1.11. UNUSUAL NOTATION

14

that

b

u

c

=

and

b

t

c

=

a

. We will prove that

b

is complementive to

a

iff

b

is

substractive from

a

and

b

v

a

.

Definition

6

.

Call

a

and

b

of a poset

A

intersecting

, denoted

a

6

b

, when

there exists a non-least element

c

such that

c

v

a

c

v

b

.

Definition

7

.

a

b

def

=

¬

(

a

6

b

).

Definition

8

.

I call elements

a

and

b

of a poset

A

joining

and denote

a

b

when there are no non-greatest element

c

such that

c

w

a

c

w

b

.

Definition

9

.

a

6≡

b

def

=

¬

(

a

b

).

Obvious

10

.

a

6

b

iff

a

u

b

is non-least, for every elements

a

,

b

of a meet-

semilattice.

Obvious

11

.

a

b

iff

a

t

b

is the greatest element, for every elements

a

,

b

of

a join-semilattice.

I extend the definitions of pseudocomplement and dual pseudocomplement to

arbitrary posets (not just lattices as it is customary):

Definition

12

.

Let

A

be a poset.

Pseudocomplement

of

a

is

max

c

A

c

a

.

If

z

is the pseudocomplement of

a

we will denote

z

=

a

.

Definition

13

.

Let

A

be a poset.

Dual pseudocomplement

of

a

is

min

c

A

c

a

.

If

z

is the dual pseudocomplement of

a

we will denote

z

=

a

+

.