicity in <i>EL</i>–Hypergroups |
| title_short | Cyclicity in <i>EL</i>–Hypergroups |
| title_sort | cyclicity in i el i hypergroups |
| topic | cyclic group cyclic hypergroup EL-hyperstructure preorder |
| url | https://www.mdpi.com/2073-8994/10/11/611 |
| work_keys_str_mv | AT michalnovak cyclicityinielihypergroups AT stepankrehlik cyclicityinielihypergroups AT irinacristea cyclicityinielihypergroups |