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F-quasigroup; Moufang loop; generalized modules
In Kepka T., Kinyon M.K., Phillips J.D., {\it The structure of F-quasigroups\/}, {\tt math.GR/0510298}, we showed that every loop isotopic to an F-quasigroup is a Moufang loop. Here we characterize, via two simple identities, the class of F-quasigroups which are isotopic to groups. We call these quasigroups FG-quasigroups. We show that FG-quasigroups are linear over groups. We then use this fact to describe their structure. This gives us, for instance, a complete description of the simple FG-quasigroups. Finally, we show an equivalence of equational classes between pointed FG-quasigroups and central generalized modules over a particular ring.
[1] Belousov V.D., Florja I.A.: On left-distributive quasigroups. Bul. Akad. Štiince RSS Moldoven 1965 (1965), no. 7, 3–13. MR 0194541
[2] Bruck R.H.: A Survey of Binary Systems. Springer, 1971. MR 0093552 | Zbl 0141.01401
[3] Golovko I.A.: F-quasigroups with idempotent elements. Mat. Issled. 4 (1969), vyp. 2 (12), 137–143. MR 0274632 | Zbl 0235.20065
[4] Kepka T.: F-quasigroups isotopic to Moufang loops. Czechoslovak Math. J. 29(104) (1979), no. 1, 62–83. MR 0518141 | Zbl 0444.20067
[5] Kepka T., Kinyon M.K., Phillips J.D.: The structure of F-quasigroups. math.GR/0510298. Zbl 1133.20051
[6] Kepka T., Kinyon M.K., Phillips J.D.: F-quasigroups and generalized modules. math.GR/0512244.
[7] Murdoch D.C.: Quasi-groups which satisfy certain generalized associative laws. Amer. J. Math. 61 (1939), 509–522. DOI 10.2307/2371517 | MR 1507391 | Zbl 0020.34702
[8] Pflugfelder H.: Quasigroups and Loops: Introduction. Sigma Series in Pure Math. 8, Helderman, Berlin, 1990. MR 1125767 | Zbl 0715.20043
[9] Sabinina L.L.: On the theory of $F$-quasigroups. in Webs and Quasigroups, pp. 127–130, Kalinin. Gos. Univ., Kalinin, 1988. MR 0949717 | Zbl 0681.20044
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