New Algebraic Properties of Middle Bol Loops

TL;DR

This paper explores new algebraic properties of middle Bol loops, introducing mappings and conditions that characterize various loop properties and classifications, including group, Moufang, and extra loops.

Contribution

It establishes novel algebraic conditions and mappings that characterize when middle Bol loops possess properties like elasticity, RIP, LIP, and are equivalent to known loop classes.

Findings

Derived necessary and sufficient conditions for properties like elasticity, RIP, LIP, RAP, and LAP.

Identified conditions under which middle Bol loops are groups, Moufang loops, or extra loops.

Connected middle Bol loops to RIF-loop, WRIF-loop, and WIP power associative conjugacy closed loops.

Abstract

A loop $(Q,\cdot,\backslash,/)$ is called a middle Bol loop if it obeys the identity $x(yz\backslash x)=(x/z)(y\backslash x)$. In this paper, some new algebraic properties of a middle Bol loop are established. Four bi-variate mappings $f_i,g_i,~i=1,2$ and four $j$-variate mappings $\alpha_j,\beta_j,\phi_j,\psi_j,~j\in\mathbb{N}$ are introduced and some interesting properties of the former are found. Neccessary and sufficient conditons in terms of $f_i,g_i,~i=1,2$, for a middle Bol loop to have the elasticity property, RIP, LIP, right alternative property (RAP) and left alternative property (LAP) are establsihed. Also, neccessary and sufficient conditons in terms of $\alpha_j,\beta_j,\phi_j,\psi_j,~j\in\mathbb{N}$, for a middle Bol loop to have power RAP and power LAP are establsihed. Neccessary and sufficient conditons in terms of $f_i,g_i,~i=1,2$ and $\alpha_j,\beta_j,\phi_j,\psi_j,~j\in\mathbb{N}$, for a middle Bol loop to be a group, Moufang loop or extra loop are established. A middle Bol loop is shown to belong to some classes of loops whose identiites are of the J.D. Phillips' RIF-loop and WRIF-loop (generalizations of Moufang and Steiner loops) and WIP power associative conjugacy closed loop types if and only if some identities defined by $g_1$ and $g_2$ are obeyed.