intersection structure

intersection structure

Intersection structures

An

intersection structure

is a set

C

such that

1.

C

is a subset of the

powerset

P

⁢

(

A

)

of some set

A

, and

2.

intersection

of a non-empty family

ℱ

of elements of

C

is again in

C

.

If order

C

by

set inclusion

, then

C

becomes a poset.

There are numerous examples of intersection structures. In

algebra

, the set of all subgroups of a group, the set of all ideals of a ring, and the set of all

subspaces

of a vector space. In

topology

, the set of all

closed sets

of a

topological space

is an intersection structure. Finally, in

functional analysis

, the set of all

convex subsets

of a

topological vector space

is also an intersection structure.

The set of all

partial orderings

on a set is also an intersection structure. A final example can be found in domain theory: let

C

be the set of all

partial functions

from a non-empty set

X

to a non-empty set

Y

. Since each partial function is a subset of

X

×

Y

,

C

is a subset of

P

⁢

(

X

×

Y

)

. Let

ℱ

:=

{

f

i

∣

i

∈

I

}

be an arbitrary

collection

of partial functions in

C

and

f

=

⋂

ℱ

.

f

is clearly a

relation

between

X

and

Y

. Suppose

x

is in the domain of

f

. Let

E

=

{

y

∈

Y

∣

x

⁢

f

⁢

y

}

. Then

x

⁢

f

i

⁢

y

for each

f

i

where

x

is in the domain of

f

i

. Since

f

i

is a partial function,

y

=

f

i

⁢

(

x

)

, so that

y

is uniquely determined. This means that

E

is a

singleton

, hence

f

is a partial function, so that

⋂

ℱ

∈

C

, meaning that

C

is an intersection structure.

The main

difference

between the last two examples and the previous examples is that in the last two examples,

C

is rarely a

complete lattice

. For example, let

≤

be a partial ordering on a set

P

. Then its dual

≤

∂

is also a partial ordering on

P

. But the join of

≤

and

≤

∂

does not exist. Here is another example: let

X

=

{

1

}

and

Y

=

{

2

,

3

}

. Then

C

=

{

∅

,

(

1

,

2

)

,

(

1

,

3

)

}

.

(

1

,

2

)

and

(

1

,

3

)

are the

maximal elements

of

C

, but the join of these two

elements

does not exist.

Topped intersection strucutres

If, in condition 2 above, we remove the requirement that

ℱ

be non-empty, then we have an intersection structure called a

topped intersection structure

.

The reason for calling them topped is because the top element of such an intersection structure always exists; it is the intersection of the empty family. In

addition

, a topped intersection structure is always a complete lattice. For a proof of this fact, see this link (

http://planetmath.org/CriteriaForAPosetToBeACompleteLattice

).

As a result, for example, to show that the subgroups of a group form a complete lattice, it is enough to observe that arbitrary intersection of subgroups is again a subgroup.

Remarks

.

•

A topped intersection structure is also called a

closure system

. The reason for calling this is that every topped intersection structure

C

⊆

P

⁢

(

X

)

induces a

closure operator

cl

on

P

⁢

(

X

)

, making

X

a

closure space

.

cl

:

P

⁢

(

X

)

→

P

⁢

(

X

)

given by

cl

⁡

(

A

)

=

⋂

{

B

∈

C

∣

A

⊆

B

}

is

well-defined

.

•

Conversely

, it is not hard to see that every closure space

(

X

,

cl

)

gives rise to a closure system

C

:=

{

cl

⁡

(

A

)

∣

A

∈

P

⁢

(

X

)

}

.

•

An intersection structure

C

is said to be

algebraic

if for every

directed set

B

⊆

C

, we have that

⋃

B

∈

C

. All of the examples above, except the set of closed sets in a topological space, are algebraic intersection structures. A topped intersection structure that is algebraic is called an

algebraic closure system

if,

•

Every algebraic closure system is an

algebraic lattice

.

References

1

B. A. Davey, H. A. Priestley,

Introduction

to Lattices and Order

, 2nd Edition, Cambridge (2003)

2

G. Grätzer:

Universal Algebra

, 2nd Edition, Springer, New York (1978).

Title

intersection structure

Canonical

name

IntersectionStructure

Date of creation

2013-03-22 17:06:28

Last modified on

2013-03-22 17:06:28

Owner

CWoo (3771)

Last modified by

CWoo (3771)

Numerical id

9

Author

CWoo (3771)

Entry type

Definition

Classification

msc 03G10

Classification

msc 06B23

Synonym

closure system

Related topic

CriteriaForAPosetToBeACompleteLattice

Defines

topped intersection structure

Defines

algebraic intersection structure

Defines

algebraic closure system

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