Holomorphic Structure of Middle Bol Loops

TL;DR

This paper investigates the holomorphic structure of middle Bol loops, establishing conditions under which their holomorphs are commutative, flexible, and invariant under isostrophies, with a focus on automorphism groups.

Contribution

It characterizes the holomorphs of middle Bol loops, linking their properties to automorphism groups and isostrophies, and identifies conditions for holomorphic invariance.

Findings

Holomorph of a commutative middle Bol loop is commutative if automorphism group is abelian.

Holomorphic invariance under isostrophy requires commutativity or flexibility.

Holomorph equality conditions depend on abelian automorphism groups and subgroup relations.

Abstract

A loop $(Q,\cdot,\backslash,/)$ is called a middle Bol loop if it obeys the identity $x(yz\backslash x)=(x/z)(y\backslash x)$. To every right (left) Bol loop corresponds a middle Bol loop via an isostrophism. In this paper, the structure of the holomorph of a middle Bol loop is explored. For some special types of automorphisms, the holomorph of a commutative loop is shown to be a commutative middle Bol loop if and only if the loop is a middle Bol loop and its automorphism group is abelian and a subgroup of both the group of middle regular mappings and the right multiplication group. It was found that commutativity (flexibility) is a necessary and sufficient condition for holomorphic invariance under the existing isostrophy between middle Bol loops and the corresponding right (left) Bol loops. The right combined holomorph of a middle Bol loop and its corresponding right (left) Bol loop was shown to be equal to the holomorph of the middle Bol loop if and only if the automorphism group is abelian and a subgroup of the multiplication group of the middle Bol loop. The obedience of an identity dependent on automorphisms was found to be a necessary and sufficient condition the left combined holomorph of a middle Bol loop and its corresponding left Bol loop to be equal to the holomorph of the middle Bol loop.