New algebraic properties of middle Bol loops II

T G Jaiyéolá,O O Oyebola,S P David

doi:10.22199/issn.0717-6279-2021-01-0006

Abstract

A loop (Q, ·, \, /) is called a middle Bol loop (MBL) if it obeys the identity x(yz\x)=(x/z)(y\x). To every MBL corresponds a right Bol loop (RBL) and a left Bol loop (LBL). In this paper, some new algebraic properties of a middle Bol loop are established in a different style. Some new methods of constructing a MBL by using a non-abelian group, the holomorph of a right Bol loop and a ring are described. Some equivalent necessary and sufficient conditions for a right (left) Bol loop to be a middle Bol loop are established. A RBL (MBL, LBL, MBL) is shown to be a MBL (RBL, MBL, LBL) if and only if it is a Moufang loop.

Highlights

In 2012, Grecu and Syrbu [7] proved that two middle Bol loops are isotopic if and only if the corresponding right Bol loops are isotopic
Must Q be a middle Bol loop? Jaiyeo. laand David [11] initiated the preparation of the ground for different reformulation of Syrbu’s open problem based on the fact that the algebraic properties and structural properties of middle Bol loops have been studied in the past relative to their corresponding right Bol loop
We introduce the set of anti-autotopisms giving by AAU T (Q, ·) = {T 0 =< U 0, V 0, W 0 >∈ SY M (Q)3|xU 0·yV 0 = (y·x)W 0 ∀ x, y ∈ Q}

Summary

Introduction

Theorem 1.1 implies that if (Q, ·) is a left Bol loop and (Q, ◦) is the corresponding middle Bol loop x ◦ y = x/y−1 and x · y = x//y−1 , where ”/” (”//”) is the right division in (Q, ·) (respectively, in (Q, ◦)). In 2012, Grecu and Syrbu [7] proved that two middle Bol loops are isotopic if and only if the corresponding right (left) Bol loops are isotopic. Laand David [11] initiated the preparation of the ground for different reformulation of Syrbu’s open problem based on the fact that the algebraic properties and structural properties of middle Bol loops have been studied in the past relative to their corresponding right (left) Bol loop.

Algebraic Properties of a Middle Bol Loop

Construction of a Middle Bol Loop from a Right Bol Loop

Construction of a Middle Bol Loop from the Ring Z2