residuated

residuated

Let

P

,

Q

be two posets. Recall that a

function

f

:

P

→

Q

is order preserving iff

x

≤

y

implies

f

⁢

(

x

)

≤

f

⁢

(

y

)

. This is

equivalent

to saying that the

preimage

of any down set under

f

is a down set.

Definition

A function

f

:

P

→

Q

between

ordered sets

P

,

Q

is said to be

residuated

if the preimage of any principal down set is a principal down set. In other words, for any

y

∈

Q

, there is some

x

∈

P

such that

f

-

1

(

↓

y

)

=

↓

x

,

where

f

-

1

⁢

(

A

)

for any

A

⊆

Q

is the set

{

a

∈

X

∣

f

⁢

(

a

)

∈

A

}

, and

↓

b

=

{

c

∣

c

≤

b

}

.

Let us make some observations on a residuated function

f

:

P

→

Q

:

1.

f

is order preserving.

Proof.

Suppose

a

≤

b

in

P

. Then there is some

c

∈

P

such that

↓

c

=

f

-

1

(

↓

f

⁢

(

b

)

)

. Since

f

(

b

)

∈

↓

f

(

b

)

,

b

∈

↓

c

, or

b

≤

c

, which means

a

≤

c

, or

a

∈

↓

c

. But this implies

a

∈

f

-

1

(

↓

f

⁢

(

b

)

)

, so that

f

(

a

)

∈

↓

f

(

b

)

, or

f

⁢

(

a

)

≤

f

⁢

(

b

)

.
∎

2.

The

x

in the definition above is unique.

Proof.

If

↓

x

=

↓

z

, then

x

∈

↓

z

, or

x

≤

z

. Similarly,

z

≤

x

, so that

x

=

z

.
∎

3.

g

:

Q

→

P

given by

g

⁢

(

y

)

=

x

is a well-defined function, with the following properties:

(a)

g

is order preserving,

(b)

1

P

≤

g

∘

f

,

(c)

f

∘

g

≤

1

Q

.

Proof.

g

is order preserving: if

r

≤

s

in

Q

, then

↓

r

⊆

↓

s

, so that

↓

u

=

f

-

1

(

↓

r

)

⊆

f

-

1

(

↓

s

)

=

↓

v

, where

u

=

g

⁢

(

r

)

and

v

=

g

⁢

(

s

)

. But

↓

u

⊆

↓

v

implies

u

≤

v

, or

g

⁢

(

r

)

≤

g

⁢

(

s

)

.

1

P

≤

g

∘

f

: let

x

∈

P

,

y

=

f

⁢

(

x

)

, and

z

=

g

⁢

(

y

)

. Then

x

∈

f

-

1

(

y

)

⊆

f

-

1

(

↓

y

)

=

↓

z

, which implies

x

≤

z

=

g

⁢

(

f

⁢

(

x

)

)

as desired.

f

∘

g

≤

1

Q

: pick

y

∈

Q

and let

x

=

g

⁢

(

y

)

. Then

f

-

1

(

↓

y

)

=

↓

x

, so that

f

(

x

)

∈

f

(

↓

x

)

=

f

(

f

-

1

(

↓

y

)

)

=

↓

y

, or

f

⁢

(

x

)

≤

y

, or

f

⁢

(

g

⁢

(

y

)

)

=

f

⁢

(

x

)

≤

y

as a result.
∎

Actually, the third observation above characterizes

f

being residuated:

Proposition 1

.

If

f

is order preserving, and

g

satisfies

3

⁢

(

a

)

through

3

⁢

(

c

)

, then

f

is residuated. In fact, such a

g

is unique.

Proof.

Suppose

y

∈

Q

, and let

x

=

g

⁢

(

y

)

. We want to show that

↓

x

=

f

-

1

(

↓

y

)

. First, suppose

a

∈

f

-

1

(

↓

y

)

. Then

f

⁢

(

a

)

≤

y

, which means

g

⁢

(

f

⁢

(

a

)

)

≤

g

⁢

(

y

)

=

x

. But then

a

≤

g

⁢

(

f

⁢

(

a

)

)

, and we get

a

≤

x

, or

a

∈

↓

x

. Next, suppose

a

≤

x

=

g

⁢

(

y

)

. Then

f

⁢

(

a

)

≤

f

⁢

(

x

)

=

f

⁢

(

g

⁢

(

y

)

)

≤

y

, so

a

∈

f

-

1

⁢

(

f

⁢

(

a

)

)

⊆

f

-

1

(

↓

y

)

.

To see uniqueness, suppose

h

:

Q

→

P

is order preserving such that

1

P

≤

h

∘

f

and

f

∘

h

≤

1

Q

. Then

g

=

1

P

∘

g

≤

(

h

∘

f

)

∘

g

=

h

∘

(

f

∘

g

)

≤

=

h

∘

1

Q

=

h

and

h

=

1

P

∘

h

≤

(

g

∘

f

)

∘

h

=

g

∘

(

f

∘

h

)

≤

g

∘

1

Q

=

g

.
∎

Definition

. Given a residuated function

f

:

P

→

Q

, the unique function

g

:

Q

→

P

defined above is called the

residual

of

f

, and is denoted by

f

+

.

For example, given any function

f

:

A

→

B

, the

induced

function

f

:

P

⁢

(

A

)

→

P

⁢

(

B

)

(by abuse of notation, we use the same notation as original function

f

), given by

f

⁢

(

S

)

=

{

f

⁢

(

a

)

∣

a

∈

S

}

is residuated. Its residual is the function

f

-

1

:

P

⁢

(

B

)

→

P

⁢

(

A

)

, given by

f

-

1

⁢

(

T

)

=

{

a

∈

A

∣

f

⁢

(

a

)

∈

T

}

.

Here are some properties of residuated functions and their residuals:

•

A

bijective

residuated function is an order isomorphism, and

conversely

. Furthermore, the residual is residuated, and is its

inverse

.

•

If

f

:

P

→

Q

is residuated, then

f

∘

f

+

∘

f

=

f

and

f

+

∘

f

∘

f

+

=

f

+

.

•

If

f

:

P

→

Q

and

g

:

Q

→

R

are residuated, so is

g

∘

f

and

(

g

∘

f

)

+

=

f

+

∘

g

+

.

•

If

f

:

P

→

Q

is residuated, then

f

+

∘

f

:

P

→

P

is a

closure map

on

P

. Conversely, any closure function can be decomposed as the

functional

composition

of a residuated function and its residual.

Remark

. Residuated functions and their residuals are closely related to

Galois connections

. If

f

:

P

→

Q

is residuated, then

(

f

,

f

+

)

forms a Galois connection between

P

and

Q

. On the other hand, if

(

f

,

g

)

is a Galois connection between

P

and

Q

, then

f

:

P

→

Q

is residuated, and

g

:

Q

→

P

is

f

+

. Note that PM defines a Galois connection as a pair of

order-preserving maps

, where as in Blyth, they are order reversing.

References

1

T.S. Blyth,

Lattices and Ordered

Algebraic Structures

, Springer, New York (2005).

2

G. Grätzer,

General

Lattice Theory

, 2nd Edition, Birkhäuser (1998)

Title

residuated

Canonical name

Residuated

Date of creation

2013-03-22 18:53:22

Last modified on

2013-03-22 18:53:22

Owner

CWoo (3771)

Last modified by

CWoo (3771)

Numerical id

12

Author

CWoo (3771)

Entry type

Definition

Classification

msc 06A99

Classification

msc 06A15

Defines

residual

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