pseudocomplement

pseudocomplement

Given an

element

a

in a

bounded lattice

L

, a

complement

of

a

is defined to be an element

b

∈

L

, if such an element exists, such that

a

∧

b

=

0

,

a

⁢

n

⁢

d

a

∨

b

=

1

.

If a complement of an element exists, it may not be unique. For example, in the middle row of the following

diagram

(called the

diamond

)

\xymatrix

⁢

&

⁢

1

⁢

\ar

⁢

@

-

[

l

⁢

d

]

⁢

\ar

⁢

@

-

[

d

]

⁢

\ar

⁢

@

-

[

r

⁢

d

]

⁢

&

⁢

a

⁢

\ar

⁢

@

-

[

r

⁢

d

]

⁢

&

⁢

b

⁢

\ar

⁢

@

-

[

d

]

⁢

&

⁢

c

⁢

\ar

⁢

@

-

[

l

⁢

d

]

⁢

&

⁢

0

⁢

&

any two of the three elements are complements of the third.

To get around the non-uniqueness issue, an alternative to a complement, called the

pseudocomplement

of an element, is defined. However, the cost of having the uniqueness is the lost of one of the equations above (in fact, the second one). The weakening of the second equation is not an arbitrary choice, but historical, when

propositional logic

was being generalized and the law of the excluded middle was dropped in order to develop non-classical logics.

An element

b

in a

lattice

L

with

0

is a pseudocomplement of

a

∈

L

if

1.

b

∧

a

=

0

2.

for any

c

such that

c

∧

a

=

0

then

c

≤

b

.

In other words,

b

is the

maximal element

in the set

{

c

∈

L

∣

c

∧

a

=

0

}

.

It is easy to see that given an element

a

∈

L

, the pseudocomplement of

a

, if it exists, is unique. If this is the case, then the psedocomplement of

a

is written as

a

*

.

The next natural question to ask is: if

a

*

is the pseudocomplement of

a

, is

a

the pseudocomplement of

a

*

? The answer is no, as the following diagram illustrates (called the

benzene

)

\xymatrix

⁢

&

⁢

1

⁢

\ar

⁢

@

-

[

r

⁢

d

]

⁢

\ar

⁢

@

-

[

l

⁢

d

]

⁢

x

⁢

\ar

⁢

@

-

[

d

]

⁢

&

⁢

&

⁢

y

⁢

\ar

⁢

@

-

[

d

]

⁢

a

⁢

\ar

⁢

@

-

[

r

⁢

d

]

⁢

&

⁢

&

⁢

b

⁢

\ar

⁢

@

-

[

l

⁢

d

]

⁢

&

⁢

0

⁢

&

The pseudocomplement of

a

is

y

, but the pseudocomplement of

y

, however, is

x

. In fact, it is possible that

a

*

*

may not even exist! A lattice

L

in which every element has a pseudocomplement is called a

pseudocomplemented lattice

. Necessarily

L

must be a bounded lattice.

From the above little discussion, it is not hard to deduce some of the basic properties of pseudocomplementation in a pseudocomplemented lattice:

1.

1

*

=

0

and

0

*

=

1

(if

c

∧

1

=

0

, then

c

=

0

, and the largest

c

such that

c

∧

0

=

0

is

1

)

2.

a

≤

a

*

*

(since

a

*

∧

a

=

0

and

a

*

∧

a

*

*

=

0

,

a

≤

a

*

*

)

3.

a

≤

b

, then

b

*

≤

a

*

(since

a

∧

b

*

≤

b

∧

b

*

=

0

, and

a

∧

a

*

=

0

,

b

*

≤

a

*

)

4.

a

*

=

a

*

⁣

*

*

(

a

≤

a

*

*

by

2

above, so

a

*

⁣

*

*

≤

a

*

by

3

, but

a

*

≤

a

*

⁣

*

*

by

2

, so

a

*

=

a

*

⁣

*

*

)

Furthermore, it can be shown that in a pseudocomplemented lattice, the subset of all pseudocomplements has the

structure

of a

Boolean lattice

.

Example

. The most common example is the lattice

L

⁢

(

X

)

of

open sets

in a

topological space

X

.

L

⁢

(

X

)

is usually not

complemented

, because the set complement of an open set is closed. However,

L

⁢

(

X

)

is pseudocomplemented, and if

U

is an open set in

X

, then its pseudocomplement is

(

U

c

)

∘

, the

interior

of the complement of

U

.

Remarks

.

•

A closely related

concept

to a pseudocomplemented lattice is that of a

pseudocomplemented algebra

, or

p-algebra

for short, which is a pseudocomplemented lattice such that

*

is considered as an

operator

. In other words, a

morphism

between two pseudocomplemented lattices is just a

lattice homomorphism

, where as a morphism between two p-algebras is a lattice homomorphism

f

preserving

*

:

f

⁢

(

a

*

)

=

f

⁢

(

a

)

*

. In the

category

of p-algebras, the morphism between any pair of objects is a

{

0

,

1

}

-lattice homomorphism, since

f

⁢

(

1

)

=

f

⁢

(

0

*

)

=

f

⁢

(

0

)

*

=

0

*

=

1

. A p-algebra is sometimes known as a

Stone algebra

.

•

The notion of a pseudocomplement can be generalized. Notice first that the definition of a pseudocomplement of an element does not involve the join

operation

. In fact, all we need is a poset with the least element. A poset

P

with the least element

0

is called a

pseudocomplemented poset

if , for each element

a

∈

P

, there is an element

a

*

∈

P

such that their

greatest lower bound

is

0

and is the largest such element with this property. By definition,

a

*

is unique for each

a

∈

P

, and that

P

itself is

bounded

, with the greatest element

1

, as it is the pseudocomplement of

0

. A pseudocomplemented poset that is also a lattice is a clearly a pseudocomplemented lattice. Examples of pseudocomplemented posets that are not lattices are found in the third reference below.

•

The notion of a pseudocomplement can be generalized in other ways. For example, we say that an element

b

in a lattice

L

is a pseudocomplement of

a

relative to

d

if

b

is a pseudocomplement in the sublattice

[

d

,

∞

)

(think of

d

as

0

in the definition of a pseudocomplement). Of course, this requires that both

a

and

b

be at least

d

. A pseudocomplement is therefore a pseudocomplement relative to

0

. See the entry on

Brouwerian lattice

for more detail.

•

In the definition of a pseudocomplement, some authors relax the first condition above. Instead, the pseudocomplement

b

(of

a

) is only required to be the

supremum

of the set

{

c

∈

L

∣

c

∧

a

=

0

}

.

References

1

T.S. Blyth,

Lattices and Ordered

Algebraic Structures

, Springer, New York (2005).

2

G. Grätzer,

General

Lattice Theory

, 2nd Edition, Birkhäuser (1998).

3

R. Halas,

http://www.emis.de/journals/AM/93-34/halas.ps

http://www.emis.de/journals/AM/93-34/halas.ps

, Archivum Mathematicum (BRNO) 1993.

4

S. Ghilardi,

http://homes.dsi.unimi.it/ ghilardi/allegati/dispcesena.pdf

http://homes.dsi.unimi.it/ ghilardi/allegati/dispcesena.pdf

, 2000.

Title

pseudocomplement

Canonical name

Pseudocomplement

Date of creation

2013-03-22 15:47:23

Last modified on

2013-03-22 15:47:23

Owner

CWoo (3771)

Last modified by

CWoo (3771)

Numerical id

21

Author

CWoo (3771)

Entry type

Definition

Classification

msc 06D15

Synonym

pseudocomplemented algebra

Synonym

Stone algebra

Synonym

Stone lattice

Related topic

BrouwerianLattice

Related topic

ComplementedLattice

Related topic

Pseudodifference

Defines

pseudocomplemented lattice

Defines

benzene

Defines

p-algebra

Defines

pseudocomplemented poset

Defines

relative pseudocomplement

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