{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,12]],"date-time":"2025-10-12T01:25:14Z","timestamp":1760232314026,"version":"build-2065373602"},"reference-count":26,"publisher":"MDPI AG","issue":"11","license":[{"start":{"date-parts":[[2022,10,28]],"date-time":"2022-10-28T00:00:00Z","timestamp":1666915200000},"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 this paper, first, we state an operator LR on an ordered semihyperring R. We show that if \u03c6:R\u27f6T is a monomorphism and K\u2286R, then LT(\u03c6(K))=\u03c6(LR(K)). Afterward, hyperatom elements in ordered semihyperrings are defined and some results in this respect are investigated. Denote by A(R) the set of all hyperatoms of R. We prove that if R is a finite ordered semihyperring and |R|\u22652, then for any q\u2208R\\{0}, there exists hq\u2208A*(R)=A(R)\\{0} such that hq\u2264q. Finally, we study the LR-graph of an ordered semihyperring and give some examples. Furthermore, we show that if \u03c6:R\u27f6T is an isomorphism, G is the LR-graph of R and G\u2032 is the LT-graph of T, then G\u2245G\u2032.<\/jats:p>","DOI":"10.3390\/sym14112265","type":"journal-article","created":{"date-parts":[[2022,10,30]],"date-time":"2022-10-30T10:47:57Z","timestamp":1667126877000},"page":"2265","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":2,"title":["Some Aspects of Hyperatom Elements in Ordered Semihyperrings"],"prefix":"10.3390","volume":"14","author":[{"given":"Zheng","family":"Kou","sequence":"first","affiliation":[{"name":"Institute of Computing Science and Technology, Guangzhou University, Guangzhou 510006, China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Mehdi","family":"Gheisari","sequence":"additional","affiliation":[{"name":"Department of Cognitive Computing, Institute of Computer Science and Engineering, Saveetha School of Engineering, Saveetha Institute of Medical and Technical Sciences, Chennai 602105, India"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Kostaq","family":"Hila","sequence":"additional","affiliation":[{"name":"Department of Mathematical Engineering, Polytechnic University of Tirana, Tirana 1001, Albania"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Nategh","family":"Abbasizadeh","sequence":"additional","affiliation":[{"name":"Department of Education, Ministry of Education Iran, Shiraz 73, Iran"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2022,10,28]]},"reference":[{"key":"ref_1","unstructured":"Marty, F. (1934). Sur une Generalization de la Notion de Groupe, University of Primorska."},{"key":"ref_2","first-page":"21","article-title":"On some representations of hypergroups","volume":"26","author":"Vougiouklis","year":"1990","journal-title":"Ann. Sci. Univ. Clermont-Ferrand II Math."},{"key":"ref_3","first-page":"85","article-title":"On ordered hyperstructures","volume":"73","author":"Heidari","year":"2011","journal-title":"Politehn. Univ. Bucharest Sci. Bull. Ser. A Appl. Math. Phys."},{"key":"ref_4","doi-asserted-by":"crossref","first-page":"5","DOI":"10.1155\/2021\/9754585","article-title":"Factorizable ordered hypergroupoids with applications","volume":"2021","author":"Shi","year":"2021","journal-title":"Math. Probl. Eng."},{"key":"ref_5","doi-asserted-by":"crossref","first-page":"208","DOI":"10.1016\/j.ejc.2014.08.006","article-title":"Relationship between ordered semihypergroups and ordered semigroups by using pseudoorder","volume":"44","author":"Davvaz","year":"2015","journal-title":"Eur. J. Combin."},{"key":"ref_6","doi-asserted-by":"crossref","first-page":"384","DOI":"10.1016\/j.jalgebra.2015.11.026","article-title":"Ordered regular equivalence relations on ordered semihypergroups","volume":"450","author":"Gu","year":"2016","journal-title":"J. Algebra"},{"key":"ref_7","doi-asserted-by":"crossref","first-page":"168","DOI":"10.1515\/math-2018-0016","article-title":"A further study on ordered regular equivalence relations in ordered semihypergroups","volume":"16","author":"Tang","year":"2018","journal-title":"Open Math."},{"key":"ref_8","first-page":"125","article-title":"Left k-bi-quasi hyperideals in ordered semihyperrings","volume":"83","author":"Rao","year":"2021","journal-title":"Politehn. Univ. Bucharest Sci. Bull. Ser. A Appl. Math. Phys."},{"key":"ref_9","first-page":"457","article-title":"Uni-soft structures applied to ordered \u0393-semihypergroups","volume":"90","author":"Khan","year":"2020","journal-title":"Proc. Natl. Acad. Sci. USA"},{"key":"ref_10","doi-asserted-by":"crossref","first-page":"1035","DOI":"10.1093\/qmath\/hay011","article-title":"On (k,n)-absorbing hyperideals in Krasner (m,n)-hyperrings","volume":"69","author":"Hila","year":"2018","journal-title":"Q. J. Math."},{"key":"ref_11","first-page":"1","article-title":"Derivations in hyperrings and prime hyperrings","volume":"8","author":"Asokkumar","year":"2013","journal-title":"Iran. J. Math. Sci. Inform."},{"key":"ref_12","doi-asserted-by":"crossref","unstructured":"Qiang, X., Guan, H., and Rashmanlou, H. (2021). A note on the w-pseudo-orders in ordered (semi)hyperrings. Symmetry, 13.","DOI":"10.3390\/sym13122371"},{"key":"ref_13","first-page":"601","article-title":"Construction of ordered regular equivalence relations on ordered semihyperrings","volume":"40","author":"Omidi","year":"2018","journal-title":"Honam Math. J."},{"key":"ref_14","doi-asserted-by":"crossref","first-page":"555573","DOI":"10.3389\/fphy.2020.555573","article-title":"Some properties of relative bi-(int-)\u0393-hyperideals in ordered \u0393-semihypergroups","volume":"8","author":"Rao","year":"2020","journal-title":"Front. Phys."},{"key":"ref_15","doi-asserted-by":"crossref","first-page":"10","DOI":"10.1155\/2021\/6683910","article-title":"A study on A-I-\u0393-hyperideals and (m,n)-\u0393-hyperfilters in ordered \u0393-Semihypergroups","volume":"2021","author":"Rao","year":"2021","journal-title":"Discrete Dyn. Nat. Soc."},{"key":"ref_16","first-page":"87","article-title":"Some properties of derivations and m-k-hyperideals in ordered semihyperrings","volume":"83","author":"Rao","year":"2021","journal-title":"Politehn. Univ. Bucharest Sci. Bull. Ser. A Appl. Math. Phys."},{"key":"ref_17","doi-asserted-by":"crossref","unstructured":"Kou, Z., Kosari, S., Monemrad, M., Akhoundi, M., and Omidi, S. (2021). A note on the connection between ordered semihyperrings. Symmetry, 13.","DOI":"10.3390\/sym13112035"},{"key":"ref_18","doi-asserted-by":"crossref","first-page":"146","DOI":"10.29020\/nybg.ejpam.v12i1.3362","article-title":"On a graph induced by a hyper BCI-algebra","volume":"12","author":"Panganduyon","year":"2019","journal-title":"Eur. J. Pure Appl. Math"},{"key":"ref_19","doi-asserted-by":"crossref","first-page":"590","DOI":"10.29020\/nybg.ejpam.v14i2.3970","article-title":"A topology on a hyper BCI-algebra generated by a hyper-order","volume":"14","author":"Panganduyon","year":"2021","journal-title":"Eur. J. Pure Appl. Math"},{"key":"ref_20","first-page":"43","article-title":"On direct sum of branches in hyper BCK-algebras","volume":"11","author":"Harizavi","year":"2016","journal-title":"Iran. J. Math. Sci. Inform."},{"key":"ref_21","first-page":"303","article-title":"Scalar elements and hyperatoms of hyper BCK-algebras","volume":"2","author":"Jun","year":"1999","journal-title":"Sci. Math."},{"key":"ref_22","doi-asserted-by":"crossref","unstructured":"Bondy, J.A., and Murty, U.S.R. (1976). Graph Theory with Applications, The Macmillan Press Ltd.","DOI":"10.1007\/978-1-349-03521-2"},{"key":"ref_23","unstructured":"West, D.B. (1996). Introduction to Graph Theory, Prentice-Hall, Inc."},{"key":"ref_24","doi-asserted-by":"crossref","first-page":"1027","DOI":"10.1007\/s12648-012-0151-x","article-title":"A physical example of algebraic hyperstructures: Leptons","volume":"86","author":"Nezhad","year":"2021","journal-title":"Indian J. Phys."},{"key":"ref_25","first-page":"169","article-title":"Algebraic hyperstructure of observable elementary particles including the Higgs boson","volume":"90","author":"Davvaz","year":"2020","journal-title":"Proc. Natl. Acad. Sci. USA"},{"key":"ref_26","doi-asserted-by":"crossref","first-page":"119","DOI":"10.2298\/FIL1401119F","article-title":"Degree hypergroupoids associated with hypergraphs","volume":"28","author":"Farshi","year":"2014","journal-title":"Filomat"}],"container-title":["Symmetry"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/2073-8994\/14\/11\/2265\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T01:04:47Z","timestamp":1760144687000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/2073-8994\/14\/11\/2265"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2022,10,28]]},"references-count":26,"journal-issue":{"issue":"11","published-online":{"date-parts":[[2022,11]]}},"alternative-id":["sym14112265"],"URL":"https:\/\/doi.org\/10.3390\/sym14112265","relation":{},"ISSN":["2073-8994"],"issn-type":[{"type":"electronic","value":"2073-8994"}],"subject":[],"published":{"date-parts":[[2022,10,28]]}}}