Brandt groupoid

Brandt groupoids, like category theoretic groupoids (http://planetmath.org/GroupoidCategoryTheoretic), 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.

Definition

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

1. For every $a\in B$, there are unique elements $e,f$ such that $ea$ and $af$ are defined, and is equal to $a$.

2. If $ae=a$ or $ea=a$ for some $a,e\in B$, then $ee$ is defined, and is equal to $e$.

3. For $a,b\in B$, $ab$ is defined iff there is an $e\in B$ such that $ae=a$ and $eb=b$.

4. For $a,b,c\in B$ such that $ab$ and $bc$ are defined, then so are $(ab)c$ and $a(bc)$ and they equal.

5. 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 $ba=f$.

6. 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 $I\ne \mathrm{\varnothing }$ by conditions 1 and 2.

Brandt Groupoids versus Categories

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 $as(a)=a$ 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, $c=t(c)c=t(b)c=(ba)c=b(ac)=b(ab)=bs(b)=b$. ∎

$b$ 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 (http://planetmath.org/GroupoidCategoryTheoretic). 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).

An Example

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 (http://planetmath.org/ConstructionOfABrandtGroupoid)).

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 $S:=B\cup \{0\}$, 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.

Title	Brandt groupoid
Canonical name	BrandtGroupoid
Date of creation	2013-03-22 18:38:32
Last modified on	2013-03-22 18:38:32
Owner	CWoo (3771)
Last modified by	CWoo (3771)
Numerical id	18
Author	CWoo (3771)
Entry type	Definition
Classification	msc 20L05
Classification	msc 18B40
Related topic	GroupoidCategoryTheoretic
Related topic	ConnectedCategory