Holomorph of generalized Bol loops II

TL;DR

This paper characterizes the holomorph of generalized Bol loops, establishing conditions under which the holomorph of a right inverse property loop becomes a GBL or a Bol loop, with algebraic properties explored.

Contribution

It provides new necessary and sufficient conditions for the holomorph of a RIPL to be a GBL or a Bol loop, expanding understanding of loop holomorphs.

Findings

Holomorph of RIPL is GBL iff the loop is GBL and certain bijections are regular.

Holomorph of RIPL is a Bol loop under specific algebraic conditions.

Algebraic properties and commutative diagrams are established for RIPL with GBL holomorph.

Abstract

The notion of the holomorph of a generalized Bol loop (GBL) is characterized afresh. The holomorph of a right inverse property loop (RIPL) is shown to be a GBL if and only if the loop is a GBL and some bijections of the loop are right (middle) regular. The holomorph of a RIPL is shown to be a GBL if and only if the loop is a GBL and some elements of the loop are right (middle) nuclear. Necessary and sufficient condition for the holomorph of a RIPL to be a Bol loop are deduced. Some algebraic properties and commutative diagrams are established for a RIPL whose holomorph is a GBL.