quasi-inverse of a function

quasi-inverse of a function

Let

f

:

X

→

Y

be a

function

from sets

X

to

Y

. A

quasi-inverse

g

of

f

is a function

g

such that

1.

g

:

Z

→

X

where

ran

⁡

(

f

)

⊆

Z

⊆

Y

, and

2.

f

∘

g

∘

f

=

f

, where

∘

denotes

functional

composition operation.

Note that

ran

⁡

(

f

)

is the range of

f

.

Examples.

1.

If

f

is a real function given by

f

⁢

(

x

)

=

x

2

. Then

g

⁢

(

x

)

=

x

defined on

[

0

,

∞

)

and

h

⁢

(

x

)

=

-

x

also defined on

[

0

,

∞

)

are both quasi-inverses of

f

.

2.

If

f

⁢

(

x

)

=

1

defined on

[

0

,

1

)

. Then

g

⁢

(

x

)

=

1

2

defined on

ℝ

is a quasi-inverse of

f

. In fact, any

g

⁢

(

x

)

=

a

where

a

∈

[

0

,

1

)

will do. Also, note that

h

⁢

(

x

)

=

x

on

[

0

,

1

)

is also a quasi-inverse of

f

.

3.

If

f

⁢

(

x

)

=

[

x

]

, the

step function

on the reals. Then by the previous example,

g

⁢

(

x

)

=

[

x

]

+

a

, any

a

∈

[

0

,

1

)

, is a quasi-inverse of

f

.

Remarks.

•

Every function has a quasi-inverse. This is just another form of the

Axiom of Choice

. In fact, if

f

:

X

→

Y

, then for

every

subset

Z

of

Y

such that

ran

⁡

(

f

)

⊆

Z

, there is a quasi-inverse

g

of

f

whose domain is

Z

.

•

However, a quasi-inverse of a function is in general not unique, as illustrated by the above examples. When it is unique, the function must be a

bijection

:

If

ran

⁡

(

f

)

≠

Y

, then there are at least two quasi-inverses, one with domain

ran

⁡

(

f

)

and one with domain

Y

. So

f

is onto. To see that

f

is

one-to-one

, let

g

be the quasi-inverse of

f

. Now suppose

f

⁢

(

x

1

)

=

f

⁢

(

x

2

)

=

z

. Let

g

⁢

(

z

)

=

x

3

and assume

x

3

≠

x

1

. Define

h

:

Y

→

X

by

h

⁢

(

y

)

=

g

⁢

(

y

)

if

y

≠

z

, and

h

⁢

(

z

)

=

x

1

. Then

h

is easily verified as a quasi-inverse of

f

that is different from

g

. This is a contradition. So

x

3

=

x

1

. Similarly,

x

3

=

x

2

and therefore

x

1

=

x

2

.

•

Conversely

, if

f

is a bijection, then the

inverse

of

f

is a quasi-inverse of

f

. In fact,

f

has only one quasi-inverse.

•

The

relation

of being quasi-inverse is not

symmetric

. In other words, if

g

is a quasi-inverse of

f

,

f

need not be a quasi-inverse of

g

. In the second example above,

h

is a quasi-inverse of

f

, but not vice versa:

h

⁢

(

0

)

=

0

, but

h

⁢

f

⁢

h

⁢

(

0

)

=

h

⁢

f

⁢

(

0

)

=

h

⁢

(

1

)

=

1

≠

h

⁢

(

0

)

.

•

Let

g

be a quasi-inverse of

f

, then the

restriction

of

g

to

ran

⁡

(

f

)

is one-to-one. If

g

and

f

are quasi-inverses of one another, and

g

strictly includes

ran

⁡

(

f

)

, then

g

is

not

one-to-one.

•

The set of real functions, with

addition

defined element-wise and

multiplication

defined as functional composition, is a ring. By remark 2, it is in fact a

Von Neumann regular ring

, as any quasi-inverse of a real function is also its pseudo-inverse as an element of the ring. Any space whose

ring of continuous functions

is Von Neumann regular is a

P-space

.

References

1

B. Schweizer, A. Sklar,

Probabilistic Metric Spaces

, Elsevier Science Publishing Company, (1983).

Title

quasi-inverse of a function

Canonical name

QuasiinverseOfAFunction

Date of creation

2013-03-22 16:22:14

Last modified on

2013-03-22 16:22:14

Owner

CWoo (3771)

Last modified by

CWoo (3771)

Numerical id

11

Author

CWoo (3771)

Entry type

Definition

Classification

msc 03E20

Synonym

quasi-inverse

Defines

quasi-inverse function

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