{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,2,28]],"date-time":"2026-02-28T04:31:07Z","timestamp":1772253067005,"version":"3.50.1"},"reference-count":15,"publisher":"MDPI AG","issue":"6","license":[{"start":{"date-parts":[[2020,6,3]],"date-time":"2020-06-03T00:00:00Z","timestamp":1591142400000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"DOI":"10.13039\/501100001871","name":"Funda\u00e7\u00e3o para a Ci\u00eancia e a Tecnologia","doi-asserted-by":"publisher","award":["project UIDB\/04106\/2020"],"award-info":[{"award-number":["project UIDB\/04106\/2020"]}],"id":[{"id":"10.13039\/501100001871","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Symmetry"],"abstract":"<jats:p>In this article, we prove an orthogonal decomposition theorem for real inner product gyrogroups, which unify some well-known gyrogroups in the literature: Einstein, M\u00f6bius, Proper Velocity, and Chen\u2019s gyrogroups. This leads to the study of left (right) coset partition of a real inner product gyrogroup induced from a subgyrogroup that is a finite dimensional subspace. As a result, we obtain gyroprojectors onto the subgyrogroup and its orthogonal complement. We construct also quotient spaces and prove an associated isomorphism theorem. The left (right) cosets are characterized using gyrolines (cogyrolines) together with automorphisms of the subgyrogroup. With the algebraic structure of the decompositions, we study fiber bundles and sections inherited by the gyroprojectors. Finally, the general theory is exemplified for the aforementioned gyrogroups.<\/jats:p>","DOI":"10.3390\/sym12060941","type":"journal-article","created":{"date-parts":[[2020,6,5]],"date-time":"2020-06-05T03:32:21Z","timestamp":1591327941000},"page":"941","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":4,"title":["Orthogonal Gyrodecompositions of Real Inner Product Gyrogroups"],"prefix":"10.3390","volume":"12","author":[{"ORCID":"https:\/\/orcid.org\/0000-0003-1816-8293","authenticated-orcid":false,"given":"Milton","family":"Ferreira","sequence":"first","affiliation":[{"name":"School of Technology and Management, Polytechnic Institute of Leiria, 2411-901 Leiria, Portugal"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-1239-5586","authenticated-orcid":false,"given":"Teerapong","family":"Suksumran","sequence":"additional","affiliation":[{"name":"Research Center in Mathematics and Applied Mathematics, Department of Mathematics, Faculty of Science, Chiang Mai University, Chiang Mai 50200, Thailand"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2020,6,3]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"57","DOI":"10.1007\/BF00661317","article-title":"Thomas rotation and the parametrization of the Lorentz transformation group","volume":"1","author":"Ungar","year":"1988","journal-title":"Found. Phys. Lett."},{"key":"ref_2","doi-asserted-by":"crossref","unstructured":"Ungar, A.A. (2005). Analytic Hyperbolic Geometry: Mathematical Foundations and Applications, World Scientific.","DOI":"10.1142\/9789812703279"},{"key":"ref_3","doi-asserted-by":"crossref","unstructured":"Ungar, A.A. (2008). Analytic Hyperbolic Geometry and Albert Einstein\u2019s Special Theory of Relativity, World Scientific.","DOI":"10.1142\/9789812772305"},{"key":"ref_4","doi-asserted-by":"crossref","first-page":"303","DOI":"10.1007\/s00006-009-0154-7","article-title":"Factorizations of M\u00f6bius gyrogroups","volume":"19","author":"Ferreira","year":"2009","journal-title":"Adv. Appl. Clifford Algebr."},{"key":"ref_5","doi-asserted-by":"crossref","first-page":"230","DOI":"10.1016\/j.jalgebra.2010.05.014","article-title":"M\u00f6bius gyrogroups: A Clifford algebra approach","volume":"328","author":"Ferreira","year":"2011","journal-title":"J. Algebra"},{"key":"ref_6","doi-asserted-by":"crossref","first-page":"212","DOI":"10.1016\/j.acha.2008.04.005","article-title":"Spherical continuous wavelet transforms arising from sections of the Lorentz group","volume":"26","author":"Ferreira","year":"2009","journal-title":"Appl. Comput. Harmon. Anal."},{"key":"ref_7","doi-asserted-by":"crossref","first-page":"77","DOI":"10.1016\/j.jmaa.2016.11.039","article-title":"Finitely generated gyrovector subspaces and orthogonal gyrodecomposition in the M\u00f6bius gyrovector space","volume":"449","author":"Abe","year":"2017","journal-title":"J. Math. Anal. Appl."},{"key":"ref_8","first-page":"67","article-title":"Isomorphism theorems for gyrogroups and L-subgyrogroups","volume":"37","author":"Suksumran","year":"2015","journal-title":"J. Geom. Symmetry Phys."},{"key":"ref_9","unstructured":"Rassias, T.M., and Pardalos, P.M. (2016). The Algebra of Gyrogroups: Cayley\u2019s Theorem, Lagrange\u2019s Theorem, and Isomorphism Theorems. Essays in Mathematics and its Applications, In Honor of Vladimir Arnold, Springer."},{"key":"ref_10","doi-asserted-by":"crossref","first-page":"1228","DOI":"10.1016\/j.camwa.2006.05.028","article-title":"Einstein\u2019s velocity addition law and its hyperbolic geometry","volume":"53","author":"Ungar","year":"2007","journal-title":"Comput. Math. Appl."},{"key":"ref_11","doi-asserted-by":"crossref","first-page":"138","DOI":"10.1080\/00029890.2008.11920506","article-title":"From M\u00f6bius to gyrogroups","volume":"115","author":"Ungar","year":"2008","journal-title":"Am. Math. Mon."},{"key":"ref_12","doi-asserted-by":"crossref","first-page":"1040","DOI":"10.1006\/jmaa.1996.0359","article-title":"Extension of the unit disk gyrogroup into the unit ball of any real inner product space","volume":"202","author":"Ungar","year":"1996","journal-title":"J. Math. Anal. Appl."},{"key":"ref_13","doi-asserted-by":"crossref","first-page":"1611","DOI":"10.1023\/A:1012694816323","article-title":"From the group SL(2,C) to gyrogroups and gyrovector spaces and hyperbolic geometry","volume":"31","author":"Chen","year":"2001","journal-title":"Found. Phys."},{"key":"ref_14","first-page":"1518254","article-title":"Orthogonal Gyroexpansion in M\u00f6bius Gyrovector Spaces","volume":"2017","author":"Watanabe","year":"2017","journal-title":"J. Funct. Spaces"},{"key":"ref_15","first-page":"111","article-title":"Bi-Gyrogroup: The Group-Like Structure Induced by Bi-Decomposition of Groups","volume":"1","author":"Teerapong","year":"2016","journal-title":"Math. Interdisc. Res."}],"container-title":["Symmetry"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/2073-8994\/12\/6\/941\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T09:35:20Z","timestamp":1760175320000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/2073-8994\/12\/6\/941"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2020,6,3]]},"references-count":15,"journal-issue":{"issue":"6","published-online":{"date-parts":[[2020,6]]}},"alternative-id":["sym12060941"],"URL":"https:\/\/doi.org\/10.3390\/sym12060941","relation":{"has-preprint":[{"id-type":"doi","id":"10.20944\/preprints202005.0371.v1","asserted-by":"object"}]},"ISSN":["2073-8994"],"issn-type":[{"value":"2073-8994","type":"electronic"}],"subject":[],"published":{"date-parts":[[2020,6,3]]}}}