background image

properties of generalized filter bases)

X ∈

S

X

up

X

:

X

F

F 6

= 0

0

6∈

F∩

F

up

X

\

F

F∩

F

up

X 6

= 0

F ∩

F

\

F

up

X 6

= 0

F ∩

F

X 6

= 0

X ∈

F

.

12. Quasidifference and quasicomplement

I’ve got quasidifference and quasicomplement (and dual quasicomplement)

replacing max and min in the definition of pseudodifference and pseudocomple-
ment (and dual pseudocomplement) with

S

and

T

. Thus quasidifference and

(dual) quasicomplement are generalizations of their pseudo- counterparts.

Remark 11 Pseudocomplements

and

pseudodifferences

is standard ter-

minology.

Quasi

- counterparts are my neologisms.

Definition 66

Let

A

be a poset,

a

A

.

Quasicomplement

of

a

is

a

=

[

{

c

A

|

c

a

}

.

Definition 67

Let

A

be a poset,

a

A

.

Dual quasicomplement

of

a

is

a

+

=

\

{

c

A

|

c

a

}

.

I will denote quasicomplement and dual quasicomplement for a specific poset

A

as

a

(

A

)

and

a

+(

A

)

.

Definition 68

Let

a, b

A

where

A

is a distributive lattice.

Quasidifference

of

a

and

b

is

a

\

b

=

\

{

z

A

|

a

b

z

}

.

Definition 69

Let

a, b

A

where

A

is a distributive lattice.

Second quasid-

ifference

of

a

and

b

is

a

#

b

def

=

[

{

z

A

|

z

a

z

b

}

.

43