{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,12]],"date-time":"2025-10-12T04:30:40Z","timestamp":1760243440214,"version":"build-2065373602"},"reference-count":30,"publisher":"MDPI AG","issue":"11","license":[{"start":{"date-parts":[[2022,11,8]],"date-time":"2022-11-08T00:00:00Z","timestamp":1667865600000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Axioms"],"abstract":"<jats:p>The concept of ternary semihypergroups can be considered as a natural generalization of arbitrary ternary semigroups. In fact, each ternary semigroup can be constructed to a ternary semihypergroup. In this article, we investigate some interesting algebraic properties of ternary semihypergroups induced by semihypergroups. Then, we extend the well-known result on group theory and semigroup theory, the so-called Cayley\u2019s theorem, to study on ternary semihypergroups. This leads us to construct the ternary semihypergroups of all multivalued full binary functions. In particular, we investigate that each element of a ternary semihypergroup induced by a semihypergroup can be represented by a multivalued full binary function. Moreover, we introduce the concept of translations for ternary semihypergroups which can be considered as a generalization of translations on ternary semigrgoups. Then, we construct ternary semihypergroups of all multivalued full functions and ternary semihypergroups via translations. So, some interesting algebraic properties are investigated. At the last section, we discover that there are ternary semihypergroups satisfying some significant conditions which can be reduced to semihypergroups. Furthermore, ternary semihypergroups with another one condition can be reduced to idempotent semihypergroups.<\/jats:p>","DOI":"10.3390\/axioms11110626","type":"journal-article","created":{"date-parts":[[2022,11,9]],"date-time":"2022-11-09T02:34:52Z","timestamp":1667961292000},"page":"626","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":0,"title":["Representations, Translations and Reductions for Ternary Semihypergroups"],"prefix":"10.3390","volume":"11","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-5198-2185","authenticated-orcid":false,"given":"Anak","family":"Nongmanee","sequence":"first","affiliation":[{"name":"Department of Mathematics, Faculty of Science, Chiang Mai University, Chiang Mai 50200, Thailand"}]},{"ORCID":"https:\/\/orcid.org\/0000-0001-8818-6134","authenticated-orcid":false,"given":"Sorasak","family":"Leeratanavalee","sequence":"additional","affiliation":[{"name":"Research Group in Mathematics and Applied Mathematics, Department of Mathematics, Faculty of Science, Chiang Mai University, Chiang Mai 50200, Thailand"}]}],"member":"1968","published-online":{"date-parts":[[2022,11,8]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"38","DOI":"10.4064\/fm-42-1-38-54","article-title":"On the extending of models I","volume":"42","author":"Los","year":"1995","journal-title":"Fund. Math."},{"key":"ref_2","doi-asserted-by":"crossref","first-page":"329","DOI":"10.2307\/2370997","article-title":"A ternary analogous of abelian groups","volume":"54","author":"Lehmer","year":"1932","journal-title":"Am. J. Math."},{"key":"ref_3","first-page":"290","article-title":"An extending of the group concept","volume":"10","author":"Kanser","year":"1904","journal-title":"Bull. Amer. Math. Soc."},{"key":"ref_4","first-page":"283","article-title":"On regular algebraic systems","volume":"39","author":"Sioson","year":"1963","journal-title":"Proc. Japan Acad."},{"key":"ref_5","first-page":"67","article-title":"Regular ternary semigroups","volume":"82","author":"Santiago","year":"1990","journal-title":"Bull. Calcutta Math. Soc."},{"key":"ref_6","doi-asserted-by":"crossref","first-page":"380","DOI":"10.1007\/s00233-010-9254-x","article-title":"Ternary semigroups","volume":"81","author":"Santiago","year":"2010","journal-title":"Semigroup Forum"},{"key":"ref_7","first-page":"103","article-title":"Congruence and Green\u2019s equivalence relation on ternary semigroup","volume":"46","author":"Dixit","year":"1997","journal-title":"Commun. Fac. Sci. Univ. Ank. Series A1"},{"key":"ref_8","first-page":"2405","article-title":"Generalized Hamiltonian dynamics","volume":"7","author":"Nambu","year":"1973","journal-title":"J. Phys. Rev."},{"key":"ref_9","doi-asserted-by":"crossref","first-page":"18500094","DOI":"10.1142\/S0219887818500949","article-title":"\u2217-Relations on ternary semihypergroups","volume":"15","author":"Ashrafi","year":"2018","journal-title":"Int. J. Geom. Methods Mod. Phys."},{"key":"ref_10","doi-asserted-by":"crossref","first-page":"121","DOI":"10.5486\/PMD.1990.37.1-2.16","article-title":"The Pexider equation on n-semigroups and n-groups","volume":"37","author":"Chronoski","year":"1990","journal-title":"Publ. Math. Debrecen"},{"key":"ref_11","doi-asserted-by":"crossref","first-page":"13","DOI":"10.1016\/j.jmaa.2007.03.073","article-title":"Homomorphismsin C*-ternary algebras and JB*-triples","volume":"337","author":"Park","year":"2008","journal-title":"J. Math. Anal. Appl."},{"key":"ref_12","doi-asserted-by":"crossref","first-page":"042303","DOI":"10.1063\/1.3093269","article-title":"Approximate ternary Jordan derivations on Banach ternary algebras","volume":"50","author":"Savadkouhi","year":"2009","journal-title":"J. Math. Phys."},{"key":"ref_13","first-page":"371","article-title":"Generalized Hyers\u2013Ulam stability of mappings on normed Lie triple systems","volume":"11","author":"Moslehian","year":"2008","journal-title":"Math. Inequal. Appl."},{"key":"ref_14","doi-asserted-by":"crossref","first-page":"1650","DOI":"10.1063\/1.531821","article-title":"Hypersymmetry: A Z3-graded generalization of supersymmetry","volume":"38","author":"Abramov","year":"1997","journal-title":"J. Math. Phys."},{"key":"ref_15","doi-asserted-by":"crossref","first-page":"1320013","DOI":"10.1142\/S0219887813200132","article-title":"Automatic continuity of 3-homomorphisms on ternary Banach algebra","volume":"10","author":"Gordji","year":"2013","journal-title":"Int. J. Geom. Methods Mod. Phys."},{"key":"ref_16","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1007\/s11005-005-0042-6","article-title":"Approximate homomorphisms of ternary semigroups","volume":"77","author":"Amyayari","year":"2006","journal-title":"Lett. Math. Phys."},{"key":"ref_17","doi-asserted-by":"crossref","unstructured":"Nongmanee, A., and Leeratanavalee, S. (2021). Ternary Menger algebras: A generalization of ternary semigroups. Mathematics, 9.","DOI":"10.3390\/math9050553"},{"key":"ref_18","doi-asserted-by":"crossref","unstructured":"Nongmanee, A., and Leeratanavalee, S. (2021). v-Regular ternary Menger algebras and left translations of ternary Menger algebras. Mathematics, 9.","DOI":"10.3390\/math9212691"},{"key":"ref_19","first-page":"163","article-title":"On n-ary semigroups with adjoint neuatral element","volume":"14","author":"Dudek","year":"2006","journal-title":"Quasigr. Relat. Syst."},{"key":"ref_20","unstructured":"Marty, F. (1934, January 14\u201318). Sur une generalization de la notion de groupe. In Proceeding of the 8th Congress Mathematics Scandenaves, Stockholm, Sweden."},{"key":"ref_21","doi-asserted-by":"crossref","unstructured":"Corsini, P., and Leoreanu, V. (2003). Applications of Hyperstructures Theory, Kluwer Academical Publications.","DOI":"10.1007\/978-1-4757-3714-1"},{"key":"ref_22","unstructured":"Davvaz, B., and Leoreanu-Fotea, V. (2007). Hyperring Theory and Applications, International Academic Press."},{"key":"ref_23","doi-asserted-by":"crossref","unstructured":"Davvaz, B. (2016). Semihypergroup Theory, Elsevier.","DOI":"10.1016\/B978-0-12-809815-8.00002-4"},{"key":"ref_24","doi-asserted-by":"crossref","first-page":"3621","DOI":"10.1080\/00927870903200935","article-title":"Binary relations on ternary semihypergroups","volume":"38","author":"Davvaz","year":"2010","journal-title":"Commun. Algebra"},{"key":"ref_25","first-page":"81","article-title":"On hyperideal structure of ternary semihypergroups","volume":"9","author":"Hila","year":"2014","journal-title":"Iran. J. Math. Sci. Inform."},{"key":"ref_26","first-page":"349","article-title":"Algebraic hyperstructure of soft sets associated to ternary semihypergroups","volume":"30","author":"Hila","year":"2013","journal-title":"Ital. J. Pure Appl. Math."},{"key":"ref_27","doi-asserted-by":"crossref","first-page":"1573","DOI":"10.1007\/s13370-014-0306-3","article-title":"On the structure of quasi-hyperideals and bi-hyperideals in ternary semihypergroups","volume":"26","author":"Naka","year":"2015","journal-title":"Afr. Mat."},{"key":"ref_28","first-page":"729","article-title":"Quaternary rectangular bands and representations of ternary semigroups","volume":"20","author":"Nongmanee","year":"2022","journal-title":"Thai J. Math."},{"key":"ref_29","doi-asserted-by":"crossref","first-page":"1513","DOI":"10.1080\/00927872.2020.1839089","article-title":"Menger hyperalgebras and their representations","volume":"49","author":"Kumduang","year":"2021","journal-title":"Commun. Algebra"},{"key":"ref_30","doi-asserted-by":"crossref","first-page":"274","DOI":"10.1016\/j.ejc.2014.08.014","article-title":"On EL-semihypergroups","volume":"44","year":"2015","journal-title":"Eur. J. 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