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Every diassociative A-loop is Moufang

Authors: Michael K. Kinyon, Kenneth Kunen and J. D. Phillips
Journal: Proc. Amer. Math. Soc. 130 (2002), 619-624
MSC (2000): Primary 20N05; Secondary 68T15
Published electronically: June 19, 2001
MathSciNet review: 1866009
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Abstract | References | Similar Articles | Additional Information


An A-loop is a loop in which every inner mapping is an automorphism. A problem which had been open since 1956 is settled by showing that every diassociative A-loop is Moufang.

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  • 1. R.H. Bruck, A Survey of Binary Systems, Springer-Verlag, 1958; third printing, 1971. MR 20:76
  • 2. R.H. Bruck and L.J. Paige, Loops whose inner mappings are automorphisms, Ann. of Math. (2) 63 (1956) 308-323. MR 17:943b
  • 3. O. Chein, H.O. Pflugfelder, and J.D.H. Smith (eds.), Quasigroups and Loops: Theory and Applications, Sigma Series in Pure Math. 8, Heldermann Verlag, Berlin, 1990. MR 93g:20133
  • 4. A. Drapal, A-loops close to code loops are groups, Comm. Math. Univ. Carolin. 41 (2000), no. 2, 245-249. CMP 2001:01
  • 5. T.S.R. Fuad, J.D. Phillips, and X.R. Shen, On diassociative A-loops, submitted.
  • 6. J. Hart and K. Kunen, Single axioms for odd exponent groups, J. Automated Reasoning 14 (1995) 383-412. MR 96h:68178
  • 7. K. Kunen, Moufang quasigroups, J. Algebra 183 (1996) 231-234. MR 97f:20096
  • 8. K. Kunen, Quasigroups, loops, and associative laws, J. Algebra 185 (1996) 194-204. MR 97g:20083
  • 9. K. Kunen, Alternative loop rings, Communications in Algebra 26 (1998) 557-564. MR 99a:17032
  • 10. K. Kunen, G-loops and permutation groups, J. Algebra 220 (1999) 694-708. MR 2000j:20133
  • 11. K. Kunen, The structure of conjugacy closed loops, Transactions Amer. Math. Soc. 352 (2000) 2889-2911. MR 2000j:20132
  • 12. W.W. McCune, OTTER 3.0 Reference Manual and Guide, Technical Report ANL-94/6, Argonne National Laboratory, 1994; or see:
  • 13. W. McCune and R. Padmanabhan, Automated Deduction in Equational Logic and Cubic Curves, Lecture Notes in Comp. Sci. #1095, Springer, Berlin, 1996. MR 98m:68238
  • 14. R. Moufang, Zur Struktur von Alternativkörpern, Math. Ann. 110 (1934) 416-430.
  • 15. H. Orlik-Pflugfelder, A special class of Moufang loops, Proc. Amer. Math. Soc. 26 (1970) 583-586. MR 42:407
  • 16. J.M. Osborn, A theorem on A-loops, Proc. Amer. Math. Soc. 9 (1958) 347-349. MR 20:79
  • 17. H.O. Pflugfelder, Quasigroups and Loops: Introduction, Sigma Series in Pure Math. 7, Heldermann Verlag, Berlin, 1990. MR 93g:20133
  • 18. J.D. Phillips, On Moufang A-loops, Comm. Math. Univ. Carolin. 41 (2000), no. 2, 371-375. CMP 2001:01
  • 19. L. Wos and G. W. Pieper, A Fascinating Country in the World of Computing -- Your Guide to Automated Reasoning, World Scientific, 1999.

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Additional Information

Michael K. Kinyon
Affiliation: Department of Mathematics & Computer Science, Indiana University, South Bend, Indiana 46634
Address at time of publication: Department of Mathematics, Western Michigan University, Kalamazoo, Michigan 49008-5248

Kenneth Kunen
Affiliation: Department of Mathematics, University of Wisconsin, Madison, Wisconsin 57306

J. D. Phillips
Affiliation: Department of Mathematics, Saint Mary’s College of California, Moraga, California 94575
Address at time of publication: Department of Mathematics and Computer Science, Wabash College, Crawfordsville, Indiana 47933

Keywords: Diassociative loop, A-loop, Moufang loop
Received by editor(s): August 3, 2000
Received by editor(s) in revised form: August 18, 2000
Published electronically: June 19, 2001
Additional Notes: The second author was partially supported by NSF Grant DMS-9704520.
Communicated by: Lance W. Small
Article copyright: © Copyright 2001 American Mathematical Society

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