New Algebraic Properties of Middle Bol 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.