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.
A Brandt groupoid is a non-empty set , together with a partial binary operation (called a multiplication) defined on it (we write for ), such that
For every , there are unique elements such that and are defined, and is equal to .
If or for some , then is defined, and is equal to .
For , is defined iff there is an such that and .
For such that and are defined, then so are and and they equal.
If for some , then there is a such that and are defined and and .
If and for some , then there is such that and are defined and are equal to .
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 , where and are the unique idempotents such that and . In addition, for all . Call the source function, the target function, and for any , the source and the target of .
The third condition says that is defined iff the source of is the equal to the target of : . The fourth condition is the associativity law for the multiplication. An easy consequence of this condition is that if exists, then and .
A morphism in is said to be an isomorphism if there is a morphism in such that . Now, is uniquely determined by , so that is an isomorphism in the category theoretic sense.
First notice that and . If , then and . So and . As a result, . ∎
is said to be the inverse of , and is often written . Condition 5 says that the category is in fact a category theoretic groupoid (http://planetmath.org/GroupoidCategoryTheoretic). Thus, a Brandt groupoid is a group if the multiplication is everywhere defined.
A Brandt groupoid may be constructed as follows: take a group and a non-empty set , set , and define multiplication on as follows:
Then with the partial multiplication is a Brandt groupoid. The idempotents in have the form , where is the group identity. And for any , its source, target, and inverse are and , 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 for all , then must be the source and target of , but then would have to be unique by condition 1, which is impossible unless is trivial. If we adjoin to a Brandt groupoid , and call , then has the structure of a semigroup. Here’s how the multiplication is defined on :
Since the multiplication on is everywhere defined, is a groupoid. To see that is a semigroup, we must show that associativity of the multiplication applies everywhere. There are four cases
If both and are defined in , they are certainly defined in , and the associativity follows from condition 4.
If neither nor is defined in , then in .
If is not defined in , but is, then , and .
Similarly, if is defined in but not , then .
Thus, is a semigroup (with ). In fact, Clifford showed that 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.
|Date of creation||2013-03-22 18:38:32|
|Last modified on||2013-03-22 18:38:32|
|Last modified by||CWoo (3771)|