Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: New Algebraic Properties of Middle Bol Loops   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.

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The categories of this paper are: nucl-ex
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The categories of this paper are: ao-sci
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The categories of this paper are: acc-phys
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The categories of this paper are: gr-qc
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The categories of this paper are: nucl-th
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The categories of this paper are: chao-dyn
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The categories of this paper are: atom-ph
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The categories of this paper are: econ
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The categories of this paper are: bayes-an
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The categories of this paper are: hep-ph
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The categories of this paper are: dg-ga
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The categories of this paper are: stat
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The categories of this paper are: supr-con
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The categories of this paper are: solv-int
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The categories of this paper are: cond-mat
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The categories of this paper are: eess