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A mixed group is a partial groupoid $G$ such that $G$ contains a non-empty subset $K$ , called the kernel of $G$ , with the following conditions:
- if $a,b\in G$ , then $ab$ is defined iff $a\in K$ ,
- if $a,b\in K$ and $c\in G$ , then $(ab)c=a(bc)$ ,
- if $a\in K$ , then $K\subseteq aK\cap Ka$ ,
- if $a\in K$ and $b\in G$ such that $ab=b$ , then $ac=c$ for all $c\in G$ .
Mixed groups are generalizations of groups, as the following proposition illustrates:
Proposition 1 If $K=G$ , then $G$ is a group.
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. 
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 $c\in G$ .
- 1
- R. H. Bruck, A Survey of Binary Systems, Springer-Verlag, 1966
- 2
- R. Baer, Zur Einordnung der Theorie der Mischgruppen in die Gruppentheorie, S.-B. Heidelberg. Akad. Wiss., Math.-naturwiss. KI. 1928, 4, 13 pp
- 2
- R. Baer, Über die Zerlegungen einer Mischgruppe nach einer Untermischgruppe, S.-B. Heidelberg. Akad. Wiss., Math.-naturwiss. KI. 1928, 5, 13 pp
- 2
- A. Loewy, Über abstrakt definierte Transmutationssysteme oder Mischgruppen, J. reine angew. Math. 157, pp 239-254, 1927
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