partially ordered algebraic system

partially ordered algebraic system

Let

A

be a poset. Recall a

function

f

on

A

is said to be

•

order-preserving

(or isotone) provided that

f

⁢

(

a

)

≤

f

⁢

(

b

)

, or

•

order-reversing (or antitone) provided that

f

⁢

(

a

)

≥

f

⁢

(

b

)

, or

whenever

a

≤

b

. Furthermore,

f

is called

monotone

if

f

is either isotone or antitone.

For every function

f

on

A

, we denote it to be

↑

,

↓

, or

↕

according to whether it is isotone, antitone, or both. The following are some easy

consequences

:

•

↑

⁣

∘

⁣

↓

⁣

=

⁣

↓

⁣

∘

⁣

↑

⁣

=

⁣

↓

(meaning that the composition of an isotone and an antitone maps is antitone),

•

↑

⁣

∘

⁣

↑

⁣

=

⁣

↓

⁣

∘

⁣

↓

⁣

=

⁣

↑

(meaning that the composition of two isotone or two antitone maps is isotone),

•

f

is

↕

iff it is a

constant

on any chain in

A

, and if this is the case, for every

a

∈

A

,

f

-

1

⁢

(

a

)

is a

maximal

chain in

A

.

The notion above can be generalized to

n

-ary

operations

on a poset

A

. An

n

-ary operation

f

on a poset

A

is said to be

isotone

,

antitone

, or

monotone

iff when

f

is isotone, antitone, or monotone with respect to each of its

n

variables

. We continue to use to arrow notations above to denote

n

-ary monotone functions. For example, a ternary function that is

(

↑

,

↓

,

↑

)

is isotone with respect to its first and third variables, and antitone with respect to its second variable.

Definition

. A

partially ordered algebraic system

is an

algebraic system

𝒜

=

(

A

,

O

)

such that

A

is a poset, and every operation

f

∈

O

on

A

is monotone. A partially ordered algebraic system is also called a partially ordered

algebra

, or a po-algebra for short.

Examples of po-algebras are

po-groups

,

po-rings

, and po-semigroups. In all three cases, the

multiplication

operations are

(

↑

,

↑

)

, as well as the

addition

operation in a po-ring.. In the case of a po-group, the

multiplicative inverse

operation is

↓

, as well as the

additive inverse

operation in a po-ring.

Another example is an

ordered vector space

V

over a field

k

. The underlying

universe

is

V

(not

k

). Addition over

V

is, like the other examples above, isotone. Each

element

r

∈

k

acts as a unary

operator

on

V

, given by

r

⁢

(

v

)

=

r

⁢

v

, the scalar multiplication of

r

and

v

. As

k

is itself a poset, it can be partitioned into three sets: the positive cone

P

⁢

(

k

)

of

k

, the

negative

cone

-

P

⁢

(

k

)

, and

{

0

}

. Then

r

∈

P

⁢

(

k

)

iff it is

↑

as a unary operator,

r

∈

-

P

⁢

(

k

)

iff it is

↓

, and

r

=

0

iff it is

↕

.

Remarks

•

A

homomorphism

from one po-algebra

𝒜

to another

ℬ

is an isotone map

ϕ

from posets

A

to

B

that is at the same time a homomorphism from the algebraic systems

𝒜

to

ℬ

.

•

A partially ordered

subalgebra

of a po-algebra

𝒜

is just a subalgebra of

𝒜

viewed as an algebra, where the

partial ordering

on the universe of the subalgebra is inherited from the partial ordering on

A

.

References

1

L. Fuchs,

Partially Ordered Algebraic Systems

, Addison-Wesley, (1963).

Title

partially ordered algebraic system

Canonical name

PartiallyOrderedAlgebraicSystem

Date of creation

2013-03-22 19:03:19

Last modified on

2013-03-22 19:03:19

Owner

CWoo (3771)

Last modified by

CWoo (3771)

Numerical id

8

Author

CWoo (3771)

Entry type

Definition

Classification

msc 06F99

Classification

msc 08C99

Classification

msc 08A99

Related topic

AlgebraicSystem

Generated on Fri Feb 9 10:07:15 2018 by

LaTeXML