Holomorph of generalized Bol loops

TL;DR

This paper characterizes the holomorphs of generalized Bol and flexible-Bol loops, establishing equivalences and describing automorphisms, thereby advancing the structural understanding of these algebraic loops.

Contribution

It provides a characterization of the holomorphs of generalized Bol and flexible-Bol loops and links automorphisms to their structure, offering new insights into their algebraic properties.

Findings

Holomorphs of generalized Bol loops are characterized.

Equivalence between a loop and its holomorph being generalized Bol.

Automorphisms are used to construct holomorphs of these loops.

Abstract

The notion of the holomorph of a generalized Bol loop and generalized flexible-Bol loop are characterized. With the aid of two self-mappings on the holomorph of a loop, it is shown that: the loop is a generalized Bol loop if and only if its holomorph is a generalized Bol loop; the loop is a generalized flexible-Bol loop if and only if its holomorph is a generalized flexible-Bol loop. Furthermore, elements of the Bryant Schneider group of a generalized Bol loop are characterized in terms of pseudo-automorphism, and the automorphisms gotten are used to build the holomorph of the generalized Bol loop.