Brandt groupoid

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)

⋅

defined on it (we write

a

⁢

b

for

a

⋅

b

), such that

1.

For every

a

∈

B

, there are unique elements

e

,

f

such that

e

⁢

a

and

a

⁢

f

are defined, and is equal to

a

.

2.

If

a

⁢

e

=

a

or

e

⁢

a

=

a

for some

a

,

e

∈

B

, then

e

⁢

e

is defined, and is equal to

e

.

3.

For

a

,

b

∈

B

,

a

⁢

b

is defined iff there is an

e

∈

B

such that

a

⁢

e

=

a

and

e

⁢

b

=

b

.

4.

For

a

,

b

,

c

∈

B

such that

a

⁢

b

and

b

⁢

c

are defined, then so are

(

a

⁢

b

)

⁢

c

and

a

⁢

(

b

⁢

c

)

and they equal.

5.

If

e

⁢

a

=

a

⁢

f

=

e

for some

a

,

e

,

f

∈

B

, then there is a

b

∈

B

such that

a

⁢

b

and

b

⁢

a

are defined and

a

⁢

b

=

e

and

b

⁢

a

=

f

.

6.

If

e

⁢

e

=

e

and

f

⁢

f

=

f

for some

e

,

f

∈

B

, then there is

a

∈

B

such that

e

⁢

a

and

a

⁢

f

are defined and are equal to

a

.

In the definition above, we see several

instances

of elements

e

such that

e

2

=

e

⁢

e

=

e

. Such elements are called

idempotents

. If we let

I

be the set of all idempotents of

B

, then

I

≠

∅

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

→

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

∈

I

. Call

s

the source

function

,

t

the target function, and for any

a

∈

B

,

s

⁢

(

a

)

,

t

⁢

(

a

)

the source and the target of

a

.

The third condition says that

a

⁢

b

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

a

⁢

b

exists, then

s

⁢

(

b

)

=

s

⁢

(

a

⁢

b

)

and

t

⁢

(

a

)

=

t

⁢

(

a

⁢

b

)

.

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

a

⁢

b

,

b

⁢

a

∈

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

⁢

(

a

⁢

b

)

=

a

⁢

b

=

t

⁢

(

a

⁢

b

)

=

t

⁢

(

a

)

and

t

⁢

(

b

)

=

t

⁢

(

b

⁢

a

)

=

b

⁢

a

=

s

⁢

(

b

⁢

a

)

=

s

⁢

(

a

)

. If

a

⁢

c

,

c

⁢

a

∈

I

, then

s

⁢

(

c

)

=

t

⁢

(

a

)

=

s

⁢

(

b

)

and

t

⁢

(

c

)

=

s

⁢

(

a

)

=

t

⁢

(

b

)

. So

a

⁢

b

=

a

⁢

c

and

b

⁢

a

=

b

⁢

c

. As a result,

c

=

t

⁢

(

c

)

⁢

c

=

t

⁢

(

b

)

⁢

c

=

(

b

⁢

a

)

⁢

c

=

b

⁢

(

a

⁢

c

)

=

b

⁢

(

a

⁢

b

)

=

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 (

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

×

G

×

I

, and define multiplication on

B

as follows:

(

p

,

x

,

q

)

⁢

(

r

,

y

,

s

)

=

{

(

p

,

x

⁢

y

,

s

)

if

⁢

q

=

r

,

undefined

otherwise.

Then

B

with the partial multiplication is a Brandt groupoid. The idempotents in

B

have the form

(

p

,

e

,

p

)

, where

e

∈

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

0

⁢

a

=

a

⁢

0

=

0

for all

a

∈

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

∪

{

0

}

, then

S

has the

structure

of a

semigroup

. Here’s how the multiplication is defined on

S

:

a

⁢

b

=

{

a

⁢

b

if

a

⁢

b

is defined in

B

,

0

otherwise, or if either

a

=

0

or

b

=

0

.

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

a

⁢

b

and

b

⁢

c

are defined in

B

, they are certainly defined in

S

, and the associativity follows from condition 4.

•

If neither

a

⁢

b

nor

b

⁢

c

is defined in

B

, then

(

a

⁢

b

)

⁢

c

=

0

⁢

c

=

0

=

a

⁢

0

=

a

⁢

(

b

⁢

c

)

in

S

.

•

If

a

⁢

b

is not defined in

B

, but

b

⁢

c

is, then

s

⁢

(

a

)

≠

t

⁢

(

b

)

=

t

⁢

(

b

⁢

c

)

, and

(

a

⁢

b

)

⁢

c

=

0

⁢

c

=

0

=

a

⁢

(

b

⁢

c

)

.

•

Similarly, if

a

⁢

b

is defined in

B

but not

b

⁢

c

, then

(

a

⁢

b

)

⁢

c

=

0

=

a

⁢

(

b

⁢

c

)

.

Thus,

S

is a semigroup (with

0

). In fact, Clifford showed that

S

is

completely simple

.

References

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.

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

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