Relation Graphs of the Sedenion Algebra
Let S denote the algebra of sedenions and let Γ O (S) denote its orthogonality graph. One can observe that every pair of zero divisors in S generates a double hexagon in Γ O (S). The set of vertices of a double hexagon can be extended to a basis of S that has a convenient multiplication table. The s...
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| Veröffentlicht in: | Journal of mathematical sciences (New York, N.Y.) N.Y.), 2021-06, Vol.255 (3), p.254-270 |
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| creator | Guterman, A. E. Zhilina, S. A. |
| description | Let S denote the algebra of sedenions and let Γ
O
(S) denote its orthogonality graph. One can observe that every pair of zero divisors in S generates a double hexagon in Γ
O
(S). The set of vertices of a double hexagon can be extended to a basis of S that has a convenient multiplication table. The set of vertices of an arbitrary connected component of Γ
O
(S) is described, and its diameter is found. Then, the bijection between the connected components of Γ
O
(S) and the lines in the imaginary part of the octonions is established. Finally, the commutativity graph of the sedenions is considered, and it is shown that all the elements whose imaginary part is a zero divisor belong to the same connected component, and its diameter lies in between 3 and 4. |
| doi_str_mv | 10.1007/s10958-021-05367-6 |
| format | Article |
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O
(S) denote its orthogonality graph. One can observe that every pair of zero divisors in S generates a double hexagon in Γ
O
(S). The set of vertices of a double hexagon can be extended to a basis of S that has a convenient multiplication table. The set of vertices of an arbitrary connected component of Γ
O
(S) is described, and its diameter is found. Then, the bijection between the connected components of Γ
O
(S) and the lines in the imaginary part of the octonions is established. Finally, the commutativity graph of the sedenions is considered, and it is shown that all the elements whose imaginary part is a zero divisor belong to the same connected component, and its diameter lies in between 3 and 4.</description><identifier>ISSN: 1072-3374</identifier><identifier>EISSN: 1573-8795</identifier><identifier>DOI: 10.1007/s10958-021-05367-6</identifier><language>eng</language><publisher>New York: Springer US</publisher><subject>Algebra ; Apexes ; Commutativity ; Mathematics ; Mathematics and Statistics ; Multiplication ; Orthogonality</subject><ispartof>Journal of mathematical sciences (New York, N.Y.), 2021-06, Vol.255 (3), p.254-270</ispartof><rights>Springer Science+Business Media, LLC, part of Springer Nature 2021</rights><rights>COPYRIGHT 2021 Springer</rights><rights>Springer Science+Business Media, LLC, part of Springer Nature 2021.</rights><lds50>peer_reviewed</lds50><woscitedreferencessubscribed>false</woscitedreferencessubscribed><citedby>FETCH-LOGICAL-c3736-d1e85f585896ea2a9d4e66ea4156994e8fa9e8084f7343def329369463fd29093</citedby><cites>FETCH-LOGICAL-c3736-d1e85f585896ea2a9d4e66ea4156994e8fa9e8084f7343def329369463fd29093</cites></display><links><openurl>$$Topenurl_article</openurl><openurlfulltext>$$Topenurlfull_article</openurlfulltext><thumbnail>$$Tsyndetics_thumb_exl</thumbnail><linktopdf>$$Uhttps://link.springer.com/content/pdf/10.1007/s10958-021-05367-6$$EPDF$$P50$$Gspringer$$H</linktopdf><linktohtml>$$Uhttps://link.springer.com/10.1007/s10958-021-05367-6$$EHTML$$P50$$Gspringer$$H</linktohtml><link.rule.ids>314,776,780,27901,27902,41464,42533,51294</link.rule.ids></links><search><creatorcontrib>Guterman, A. E.</creatorcontrib><creatorcontrib>Zhilina, S. A.</creatorcontrib><title>Relation Graphs of the Sedenion Algebra</title><title>Journal of mathematical sciences (New York, N.Y.)</title><addtitle>J Math Sci</addtitle><description>Let S denote the algebra of sedenions and let Γ
O
(S) denote its orthogonality graph. One can observe that every pair of zero divisors in S generates a double hexagon in Γ
O
(S). The set of vertices of a double hexagon can be extended to a basis of S that has a convenient multiplication table. The set of vertices of an arbitrary connected component of Γ
O
(S) is described, and its diameter is found. Then, the bijection between the connected components of Γ
O
(S) and the lines in the imaginary part of the octonions is established. Finally, the commutativity graph of the sedenions is considered, and it is shown that all the elements whose imaginary part is a zero divisor belong to the same connected component, and its diameter lies in between 3 and 4.</description><subject>Algebra</subject><subject>Apexes</subject><subject>Commutativity</subject><subject>Mathematics</subject><subject>Mathematics and Statistics</subject><subject>Multiplication</subject><subject>Orthogonality</subject><issn>1072-3374</issn><issn>1573-8795</issn><fulltext>true</fulltext><rsrctype>article</rsrctype><creationdate>2021</creationdate><recordtype>article</recordtype><recordid>eNp9kU1Lw0AQhoMoWKt_wFPAg3jYuh_Zr2MpWgsFodXzsiazaUqa1N0U9N-7NUIpFNnDDMPz7MC8SXJL8IhgLB8DwZorhClBmDMhkThLBoRLhpTU_Dz2WFLEmMwuk6sQ1jhKQrFBcr-A2nZV26RTb7erkLYu7VaQLqGAZj8e1yV8eHudXDhbB7j5q8Pk_fnpbfKC5q_T2WQ8RzmTTKCCgOKOK660AEutLjIQscsIF1pnoJzVoLDKnGQZK8AxqpnQmWCuoBprNkzu-n-3vv3cQejMut35Jq40lFNKKM4YOVClrcFUjWs7b_NNFXIzllQoJQRhkUInqBIa8LZuG3BVHB_xoxN8fAVsqvyk8HAkRKaDr660uxDMbLk4ZmnP5r4NwYMzW19trP82BJt9hqbP0MQMzW-GRkSJ9VKIcFOCP1zjH-sHd6CY7w</recordid><startdate>20210601</startdate><enddate>20210601</enddate><creator>Guterman, A. E.</creator><creator>Zhilina, S. A.</creator><general>Springer US</general><general>Springer</general><general>Springer Nature B.V</general><scope>AAYXX</scope><scope>CITATION</scope><scope>ISR</scope></search><sort><creationdate>20210601</creationdate><title>Relation Graphs of the Sedenion Algebra</title><author>Guterman, A. E. ; Zhilina, S. A.</author></sort><facets><frbrtype>5</frbrtype><frbrgroupid>cdi_FETCH-LOGICAL-c3736-d1e85f585896ea2a9d4e66ea4156994e8fa9e8084f7343def329369463fd29093</frbrgroupid><rsrctype>articles</rsrctype><prefilter>articles</prefilter><language>eng</language><creationdate>2021</creationdate><topic>Algebra</topic><topic>Apexes</topic><topic>Commutativity</topic><topic>Mathematics</topic><topic>Mathematics and Statistics</topic><topic>Multiplication</topic><topic>Orthogonality</topic><toplevel>peer_reviewed</toplevel><toplevel>online_resources</toplevel><creatorcontrib>Guterman, A. E.</creatorcontrib><creatorcontrib>Zhilina, S. A.</creatorcontrib><collection>CrossRef</collection><collection>Gale In Context: Science</collection><jtitle>Journal of mathematical sciences (New York, N.Y.)</jtitle></facets><delivery><delcategory>Remote Search Resource</delcategory><fulltext>fulltext</fulltext></delivery><addata><au>Guterman, A. E.</au><au>Zhilina, S. A.</au><format>journal</format><genre>article</genre><ristype>JOUR</ristype><atitle>Relation Graphs of the Sedenion Algebra</atitle><jtitle>Journal of mathematical sciences (New York, N.Y.)</jtitle><stitle>J Math Sci</stitle><date>2021-06-01</date><risdate>2021</risdate><volume>255</volume><issue>3</issue><spage>254</spage><epage>270</epage><pages>254-270</pages><issn>1072-3374</issn><eissn>1573-8795</eissn><abstract>Let S denote the algebra of sedenions and let Γ
O
(S) denote its orthogonality graph. One can observe that every pair of zero divisors in S generates a double hexagon in Γ
O
(S). The set of vertices of a double hexagon can be extended to a basis of S that has a convenient multiplication table. The set of vertices of an arbitrary connected component of Γ
O
(S) is described, and its diameter is found. Then, the bijection between the connected components of Γ
O
(S) and the lines in the imaginary part of the octonions is established. Finally, the commutativity graph of the sedenions is considered, and it is shown that all the elements whose imaginary part is a zero divisor belong to the same connected component, and its diameter lies in between 3 and 4.</abstract><cop>New York</cop><pub>Springer US</pub><doi>10.1007/s10958-021-05367-6</doi><tpages>17</tpages></addata></record> |
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| identifier | ISSN: 1072-3374 |
| ispartof | Journal of mathematical sciences (New York, N.Y.), 2021-06, Vol.255 (3), p.254-270 |
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| source | Springer Nature - Complete Springer Journals |
| subjects | Algebra Apexes Commutativity Mathematics Mathematics and Statistics Multiplication Orthogonality |
| title | Relation Graphs of the Sedenion Algebra |
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