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operator in
H(W)
). In fact,
H(W)
. Consequently, we
have the following surjective homomorphism between lattices,
(e)
Γ
:
(H(W),
⊆
,
∩
,
∪
,
{
{
0
,
⊥}∪{
0
,
}={
0
,
⊥
,
}∈
0
}
,W)
→
(
F
(W),
⊆
,
∩
,
,
{
0
}
,W)
,
such that
Γ(
{
0
}
)
={
0
}
(bottom element in
F
(W)
),
Γ(
{
0
,
⊥}
)
={
0
,
⊥}
,
Γ(
{
0
,
}
)
=
{
0
,
}
, and
Γ(
{
0
,
⊥
,
}
)
=
Γ(W)
=
W
. We have also that for any two hereditary
sets
S
1
,S
2
∈
H(W)
,
•
Γ(
∩
(S
1
,S
2
))
=
(Γ
∩
)(Γ (S
1
),Γ (S
2
))
=∩
(Γ (S
1
),Γ (S
2
))
, and
•
(Γ (S
1
),Γ (S
2
))
,
and hence
Γ
is a homomorphism between these two lattices.
Γ(
∪
(S
1
,S
2
))
=
(Γ
∪
)(Γ (S
1
),Γ (S
2
))
=
It is easy to verify that the join operator
, in the compact set-based representa-
tion
)
is
a total ordering (as, for example, in the fuzzy logic with the closed interval of reals
W
=[
F
(W)
, reduces to the standard set union
∪
when the complete lattice
(W,
≤
for the set of truth values).
Let us extend Example
43
to relative pseudo-complements, so that for two el-
ements (sets) in
H(W)
,
S
1
=
0
,
1
]
,
S
1
⇒
h
S
2
=
{
W
and
S
2
={
0
,
⊥
,
}
Z
∈
H(W)
|
S
2
}=
{
Z
∩
S
2
.
Let us show that
Γ(S
1
⇒
h
S
2
)
=
(Γ S
1
)
⇒
(Γ S
2
)
. In fact,
Γ(S
1
⇒
h
S
2
)
=
S
1
⊆
Z
∈
H(W)
|
Z
⊆
S
2
}={
0
,
⊥
,
}=
Γ
{
0
,
⊥
,
}=
W
, while
⇒
Γ
}
=
(Γ S
1
)
⇒
(Γ S
2
)
=
W
{
0
,
⊥
,
W
⇒
W
=
Γ(W
⇒
h
W)
Γ
Z
W
=
∈
H(W)
|
Z
∩
W
⊆
Γ
Z
W
=
=
∈
H(W)
|
Z
⊆
ΓW
=
W.
We can generalize these results as follows:
Corollary 34
There is the following homomorphism between the distributive com-
plete
(
r
.
p
.
c
.)
lattice (H(W),
⊆
∩
∪
{
}
≤
,
,
,
0
,W) of hereditary subsets in (W,
)
,
and
its “closed” version complete lattice
:
1.
Γ
:
(H(W),
⊆
,
∩
,
∪
,
{
0
}
,W)
→
(
F
(W),
⊆
,Γ
∩
,Γ
∪
,
{
0
}
,W)
,
where Γ
∩=∩
(W)
.
Thus
,
we can define the following Heyting algebra of closed hereditary subsets
:
2.
F
(W)
and Γ
∪=
are the meet and join lattice operators in
F
,
¬
and
¬
are
(
relative
)
pseudo-complements in this r
.
p
.
c
.
lattice
,
and hence there is the following ho-
momorphisms of Heyting algebras
H
(W)
=
(
F
(W),
⊆
,
∩
,Γ
∪
,
⇒
,
{
0
}
,W)
,
where
⇒
=
(H(W),
⊆
,
∩
,
∪
,
⇒
h
,
¬
h
,
{
0
}
,W)
and
F
(W)
:
3.
Γ
:
→
F
(W)
.
Proof
By composition of the homomorphism
:
H
(W)
(H(W),
⊆
,
∩
,
∪
,
{
0
}
,W)
→
(W,
≤
,
∧
,
∨
,
0
,
1
)
and homomorphism
↓
in Proposition
68
, we obtain the homo-
morphism
→
F
,W
,
Γ
=↓
:
H
(W)
(W),
⊆
,Γ
∩
,Γ
∪
,
{
0
}
with
{
0
}=
Γ(
{
0
}
)
,
W
=
Γ(W)
, and, for any two
S
1
,S
2
∈
H(W)
,
