{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,3,11]],"date-time":"2026-03-11T15:12:51Z","timestamp":1773241971463,"version":"3.50.1"},"reference-count":14,"publisher":"American Mathematical Society (AMS)","issue":"5","license":[{"start":{"date-parts":[[2008,11,27]],"date-time":"2008-11-27T00:00:00Z","timestamp":1227744000000},"content-version":"am","delay-in-days":366,"URL":"https:\/\/www.ams.org\/publications\/copyright-and-permissions"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Trans. Amer. Math. Soc."],"abstract":"<p>\n                    A left Bol loop is a loop satisfying\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"x left-parenthesis y left-parenthesis x z right-parenthesis right-parenthesis equals left-parenthesis x left-parenthesis y x right-parenthesis right-parenthesis z\">\n                        <mml:semantics>\n                          <mml:mrow>\n                            <mml:mi>x<\/mml:mi>\n                            <mml:mo stretchy=\"false\">(<\/mml:mo>\n                            <mml:mi>y<\/mml:mi>\n                            <mml:mo stretchy=\"false\">(<\/mml:mo>\n                            <mml:mi>x<\/mml:mi>\n                            <mml:mi>z<\/mml:mi>\n                            <mml:mo stretchy=\"false\">)<\/mml:mo>\n                            <mml:mo stretchy=\"false\">)<\/mml:mo>\n                            <mml:mo>=<\/mml:mo>\n                            <mml:mo stretchy=\"false\">(<\/mml:mo>\n                            <mml:mi>x<\/mml:mi>\n                            <mml:mo stretchy=\"false\">(<\/mml:mo>\n                            <mml:mi>y<\/mml:mi>\n                            <mml:mi>x<\/mml:mi>\n                            <mml:mo stretchy=\"false\">)<\/mml:mo>\n                            <mml:mo stretchy=\"false\">)<\/mml:mo>\n                            <mml:mi>z<\/mml:mi>\n                          <\/mml:mrow>\n                          <mml:annotation encoding=\"application\/x-tex\">x(y(xz)) = (x(yx))z<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    . The commutant of a loop is the set of elements which commute with all elements of the loop. In a finite Bol loop of odd order or of order\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"2 k\">\n                        <mml:semantics>\n                          <mml:mrow>\n                            <mml:mn>2<\/mml:mn>\n                            <mml:mi>k<\/mml:mi>\n                          <\/mml:mrow>\n                          <mml:annotation encoding=\"application\/x-tex\">2k<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    ,\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"k\">\n                        <mml:semantics>\n                          <mml:mi>k<\/mml:mi>\n                          <mml:annotation encoding=\"application\/x-tex\">k<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    odd, the commutant is a subloop. We investigate conditions under which the commutant of a Bol loop is not a subloop. In a finite Bol loop of order relatively prime to\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"3\">\n                        <mml:semantics>\n                          <mml:mn>3<\/mml:mn>\n                          <mml:annotation encoding=\"application\/x-tex\">3<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    , the commutant generates an abelian group of order dividing the order of the loop. This generalizes a well-known result for Moufang loops. After describing all extensions of a loop\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"upper K\">\n                        <mml:semantics>\n                          <mml:mi>K<\/mml:mi>\n                          <mml:annotation encoding=\"application\/x-tex\">K<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    such that\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"upper K\">\n                        <mml:semantics>\n                          <mml:mi>K<\/mml:mi>\n                          <mml:annotation encoding=\"application\/x-tex\">K<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    is in the left and middle nuclei of the resulting loop, we show how to construct classes of Bol loops with a non-subloop commutant. In particular, we obtain all Bol loops of order\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"16\">\n                        <mml:semantics>\n                          <mml:mn>16<\/mml:mn>\n                          <mml:annotation encoding=\"application\/x-tex\">16<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    with a non-subloop commutant.\n                  <\/p>","DOI":"10.1090\/s0002-9947-07-04391-7","type":"journal-article","created":{"date-parts":[[2008,1,29]],"date-time":"2008-01-29T07:44:19Z","timestamp":1201592659000},"page":"2393-2408","source":"Crossref","is-referenced-by-count":10,"special_numbering":"876","title":["When is the commutant of a Bol loop a subloop?"],"prefix":"10.1090","volume":"360","author":[{"given":"Michael","family":"Kinyon","sequence":"first","affiliation":[]},{"given":"J.","family":"Phillips","sequence":"additional","affiliation":[]},{"given":"Petr","family":"Vojt\u011bchovsk\u00fd","sequence":"additional","affiliation":[]}],"member":"14","published-online":{"date-parts":[[2007,11,27]]},"reference":[{"key":"1","series-title":"Reihe: Gruppentheorie","doi-asserted-by":"crossref","DOI":"10.1007\/978-3-662-35338-7","volume-title":"A survey of binary systems","author":"Bruck, Richard Hubert","year":"1958"},{"issue":"3","key":"2","doi-asserted-by":"publisher","first-page":"377","DOI":"10.1017\/S0305004100055213","article-title":"Finite Bol loops","volume":"84","author":"Burn, R. P.","year":"1978","journal-title":"Math. Proc. Cambridge Philos. Soc.","ISSN":"https:\/\/id.crossref.org\/issn\/0305-0041","issn-type":"print"},{"issue":"3","key":"3","doi-asserted-by":"publisher","first-page":"445","DOI":"10.1017\/S0305004100058357","article-title":"Finite Bol loops. II","volume":"89","author":"Burn, R. P.","year":"1981","journal-title":"Math. Proc. Cambridge Philos. Soc.","ISSN":"https:\/\/id.crossref.org\/issn\/0305-0041","issn-type":"print"},{"issue":"3-4","key":"4","first-page":"573","article-title":"Semi-direct products and Bol loop","volume":"27","author":"Goodaire, Edgar G.","year":"1994","journal-title":"Demonstratio Math.","ISSN":"https:\/\/id.crossref.org\/issn\/0420-1213","issn-type":"print"},{"key":"5","series-title":"Lecture Notes in Mathematics","isbn-type":"print","doi-asserted-by":"publisher","DOI":"10.1007\/b83276","volume-title":"Theory of $K$-loops","volume":"1778","author":"Kiechle, Hubert","year":"2002","ISBN":"https:\/\/id.crossref.org\/isbn\/3540432620"},{"key":"6","doi-asserted-by":"publisher","first-page":"235","DOI":"10.1007\/BF02941674","article-title":"On the extension of involutorial Bol loops","volume":"72","author":"Kiechle, H.","year":"2002","journal-title":"Abh. Math. Sem. Univ. Hamburg","ISSN":"https:\/\/id.crossref.org\/issn\/0025-5858","issn-type":"print"},{"issue":"3","key":"7","doi-asserted-by":"publisher","first-page":"617","DOI":"10.1090\/S0002-9939-03-07211-3","article-title":"Commutants of Bol loops of odd order","volume":"132","author":"Kinyon, Michael K.","year":"2004","journal-title":"Proc. Amer. Math. Soc.","ISSN":"https:\/\/id.crossref.org\/issn\/0002-9939","issn-type":"print"},{"key":"8","unstructured":"G. P. Nagy and P. Vojt\u011bchovsk\u00fd, LOOPS: Computing with quasigroups and loops in GAP, version 1.0.0, computational package for GAP; http:\/\/www.math.du.edu\/loops"},{"key":"9","doi-asserted-by":"crossref","unstructured":"W. W. McCune, OTTER 3.3 Reference Manual and Guide, Argonne National Laboratory Technical Memorandum ANL\/MCS-TM-263, 2003; http:\/\/www.mcs.anl.gov\/AR\/otter\/","DOI":"10.2172\/822573"},{"key":"10","unstructured":"W. W. McCune, Prover9, automated reasoning software, Argonne National Laboratory, 2005; http:\/\/www.mcs.anl.gov\/AR\/prover9\/"},{"key":"11","doi-asserted-by":"crossref","unstructured":"W. W. McCune, Mace 4.0 Reference Manual and Guide, Argonne National Laboratory Technical Memorandum ANL\/MCS-TM-264, 2003; http:\/\/www.mcs.anl.gov\/AR\/mace4\/","DOI":"10.2172\/822574"},{"key":"12","unstructured":"G. Eric Moorhouse, Bol Loops of Small Order; http:\/\/www.uwyo.edu\/moorhouse\/pub\/bol\/ index.html"},{"key":"13","series-title":"Sigma Series in Pure Mathematics","isbn-type":"print","volume-title":"Quasigroups and loops: introduction","volume":"7","author":"Pflugfelder, Hala O.","year":"1990","ISBN":"https:\/\/id.crossref.org\/isbn\/3885380072"},{"key":"14","doi-asserted-by":"publisher","first-page":"341","DOI":"10.2307\/1994661","article-title":"Bol loops","volume":"123","author":"Robinson, D. A.","year":"1966","journal-title":"Trans. Amer. Math. Soc.","ISSN":"https:\/\/id.crossref.org\/issn\/0002-9947","issn-type":"print"}],"container-title":["Transactions of the American Mathematical Society"],"original-title":[],"language":"en","link":[{"URL":"http:\/\/www.ams.org\/tran\/2008-360-05\/S0002-9947-07-04391-7\/S0002-9947-07-04391-7.pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"},{"URL":"https:\/\/www.ams.org\/tran\/2008-360-05\/S0002-9947-07-04391-7\/S0002-9947-07-04391-7.pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2026,3,2]],"date-time":"2026-03-02T17:19:18Z","timestamp":1772471958000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.ams.org\/tran\/2008-360-05\/S0002-9947-07-04391-7\/"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2007,11,27]]},"references-count":14,"journal-issue":{"issue":"5","published-print":{"date-parts":[[2008,5]]}},"alternative-id":["S0002-9947-07-04391-7"],"URL":"https:\/\/doi.org\/10.1090\/s0002-9947-07-04391-7","archive":["CLOCKSS","Portico"],"relation":{},"ISSN":["1088-6850","0002-9947"],"issn-type":[{"value":"1088-6850","type":"electronic"},{"value":"0002-9947","type":"print"}],"subject":[],"published":{"date-parts":[[2007,11,27]]}}}