properties of certain monotone functions

properties of certain monotone functions

In the definitions of some

partially ordered algebraic systems

such as

po-groups

and

po-rings

, the

multiplication

is set to be

compatible

with the

partial ordering

on the

universe

in the following sense:

a

⁢

b

≤

a

⁢

c

iff

b

≤

c

and

a

⁢

b

≤

c

⁢

b

iff

a

≤

c

This is no coincidence. In fact, these “definitions” are actually

consequences

of properties concerning

monotone functions

satisfying certain

algebraic

rules, which is the focus of this entry.

Recall that an

n

-ary

function

f

on a set

A

is said to be

monotone

if it is monotone in each of its

variables

. In other words, for every

i

=

1

,

2

,

…

,

n

, the function

f

⁢

(

a

1

,

…

,

a

i

-

1

,

x

,

a

i

+

1

,

…

,

a

n

)

is monotone in

x

, where each of the

a

j

is a fixed but arbitrary element of

A

. We use the notation

↑

,

↓

,

↕

to denote the monotonicity of each variable in

f

. For example,

(

↑

,

↑

,

↑

)

denotes a ternary isotone function, whereas

(

↓

,

↕

)

denotes a binary function which is antitone with respect to its first variable, and both isotone/antitone with respect to the second.

Proposition 1

.

Let

f

be an

n

-ary

commutative

monotone

operation

on a set

A

. Then

f

is either isotone or antitone.

Proof.

Suppose

f

is isotone (or antitone) in its first variable. Since

f

⁢

(

x

,

a

1

,

…

,

a

n

-

1

)

=

f

⁢

(

a

1

,

x

,

…

,

a

n

-

1

)

=

⋯

=

f

⁢

(

a

1

,

…

,

a

n

-

1

,

x

)

,

f

is isotone (or antitone) in each of its remaining variables.
∎

Proposition 2

.

Let

f

be an

n

-ary monotone operation on a set

A

with an

identity element

e

∈

A

. In other words,

f

⁢

(

x

,

e

,

…

,

e

)

=

f

⁢

(

e

,

x

,

…

,

e

)

=

⋯

=

f

⁢

(

e

,

e

,

…

,

x

)

=

x

. Then

f

is either strictly isotone or strictly antitone.

Proof.

The proof is the same as the one before. Furthermore, if

f

is isotone and

a

<

b

, then

f

⁢

(

a

,

e

,

…

,

e

)

=

a

<

b

=

f

⁢

(

b

,

e

,

…

,

e

)

, so the strict ordering is preserved. The same holds true if

f

is antitone.
∎

Proposition 3

.

Let

f

be a binary monotone operation on a set

A

such that it is isotone (antitone) with respect to its first variable. Suppose

g

is a unary operation on

A

such that

f

⁢

(

x

,

g

⁢

(

x

)

)

is a fixed element of

A

. Then

g

is antitone (isotone).

Proposition 4

.

Let

f

be an

n

-ary

associative

monotone operation on a set

A

. Then

•

f

is isotone if

n

is even

•

f

is either isotone, or is

(

↑

,

…

,

↑

⏟

m

,

↓

,

↑

,

…

,

↑

⏟

m

)

, if

n

is odd, say

n

=

2

⁢

m

+

1

.

Proof.

Suppose first that

n

=

2

⁢

m

,

i

≤

m

, and

g

⁢

(

x

)

=

f

⁢

(

a

1

,

…

,

a

i

-

1

,

x

,

…

,

a

m

+

1

,

…

,

a

2

⁢

m

)

is antitone. Then

g

⁢

(

g

⁢

(

x

)

)

is isotone. By the associativity of

f

,

g

⁢

(

g

⁢

(

x

)

)

is

f

⁢

(

a

1

,

…

,

a

i

-

1

,

f

⁢

(

a

1

,

…

,

a

i

-

1

,

x

,

a

i

+

1

,

…

,

a

m

,

…

,

a

2

⁢

m

)

,

…

,

a

m

+

1

,

…

,

a

2

⁢

m

)

=

f

⁢

(

a

1

,

…

,

a

i

-

1

,

a

1

,

…

,

a

i

-

1

,

x

,

a

i

+

1

,

…

,

f

⁢

(

a

2

⁢

m

-

i

+

1

,

…

,

a

2

⁢

m

,

a

i

+

1

,

…

,

a

m

+

1

,

…

,

a

2

⁢

m

)

)

.

In the second expression, the position of

x

is

2

⁢

i

-

2

≤

2

⁢

m

-

1

<

2

⁢

m

, therefore implying that

g

⁢

(

g

⁢

(

x

)

)

is antitone, which is a

contradiction

! Therefore,

g

⁢

(

x

)

is isotone.
Now, if

i

>

m

, and

h

⁢

(

x

)

=

f

⁢

(

b

1

,

…

,

b

m

,

…

,

b

i

-

1

,

x

,

b

i

+

1

,

…

,

b

2

⁢

m

)

is antitone, then

h

⁢

(

h

⁢

(

x

)

)

is isotone. But

f

⁢

(

b

1

,

…

,

b

m

,

…

,

b

i

-

1

,

f

⁢

(

b

1

,

…

,

b

m

,

…

,

b

i

-

1

,

x

,

b

i

+

1

,

…

,

b

2

⁢

m

)

,

b

i

+

1

,

…

,

b

2

⁢

m

)

=

f

(

f

(

b

1

,

…

,

b

m

,

…

,

b

i

-

1

,

b

1

,

…

,

b

2

⁢

m

-

i

+

1

)

,

…

,

b

i

-

1

,

x

,

b

i

+

1

,

…

,

b

2

⁢

m

,

b

i

+

1

,

…

,

b

2

⁢

m

)

)

,

and the position of

x

is the second expression is

(

i

-

1

)

-

(

2

⁢

m

-

i

+

1

)

+

2

=

2

⁢

i

-

2

⁢

m

>

1

, therefore implying that

h

⁢

(

h

⁢

(

x

)

)

is antitone, again a contradiction. As a result,

f

is isotone for all

i

=

1

,

…

,

n

.

The

argument

above also works when

n

is odd, say

n

=

2

⁢

m

+

1

and

i

≠

m

+

1

. Finally, since

f

is monotone, it is monotone with respect to the

i

-th variable when

i

=

m

+

1

, so

f

is one of the following three forms:

(

↑

,

…

,

↑

⏟

2

⁢

m

+

1

)

,

(

↑

,

…

,

↑

⏟

m

,

↕

,

↑

,

…

,

↑

⏟

m

)

,

(

↑

,

…

,

↑

⏟

m

,

↓

,

↑

,

…

,

↑

⏟

m

)

,

the first two of which imply that

f

is isotone.
∎

An example of an associative function that is, say

(

↑

,

↓

,

↑

)

, is given by

f

:

ℤ

3

→

ℤ

where

f

⁢

(

x

,

y

,

z

)

=

x

-

y

+

z

.

f

is associative since

f

⁢

(

f

⁢

(

r

,

s

,

t

)

,

u

,

v

)

=

f

⁢

(

r

,

f

⁢

(

s

,

t

,

u

)

,

v

)

=

f

⁢

(

r

,

s

,

f

⁢

(

t

,

u

,

v

)

)

=

r

-

s

+

t

-

u

+

v

.

Title

properties of certain monotone functions

Canonical name

PropertiesOfCertainMonotoneFunctions

Date of creation

2013-03-22 19:03:31

Last modified on

2013-03-22 19:03:31

Owner

CWoo (3771)

Last modified by

CWoo (3771)

Numerical id

9

Author

CWoo (3771)

Entry type

Result

Classification

msc 06F99

Classification

msc 08C99

Classification

msc 08A99

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