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Brandt groupoids, like category theoretic groupoids, are generalizations of groups, where a multiplication is defined, and inverses with respect to the multiplication exist for each element. However, unlike elements of a group, each element in a Brandt groupoid behaves like an arrow, with a source and target, and
multiplication of two elements only work when the target of the first element coincides with the source of the second element.
A Brandt groupoid is a non-empty set $B$ , together with a partial binary operation (called a multiplication) $\cdot$ defined on it (we write $ab$ for $a\cdot b$ ), such that
- For every $a\in B$ , there are unique elements $e,f$ such that $ea$ and $af$ are defined, and is equal to $a$ .
- If $ae=a$ or $ea=a$ for some $a,e\in B$ , then $ee$ is defined, and is equal to $e$ .
- For $a,b\in B$ , $ab$ is defined iff there is an $e\in B$ such that $ae=a$ and $eb=b$ .
- For $a,b,c\in B$ such that $ab$ and $bc$ are defined, then so are $(ab)c$ and $a(bc)$ and they equal.
- If $ea=af=e$ for some $a,e,f\in B$ , then there is a $b\in B$ such that $ab$ and $ba$ are defined and $ab=e$ and $ba=f$ .
- If $ee=e$ and $ff=f$ for some $e,f\in B$ , then there is $a\in B$ such that $ea$ and $af$ are defined and are equal to $a$ .
In the definition above, we see several instances of elements $e$ such that $e^2=ee=e$ . Such elements are called idempotents. If we let $I$ be the set of all idempotents of $B$ , then $I\ne \varnothing$ by conditions 1 and 2.
Brandt groupoids are intimately related to categories, as we will presently discuss.
The first two conditions above imply that there are two surjective functions $s,t:B\to I$ , where $t(a)$ and $s(a)$ are the unique idempotents such that $a s(a)=a$ and $t(a) a=a$ . In addition, $s(e)=t(e)=e$ for all $e\in I$ . Call $s$ the source function, $t$ the target function, and for any $a\in B$ , $s(a),t(a)$ the source and the target of $a$ .
The third condition says that $ab$ is defined iff the source of $a$ is the equal to the target of $b$ : $s(a)=t(b)$ . The fourth condition is the associativity law for the multiplication. An easy consequence of this condition is that if $ab$ exists, then $s(b)=s(ab)$ and $t(a)=t(ab)$ .
Altogether, the first four conditions say that a $B$ is a small category, with $I$ its set of objects, and $G$ the set of morphisms, and composition of morphisms is just the multiplication.
A morphism $a$ in $B$ is said to be an isomorphism if there is a morphism $b$ in $G$ such that $ab,ba \in I$ . Now, $b$ is uniquely determined by $a$ , so that $a$ is an isomorphism in the category theoretic sense.
Proof. First notice that $s(b)=s(ab)=ab=t(ab)=t(a)$ and $t(b)=t(ba)=ba=s(ba)=s(a)$ . If $ac, ca \in I$ , then $s(c)=t(a)=s(b)$ and $t(c)=s(a)=t(b)$ . So $ab=ac$ and $ba=bc$ . As a result, $c=t(c)c = t(b)c= (ba)c=b(ac)= b(ab)= b s(b)=b$ . 
$b$ is said to be the inverse of $a$ , and is often written $a^{-1}$ . Condition 5 says that the category $B$ is in fact a category theoretic groupoid. Thus, a Brandt groupoid is a group if the multiplication is everywhere defined.
Finally, condition 6 says that between every pair of objects, there is a morphism from one to the other, this is equivalent to saying that $B$ is strongly connected. As a result, a Brandt groupoid may be equivalently defined as a small strongly connected groupoid (in the category theoretic sense).
A Brandt groupoid may be constructed as follows: take a group $G$ and a non-empty set $I$ , set $B:=I\times G\times I$ , and define multiplication on $B$ as follows:
Then $B$ with the partial multiplication is a Brandt groupoid. The idempotents in $B$ have the form $(p,e,p)$ , where $e\in G$ is the group identity. And for any $(p,x,q)$ , its source, target, and inverse are $(q,e,q)$ and $(p,e,p)$ , $(q,x^{-1},p)$ respectively.
In fact, it may be shown that every Brandt groupoid is isomorphic to one constructed above (for a proof, see here).
Remark. A non-trivial Brandt groupoid can not have a zero element, for if $0a=a0=0$ for all $a\in B$ , then $a$ must be the source and target of $0$ , but then $a$ would have to be unique by condition 1, which is impossible unless $B$ is trivial. If we adjoin $0$ to a Brandt groupoid $B$ , and call $S:=B\cup \lbrace 0\rbrace$ , then $S$ has the structure of a semigroup. Here's how the multiplication is
defined on $S$ :
Since the multiplication on $S$ is everywhere defined, $S$ is a groupoid. To see that $S$ is a semigroup, we must show that associativity of the multiplication applies everywhere. There are four cases
- If both $ab$ and $bc$ are defined in $B$ , they are certainly defined in $S$ , and the associativity follows from condition 4.
- If neither $ab$ nor $bc$ is defined in $B$ , then $(ab)c=0c=0=a0=a(bc)$ in $S$ .
- If $ab$ is not defined in $B$ , but $bc$ is, then $s(a)\ne t(b)=t(bc)$ , and $(ab)c=0c=0=a(bc)$ .
- Similarly, if $ab$ is defined in $B$ but not $bc$ , then $(ab)c=0=a(bc)$ .
Thus, $S$ is a semigroup (with $0$ ). In fact, Clifford showed that $S$ is completely simple.
- 1
- H. Brandt, Uber die Axiome des Gruppoids, Vierteljschr. naturforsch. Ges. Zurich 85, Beiblatt (Festschrift Rudolph Fueter), pp. 95-104, MR2, 218, 1940.
- 2
- R. H. Bruck, A Survey on Binary Systems, Springer-Verlag, New York, 1966.
- 3
- N. Jacobson, Theory of Rings, American Mathematical Society, New York, 1943.
- 4
- A. H. Clifford, Matrix Representations of Completely Simple Semigroups, Amer. J. Math. 70. pp. 521-526, 1948.
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