isotope of a groupoid

isotope of a groupoid

Let

G

,

H

be

groupoids

(

http://planetmath.org/Groupoid

). An

isotopy

ϕ

from

G

to

H

is an ordered triple:

ϕ

=

(

f

,

g

,

h

)

, of

bijections

from

G

to

H

, such that

f

⁢

(

a

)

⁢

g

⁢

(

b

)

=

h

⁢

(

a

⁢

b

)

for all

⁢

a

,

b

∈

G

.

H

is called an

isotope

of

G

(or

H

is

isotopic

to

G

) if there is an isotopy

ϕ

:

G

→

H

.

Some easy examples of isotopies:

1.

If

f

:

G

→

H

is an

isomorphism

,

(

f

,

f

,

f

)

:

G

→

H

is an isotopy. By abuse of

language

, we write

f

=

(

f

,

f

,

f

)

. In particular,

(

1

G

,

1

G

,

1

G

)

:

G

→

G

is an isotopy.

2.

If

ϕ

=

(

f

,

g

,

h

)

:

G

→

H

is an isotopy, then so is

ϕ

-

1

:=

(

f

-

1

,

g

-

1

,

h

-

1

)

:

H

→

G

,

for if

f

-

1

⁢

(

a

)

=

c

and

g

-

1

⁢

(

b

)

=

d

, then

a

⁢

b

=

f

⁢

(

c

)

⁢

g

⁢

(

d

)

=

h

⁢

(

c

⁢

d

)

, so that

f

-

1

⁢

(

a

)

⁢

g

-

1

⁢

(

b

)

=

c

⁢

d

=

h

-

1

⁢

(

a

⁢

b

)

3.

If

ϕ

=

(

f

,

g

,

h

)

:

G

→

H

and

γ

=

(

r

,

s

,

t

)

:

H

→

K

are isotopies, then so is

γ

∘

ϕ

:=

(

r

∘

f

,

s

∘

g

,

t

∘

h

)

:

G

→

K

,

for

(

r

∘

f

)

⁢

(

a

)

⁢

(

s

∘

g

)

⁢

(

b

)

=

r

⁢

(

f

⁢

(

a

)

)

⁢

s

⁢

(

g

⁢

(

b

)

)

=

t

⁢

(

f

⁢

(

a

)

⁢

g

⁢

(

b

)

)

=

t

⁢

(

h

⁢

(

a

⁢

b

)

)

=

(

t

∘

h

)

⁢

(

a

⁢

b

)

.

From the examples above, it is easy to see that “groupoids being isotopic” on the class of groupoids is an

equivalence relation

, and that an isomorphism class is

contained

in an isotopic class. In fact, the containment is strict. For an example of non-isomorphic isotopic groupoids, see the reference below. However, if

G

is a groupoid with unity and

G

is isotopic to a

semigroup

S

, then it is isomorphic to

S

. Other conditions making isotopic groupoids isomorphic can be found in the reference below.

An isotopy of the form

(

f

,

g

,

1

H

)

:

G

→

H

is called a

principal isotopy

, where

1

H

is the

identity function

on

H

.

H

is called a

principal isotope

of

G

. If

H

is isotopic to

G

, then

H

is isomorphic to a principal isotope

K

of

G

.

Proof.

Suppose

(

f

,

g

,

h

)

:

G

→

H

is an isotopy. To

construct

K

, start with elements of

G

, which will form the underlying set of

K

. The

binary operation

on

K

is defined by

a

⋅

b

:=

(

f

-

1

∘

h

)

⁢

(

a

)

⁢

(

g

-

1

∘

h

)

⁢

(

b

)

.

Then

⋅

is well-defined, since

f

,

g

are

bijective

, for all pairs of elements of

G

. Hence

K

is a groupoid. Furthermore,

(

f

-

1

∘

h

,

g

-

1

∘

h

,

1

K

)

:

G

→

K

is an isotopy by definition, so that

K

is a principal isotope of

G

. Finally,

h

⁢

(

a

⋅

b

)

=

h

⁢

(

f

-

1

⁢

(

h

⁢

(

a

)

)

⁢

g

-

1

⁢

(

h

⁢

(

b

)

)

)

=

f

⁢

(

f

-

1

⁢

(

h

⁢

(

a

)

)

)

⁢

g

⁢

(

g

-

1

⁢

(

h

⁢

(

b

)

)

)

=

h

⁢

(

a

)

⁢

h

⁢

(

b

)

, showing that

h

:

K

→

H

is a bijective

homomorphism

, and hence an isomorphism.
∎

Remark

. In the literature, the definition of an isotope is sometimes limited to

quasigroups

. However, this is not necessary, as the follow

proposition

suggests:

Proposition 1

.

Any isotope of a quasigroup is a quasigroup.

Proof.

Suppose

(

f

,

g

,

h

)

:

G

→

H

is an isotopy, and

G

a quasigroup. Pick

x

,

z

∈

H

. Let

a

,

c

∈

G

be such that

f

⁢

(

a

)

=

x

and

h

⁢

(

c

)

=

z

. Let

b

∈

G

be such that

a

⁢

b

=

c

. Set

y

=

g

⁢

(

b

)

∈

H

. Then

x

⁢

y

=

f

⁢

(

a

)

⁢

g

⁢

(

b

)

=

h

⁢

(

a

⁢

b

)

=

h

⁢

(

c

)

=

z

. Similarly, there is

t

∈

H

such that

t

⁢

x

=

z

. Hence

H

is a quasigroup. ∎

On the other hand, an isotope of a loop may not be a loop. Nevertheless, we sometimes say that an isotope of a loop

L

as a loop isotopic to

L

.

References

1

R. H. Bruck:

A Survey of Binary Systems

.  Springer-Verlag. New York (1966).

Title

isotope of a groupoid

Canonical name

IsotopeOfAGroupoid

Date of creation

2013-03-22 18:35:54

Last modified on

2013-03-22 18:35:54

Owner

CWoo (3771)

Last modified by

CWoo (3771)

Numerical id

8

Author

CWoo (3771)

Entry type

Definition

Classification

msc 20N02

Classification

msc 20N05

Synonym

isotopism

Synonym

homotopism

Defines

isotopy

Defines

isotope

Defines

homotopy

Defines

homotope

Defines

isotopic

Defines

homotopic

Defines

principal isotopy

Defines

principal isotope

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