On the structure of left and right F-, SM- and E-quasigroups
Abstract
Description
It is proved that any left F-quasigroup is isomorphic to the direct product of a left F-quasigroup with a unique idempotent element and isotope of a special form of a left distributive quasigroup. The similar theorems are proved for right F-quasigroups, left and right SM- and E-quasigroups. Information on simple quasigroups from these quasigroup classes is given, for example, finite simple F-quasigroup is a simple group or a simple medial quasigroup. It is proved that any left F-quasigroup is isotopic to the direct product of a group and a left S-loop. Some properties of loop isotopes of F-quasigroups (including M-loops) are pointed out. A left special loop is an isotope of a left F-quasigroup if and only if this loop is isomorphic the direct product of a group and a left S-loop (this is an answer to Belousov "1a", problem). Any left FESM-quasigroup is isotopic to the direct product of an abelian group and a left S-loop (this is an answer to Kinyon-Phillips 2.8(2) problem). New proofs of some known results on the structure of commutative Moufang loops are presented.
67 pages, Keywords: quasigroup, left F-quasigroup, F-quasigroup, left SM-quasigroup, SM-quasigroup, left S-loop, left M-loop, M-loop, left E-quasigroup, E-quasigroup, linear quasigroup, left special loop, special oop, commutative Moufang loop (CML), group isotope, Sushkevich postulate, Belousov problem, Kinyon-Phillips problems
67 pages, Keywords: quasigroup, left F-quasigroup, F-quasigroup, left SM-quasigroup, SM-quasigroup, left S-loop, left M-loop, M-loop, left E-quasigroup, E-quasigroup, linear quasigroup, left special loop, special oop, commutative Moufang loop (CML), group isotope, Sushkevich postulate, Belousov problem, Kinyon-Phillips problems