\Chapter{Mathematical background}

We assume that you are familiar with the theory of quasigroups and loops, for
instance with the textbook of Bruck \cite{Br} or Pflugfelder \cite{Pf}.
Nevertheless, we did include definitions and results in this manual in order to
unify the terminology and improve the intelligibility of the text. Some general
concepts of quasigroups and loops can be found in this chapter. More special
concepts are defined throughout the text as needed.

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\Section{Quasigroups and loops}

A set with one binary operation (denoted $\cdot$ here) is called
<groupoid>
%
\index{groupoid}
%
 or <magma>
%
\index{magma}
%
, the latter name being used in {\GAP}.
Associative groupoid is known as <semigroup>
%
\index{semigroup}
%
.

An element $1$ of a groupoid $G$ is a <neutral element>
%
\index{neutral element}
%
or an
<identity element>
%
\index{identity element}
%
 if $1\cdot x = x\cdot 1 = x$ for every $x$ in $G$.
Semigroup with a neutral element is a <monoid>
%
\index{monoid}
%
.

Let $G$ be a groupoid with neutral element $1$. Then an element $y$ is called a
<two-sided inverse>
%
\index{two-sided inverse}
%
 of $x$ in $G$ if $x\cdot y = y\cdot x = 1$. A monoid
in which every element has a two-sided inverse is called a <group>
%
\index{group}
%
.

Groups can be reached in another way from groupoids, namely through quasigroups
and loops.

A <quasigroup>
%
\index{quasigroup}
%
 $Q$ is a groupoid such that the equation $x\cdot y=z$ has a
unique solution in $Q$ whenever two of the three elements $x$, $y$, $z$ of $Q$
are specified. Note that multiplication tables of finite quasigroups are
precisely <Latin squares>\index{Latin square}, i.e., a square arrays with
symbols arranged so that each symbol occurs in each row and in each column
exactly once. A <loop>
%
\index{loop}
%
 $L$ is a quasigroup with a neutral element.

Groups are clearly loops, and one can show easily that an associative
quasigroup is a group. Hence the theory of quasigroups and loops is in a sense
complementary to the theory of semigroups and monoids.

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\Section{Translations}

Given an element $x$ of a quasigroup $Q$ we can associative two permutations of
$Q$ with it: the <left translation>
%
\index{left translation}
%
 $L_x:Q\to Q$ defined by $y\mapsto
x\cdot y$, and the <right translation>
%
\index{right translation}
%
 $R_x:Q\to Q$ defined by $y\mapsto
y\cdot x$.

Although it is possible to compose two right (left) translations, the resulting
permutation is not necessarily a right (left) translation. The set $\{L_x;x\in
Q\}$ is called the <left section>
%
\index{left section}
%
 of $Q$, and $\{R_x;x\in Q\}$ is the
<right section>
%
\index{right section}
%
 of $Q$.

Let $S_Q$ be the symmetric group on $Q$. Then the subgroup ${\rm{LMlt}}(Q)=\langle
L_x|x\in Q\rangle$ of $S_Q$ generated by all left translations is the
<left multiplication group>
%
\index{left multiplication group}
%
 of $Q$. Similarly, ${\rm{RMlt}}(Q)= \langle R_x|x\in
Q\rangle$ is the <right multiplication group>
%
\index{right multiplication group}
%
 of $Q$. The smallest group containing both ${\rm{LMlt}}(Q)$ and
${\rm{RMlt}}(Q)$ is called the <multiplication group>
%
\index{multiplication group}
%
 of $Q$ and is denoted by ${\rm{Mlt}}(Q)$.

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\Section{Homomorphisms and homotopisms}

Let $K$, $H$ be two quasigroups. Then a map $f:K\to H$ is a <homomorphism>
%
\index{homomorphism}
%
 if $f(x)\cdot f(y)=f(x\cdot y)$ for every $x$, $y\in K$. If
$f$ is also a bijection, we speak of an <isomorphism>
%
\index{isomorphism}
%
, and the two quasigroups are called <isomorphic>.

The ordered triple $(\alpha,\beta,\gamma)$ of maps $\alpha$, $\beta$,
$\gamma:K\to H$ is a <homotopism>
%
\index{homotopism}
%
 if $\alpha(x)\cdot\beta(y) =
\gamma(x\cdot y)$ for every $x$, $y\in K$. If the three maps are bijections,
$(\alpha,\beta,\gamma)$ is an <isotopism>
%
\index{isotopism}
%
, and the two quasigroups are
<isotopic>.

Isotopic groups are necessarily isomorphic, but this is certainly not true for
nonassociative quasigroups or loops. In fact, every quasigroup is isotopic to a
loop, as we shall see.

Let $(K,\cdot)$, $(K,\circ)$ be two quasigroups defined on the same set $K$.
Then an isotopism $(\alpha,\beta,{\rm{id}}_K)$ is called a <principal isotopism>
%
\index{principal isotopism}
%
. An important class of principal isotopisms is obtained as follows:

Let $(K,\cdot)$ be a quasigroup, and let $f$, $g$ be elements of $K$. Define a
new operation $\circ$ on $K$ by
$$
    x\circ y = R_g^{-1}(x)\cdot L_f^{-1}(y),
$$
where $R_g$, $L_f$ are translations. Then $(K,\circ)$ is a quasigroup isotopic
to $(K,\cdot)$, in fact a loop with neutral element $f\cdot g$. We call
$(K,\circ)$ a <principal loop isotope>
%
\index{principal loop isotope}
%
 of $(K,\cdot)$.

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\Section{Extensions}

Let $K$, $F$ be loops. Then a loop $Q$ is an <extension>
%
\index{extension of loops}
%
of $K$ by $F$ if $K$ is a normal subloop of $Q$ such that $Q/K$ is isomorphic
to $F$.

An extension $Q$ of $K$ by $F$ is <nuclear>
%
\index{nuclear extension}
%
if $K$ is an abelian group and $K\le N(Q)$. A map $\theta:F\times F\to K$ is a
<cocycle>
%
\index{cocycle}
%
if $\theta(1,x) = \theta(x,1) = 1$ for every $x\in F$.

The following theorem holds for loops $Q$, $F$ and an abelian group $K$: $Q$ is
a nuclear extension of $K$ by $F$ if and only if there is a cocycle
$\theta:F\times F\to K$ and a homomorphism $\varphi:F\to{\rm{Aut}}{Q}$ such
that $K\times F$ with multiplication $(a,x)(b,y) =
(a\varphi_x(b)\theta(x,y),xy)$ is isomorphic to $Q$.
