{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,14]],"date-time":"2025-10-14T00:43:20Z","timestamp":1760402600456,"version":"build-2065373602"},"reference-count":32,"publisher":"MDPI AG","issue":"1","license":[{"start":{"date-parts":[[2020,1,15]],"date-time":"2020-01-15T00:00:00Z","timestamp":1579046400000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Symmetry"],"abstract":"<jats:p>In every hypergroup, the equivalence classes modulo the fundamental relation    \u03b2    are the union of hyperproducts of element pairs. Making use of this property, we introduce the notion of height of a    \u03b2   -class and we analyze properties of hypergroups where the height of a    \u03b2   -class coincides with its cardinality. As a consequence, we obtain a new characterization of 1-hypergroups. Moreover, we define a hierarchy of classes of hypergroups where at least one    \u03b2   -class has height 1 or cardinality 1, and we enumerate the elements in each class when the size of the hypergroups is     n \u2264 4    , apart from isomorphisms.<\/jats:p>","DOI":"10.3390\/sym12010168","type":"journal-article","created":{"date-parts":[[2020,1,15]],"date-time":"2020-01-15T11:50:28Z","timestamp":1579089028000},"page":"168","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":8,"title":["On Hypergroups with a \u03b2-Class of Finite Height"],"prefix":"10.3390","volume":"12","author":[{"ORCID":"https:\/\/orcid.org\/0000-0001-5271-1742","authenticated-orcid":false,"given":"Mario","family":"De Salvo","sequence":"first","affiliation":[{"name":"Dipartimento di Scienze Matematiche e Informatiche, Scienze Fisiche e Scienze della Terra, Universit\u00e0 di Messina, 98166 Messina, Italy"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0001-7682-0660","authenticated-orcid":false,"given":"Dario","family":"Fasino","sequence":"additional","affiliation":[{"name":"Dipartimento di Scienze Matematiche, Informatiche e Fisiche, Universit\u00e0 di Udine, 33100 Udine, Italy"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0003-2895-7632","authenticated-orcid":false,"given":"Domenico","family":"Freni","sequence":"additional","affiliation":[{"name":"Dipartimento di Scienze Matematiche, Informatiche e Fisiche, Universit\u00e0 di Udine, 33100 Udine, Italy"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0001-6174-8627","authenticated-orcid":false,"given":"Giovanni","family":"Lo Faro","sequence":"additional","affiliation":[{"name":"Dipartimento di Scienze Matematiche e Informatiche, Scienze Fisiche e Scienze della Terra, Universit\u00e0 di Messina, 98166 Messina, Italy"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2020,1,15]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","unstructured":"Corsini, P., and Leoreanu-Fotea, V. (2003). Applications of Hyperstructures Theory, Kluwer Academic Publisher.","DOI":"10.1007\/978-1-4757-3714-1"},{"key":"ref_2","doi-asserted-by":"crossref","unstructured":"Cristea, I., Kocijan, J., and Nov\u00e1k, M. (2019). Introduction to Dependence Relations and Their Links to Algebraic Hyperstructures. Mathematics, 7.","DOI":"10.3390\/math7100885"},{"key":"ref_3","doi-asserted-by":"crossref","unstructured":"Nov\u00e1k, M., K\u0159ehl\u00edk, \u0160., and Ovaliadis, K. (2019). Elements of hyperstructure theory in UWSN design and data aggregation. Symmetry, 11.","DOI":"10.3390\/sym11060734"},{"key":"ref_4","doi-asserted-by":"crossref","first-page":"4389","DOI":"10.1080\/00927870600938316","article-title":"A realization of hyperrings","volume":"34","author":"Davvaz","year":"2006","journal-title":"Commun. Algebra"},{"key":"ref_5","doi-asserted-by":"crossref","first-page":"65","DOI":"10.1016\/j.jalgebra.2014.05.033","article-title":"Fully simple semihypergroups, transitive digraphs, and Sequence A000712","volume":"415","author":"Fasino","year":"2014","journal-title":"J. Algebra"},{"key":"ref_6","first-page":"553","article-title":"A family of 0-simple semihypergroups related to sequence A00070","volume":"27","author":"Fasino","year":"2016","journal-title":"J. Mult. Valued Log. Soft Comput."},{"key":"ref_7","doi-asserted-by":"crossref","first-page":"4177","DOI":"10.2298\/FIL1812177S","article-title":"Semihypergroups obtained by merging of 0-semigroups with groups","volume":"32","author":"Fasino","year":"2018","journal-title":"Filomat"},{"key":"ref_8","doi-asserted-by":"crossref","first-page":"358","DOI":"10.1016\/j.jalgebra.2013.09.046","article-title":"Fully simple semihypergroups","volume":"399","author":"Freni","year":"2014","journal-title":"J. Algebra"},{"key":"ref_9","first-page":"595","article-title":"Hypercyclic subhypergroups of finite fully simple semihypergroups","volume":"29","author":"Freni","year":"2017","journal-title":"J. Mult. Valued Log. Soft Comput."},{"key":"ref_10","first-page":"1905","article-title":"On hypercyclic fully zero-simple semihypergroups","volume":"4","author":"Freni","year":"2019","journal-title":"Turk. J. Math."},{"key":"ref_11","doi-asserted-by":"crossref","first-page":"177","DOI":"10.1016\/S0012-365X(99)00071-0","article-title":"On the n*-complete hypergroups","volume":"208\/209","year":"1999","journal-title":"Discret. Math."},{"key":"ref_12","first-page":"153","article-title":"Une note sur le c\u0153ur d\u2019un hypergroup et sur la cl\u00f4ture transitive \u03b2* de \u03b2","volume":"8","author":"Freni","year":"1991","journal-title":"Riv. Mat. Pura Appl."},{"key":"ref_13","doi-asserted-by":"crossref","first-page":"969","DOI":"10.1081\/AGB-120027961","article-title":"Strongly transitive geometric spaces: Applications to hypergroups and semigroups theory","volume":"32","author":"Freni","year":"2004","journal-title":"Commun. Algebra"},{"key":"ref_14","doi-asserted-by":"crossref","first-page":"189","DOI":"10.1007\/BF02844485","article-title":"On the transitivity of the relation \u03b2 in semihypergroups","volume":"45","author":"Gutan","year":"1996","journal-title":"Rendiconti del Circolo Matematico di Palermo"},{"key":"ref_15","doi-asserted-by":"crossref","first-page":"519","DOI":"10.1016\/j.camwa.2009.06.013","article-title":"On enumeration of hypergroups of order 3","volume":"59","author":"Tsitouras","year":"2010","journal-title":"Comput. Math. Appl."},{"key":"ref_16","first-page":"113","article-title":"Fundamental relations in hyperstructures","volume":"42","author":"Vougiouklis","year":"1999","journal-title":"Bull. Greek Math. Soc."},{"key":"ref_17","first-page":"155","article-title":"Groupo\u00efdes, demi-hypergroupes et hypergroupes","volume":"49","author":"Koskas","year":"1970","journal-title":"J. Math. Pures Appl."},{"key":"ref_18","first-page":"361","article-title":"A new class of hypergroupoids associated with binary relations","volume":"9","year":"2003","journal-title":"J. Mult. Valued Log. Soft Comput."},{"key":"ref_19","doi-asserted-by":"crossref","first-page":"175","DOI":"10.1007\/s40065-012-0025-2","article-title":"Fundamental relations in simple and 0-simple semi-hypergroups of small size","volume":"1","author":"Fasino","year":"2012","journal-title":"Arab. J. Math."},{"key":"ref_20","doi-asserted-by":"crossref","first-page":"77","DOI":"10.1016\/j.jalgebra.2011.05.015","article-title":"Minimal order semi-hypergroups of type U on the right. II","volume":"340","author":"Freni","year":"2011","journal-title":"J. Algebra"},{"key":"ref_21","doi-asserted-by":"crossref","first-page":"4183","DOI":"10.1080\/00927872.2010.521932","article-title":"On quasi-hyperideals in semihypergroups","volume":"39","author":"Hila","year":"2011","journal-title":"Commun. Algebra"},{"key":"ref_22","doi-asserted-by":"crossref","first-page":"2203","DOI":"10.3233\/IFS-130894","article-title":"On soft semihypergroups","volume":"26","author":"Naz","year":"2014","journal-title":"J. Intell. Fuzzy Syst."},{"key":"ref_23","unstructured":"Vougiouklis, T. (2003). Fundamental relations in special extensions. Algebraic Hyperstructures and Applications, Spanidis Press."},{"key":"ref_24","doi-asserted-by":"crossref","unstructured":"Davvaz, B. (2016). Semihypergroup Theory, Academic Press.","DOI":"10.1016\/B978-0-12-809815-8.00002-4"},{"key":"ref_25","first-page":"493","article-title":"Bi-hyperideals and quasi-hyperideals in ordered semihypergroups","volume":"35","author":"Changphas","year":"2015","journal-title":"Ital. J. Pure Appl. Math."},{"key":"ref_26","unstructured":"Corsini, P. (1993). Prolegomena of Hypergroup Theory, Aviani Editore."},{"key":"ref_27","unstructured":"Davvaz, B., and Leoreanu-Fotea, V. (2007). Hyperring Theory and Applications, International Academic Press."},{"key":"ref_28","doi-asserted-by":"crossref","first-page":"48","DOI":"10.1007\/s00009-019-1324-z","article-title":"On further properties of fully zero-simple semihypergroups","volume":"16","author":"Freni","year":"2019","journal-title":"Mediterr. J. Math."},{"key":"ref_29","first-page":"44","article-title":"Semi-ipergruppi e ipergruppi ciclici","volume":"30","author":"Freni","year":"1981","journal-title":"Atti Sem. Mat. Fis. Univ. Modena"},{"key":"ref_30","first-page":"37","article-title":"An algorithm on number of isomorphism classes of hypergroups of order 3","volume":"2","author":"Nordo","year":"1997","journal-title":"Ital. J. Pure Appl. Math."},{"key":"ref_31","doi-asserted-by":"crossref","first-page":"97","DOI":"10.1006\/jabr.1997.6789","article-title":"Transposition hypergroups: Noncommutative join spaces","volume":"187","author":"Jantosciak","year":"1997","journal-title":"J. Algebra"},{"key":"ref_32","doi-asserted-by":"crossref","first-page":"101","DOI":"10.1215\/S0012-7094-40-00609-3","article-title":"Theory of cogroups","volume":"6","author":"Eaton","year":"1940","journal-title":"Duke Math."}],"container-title":["Symmetry"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/2073-8994\/12\/1\/168\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,13]],"date-time":"2025-10-13T13:43:03Z","timestamp":1760362983000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/2073-8994\/12\/1\/168"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2020,1,15]]},"references-count":32,"journal-issue":{"issue":"1","published-online":{"date-parts":[[2020,1]]}},"alternative-id":["sym12010168"],"URL":"https:\/\/doi.org\/10.3390\/sym12010168","relation":{},"ISSN":["2073-8994"],"issn-type":[{"type":"electronic","value":"2073-8994"}],"subject":[],"published":{"date-parts":[[2020,1,15]]}}}