On some isostrophy invariants of Bol loops

Abstract

A loop (Q, •) which satisfies the identity x(yz \ x) = (x/z)(y \ x) is called a middle Bol loop. It is known that a loop (Q, •) is middle Bol if all its loop isotopes satisfy the anti-automorphic inverse property (xy)⁻¹ = y⁻¹x⁻¹. A. Gwaramija proved that middle Bol loops are isostrophs of left (right) Bol loops. Invariant properties under this isostrophy are studied in the present work. It is shown that two middle Bol loops are isotopic (isomorphic) if and only if the corresponding right (left) Bol loops are isotopic (isomorphic). So, the isotopy-isomorphism property is invariant. Also, it is proved that the group of autotopisms (left pseudo-automorphisms) of a middle Bol loop is isomorphic to the group of autotopisms (left (right) pseudo-automorphisms) of the corresponding right (left) Bol loop, the group of automorphisms is invariant and that every congruence (normal congruence, normal subloop) of a right Bol loop is a congruence (normal congruence, normal subloop) of the corresponding middle Bol loop.