On the structure of left and right F-, SM- and E-quasigroups

TL;DR

This paper characterizes the structure of various classes of quasigroups, showing they can be decomposed into products of groups and loops, and provides new insights into their properties and simple forms.

Contribution

It establishes structural theorems for left and right F-, SM-, and E-quasigroups, including their decomposition into groups and loops, and offers new proofs for properties of commutative Moufang loops.

Findings

Left F-quasigroups are isomorphic to a product of a group and a left S-loop.

Finite simple F-quasigroup is either a simple group or a simple medial quasigroup.

Any left FESM-quasigroup is isotopic to a product of an abelian group and a left S-loop.

Abstract

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.