mixed group
A mixed group is a partial groupoid $G$ such that $G$ contains a nonempty subset $K$, called the kernel of $G$, with the following conditions:

1.
if $a,b\in G$, then $ab$ is defined iff $a\in K$,

2.
if $a,b\in K$ and $c\in G$, then $(ab)c=a(bc)$,

3.
if $a\in K$, then $K\subseteq aK\cap Ka$,

4.
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 $K\mathrm{=}G$, then $G$ is a group.
Proof.
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 twosided 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 twosided inverse^{}. This means that $G$ is a group. ∎
For a nontrivial 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 $c\in G$.
References
 1 R. H. Bruck, A Survey of Binary Systems, SpringerVerlag, 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
 3 R. Baer, Über die Zerlegungen einer Mischgruppe nach einer Untermischgruppe, S.B. Heidelberg. Akad. Wiss., Math.naturwiss. KI. 1928, 5, 13 pp
 4 A. Loewy, Über abstrakt definierte Transmutationssysteme oder Mischgruppen, J. reine angew. Math. 157, pp 239254, 1927
Title  mixed group 

Canonical name  MixedGroup 
Date of creation  20130322 18:42:32 
Last modified on  20130322 18:42:32 
Owner  CWoo (3771) 
Last modified by  CWoo (3771) 
Numerical id  7 
Author  CWoo (3771) 
Entry type  Definition 
Classification  msc 20N99 
Defines  kernel 