mixed group
MixedGroup
2009-01-06 21:43:58
2009-01-07 00:42:49
Definition
kernel
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A \emph{mixed group} is a partial groupoid $G$ such that $G$ contains a non-empty subset $K$, called the \emph{kernel} of $G$, with the following conditions:
\begin{enumerate}
\item if $a,b\in G$, then $ab$ is defined iff $a\in K$,
\item if $a,b\in K$ and $c\in G$, then $(ab)c=a(bc)$,
\item if $a\in K$, then $K\subseteq aK\cap Ka$,
\item if $a\in K$ and $b\in G$ such that $ab=b$, then $ac=c$ for all $c\in G$.
\end{enumerate}
Mixed groups are generalizations of groups, as the following proposition illustrates:
\begin{prop} If $K=G$, then $G$ is a group. \end{prop}
\begin{proof}
$G$ is a groupoid by condition 1, and a semigroup by condition 2.
Now, by condition 3, given $a\in G$, there is $b\in G$ such that $ba=a$, so that $bc=c$ for all $c\in G$ by condition 4. In other words, $b$ is a left identity of $G$. Again, by condition 3, for every $a\in G$, there is a $d\in G$ such that $b=da$. So $ad= a(bd)=a(da)d=(ad)^2$, so, by condition 4, $adx=x$ for all $x\in G$. In particular, set $x=a$, we get $a=(ad)a=a(da)=ab$. Hence, $b$ is a two-sided identity, and $G$ is a monoid.
Finally, by condition 3, for every $a\in G$, there are $c,d\in G$, such that $b=ac=da$. So, $c=bc=(da)c=d(ac)=db=d$, showing that $a$ has a two-sided inverse. This means that $G$ is a group.
\end{proof}
For a non-trivial example of a mixed group, let $G$ be a group and $H$ a subgroup of $G$. Define a new multiplication $\cdot$ on $G$ as follows: $a\cdot b$ is defined iff $a\in H$, and if $a\cdot b$ is defined, it is defined as $ab$, the group multiplication of $a$ and $b$. Then $(G,\cdot)$ is a mixed group. Clearly, associativity of $\cdot$ is automatically satisfied. Next, pick any $a\in H$, then, for any $b\in H$, $a^{-1}\cdot b$ and $b\cdot a^{-1}$ are both elements of $H$, so that $b\in a\cdot H\cap H\cdot a$, and condition 3 is also satisfied. Finally, if $a\in H$ and $b\in G$ such that $a\cdot b=b$, then $a$ is the multiplicative identity of $G$, clearly $a\cdot c=c$ for all $c\in G$.
\begin{thebibliography}{0}
\bibitem{HB}
R. H. Bruck,
{\it A Survey of Binary Systems}, Springer-Verlag, 1966
\bibitem{HP}
R. Baer,
{\it Zur Einordnung der Theorie der Mischgruppen in die Gruppentheorie}, S.-B. Heidelberg. Akad. Wiss., Math.-naturwiss. KI. 1928, 4, 13 pp
\bibitem{HP}
R. Baer,
{\it \"{U}ber die Zerlegungen einer Mischgruppe nach einer Untermischgruppe}, S.-B. Heidelberg. Akad. Wiss., Math.-naturwiss. KI. 1928, 5, 13 pp
\bibitem{HP}
A. Loewy,
{\it \"{U}ber abstrakt definierte Transmutationssysteme oder Mischgruppen}, J. reine angew. Math. 157, pp 239-254, 1927
\end{thebibliography}