  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.

Group Theory