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2.1. Intersecting and joining elements

Let

A

be a poset.

Definition 1

I will call elements

a

and

b

of

A

intersecting

and denote

a

6≍

b

when exists not least element

c

such that

c

a

c

b

.

Definition 2

a

b

def

=

¬

(

a

6≍

b

)

.

Obvious 1

If

A

is a meet-semilattice then

a

6≍

b

iff

a

b

is non-least.

Obvious 2

a

0

6≍

b

0

a

1

a

0

b

1

b

0

a

1

6≍

b

1

.

Definition 3

I will call elements

a

and

b

of

A

joining

and denote

a

b

when

not exists not greatest element

c

such that

c

a

c

b

.

Definition 4

a

6≡

b

def

=

¬

(

a

b

)

.

Obvious 3

Intersecting is the dual of non-joining.

Obvious 4

If

A

is a join-semilattice then

a

b

iff

a

b

is its greatest element.

Obvious 5

a

0

b

0

a

1

a

0

b

1

b

0

a

1

b

1

.

2.2. Atoms of a poset

Definition 5

An

atom

of the poset is an element which has no non-least subele-

ments.

Remark 1

This definition is valid even for posets without least element.

I will denote (atoms

A

a

) or just (atoms

a

) the set of atoms contained in

element

a

of a poset

A

.

Definition 6

A poset

A

is called

atomic

when

atoms

a

6

=

for every non-least

element

a

A

.

Definition 7

Atomistic poset

is such poset that

a

=

S

atoms

a

for every

element

a

of this poset.

Proposition 1

Let

A

be a poset. If

a

is an atom of

A

and

B

A

then

a

B

a

6≍

B

.

Proof

a

B

a

a

a

B

, thus

a

6≍

B

because

a

is not least.

a

6≍

B

implies existence of non-least element

x

such that

x

B

and

x

a

.

Because

a

is an atom, we have

x

=

a

. So

a

B

.

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