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=↓
(S
1
∪
(
S
1
∨
S
2
)
(
from
↓=
((
↓
1.
Γ(S
1
∪
S
2
)
S
2
)
=↓
=
id
W
)
=↓
)
S
1
∨
(
↓
)
S
2
)
=
↓
↓
S
1
∪↓
S
2
)
(
=
Γ(ΓS
1
∪
ΓS
2
)
∈
F
(W)
.
Thus,
Γ
is a homomorphism for join operators, and, analogously,
2.
Γ(S
1
∩
=
Γ(ΓS
1
∩
=
S
2
)
ΓS
2
)
(from the fact that
Γ
is a closure operator)
(W)
.
Thus,
Γ
is a homomorphism for meet operators. For each
a
∈
W
,theset
=
ΓS
1
∩
ΓS
2
∈
F
↓
a
is
closed, so that
Γ(
↓
a)
=↓
a
, and for any
S
∈
H(W)
,
a,b
∈
W
,
3.
S
∩↓
a
⊆↓
b
iff
ΓS
∩↓
a
⊆↓
b
.
(the implication from right to left is obvious, based on the closure property of
Γ
,
S
⊆
ΓS
; for the implication from left to right, we can use the monotonicity
property of the closure operator
Γ
, and hence if
S
∩↓
a
⊆↓
b
then
Γ(S
∩
↓
a)
=
(
from 2
.)
=
ΓS
∩
Γ(
↓
a)
=
ΓS
∩↓
a
⊆
Γ(
↓
b)
=↓
b
.
Let us show that the complete lattice
(
,W)
is also dis-
tributive and hence an r.p.c. lattice, so that we can define the (relative) pseudo-
complements
F
(W),
⊆
,Γ
∩
,Γ
∪
,
{
0
}
and
¬
⇒
. In fact, for any
S
1
=
ΓS
1
=↓
a,S
2
=
ΓS
2
=↓
b,S
3
=
ΓS
3
∈
F
(W)
⊆
H(W)
,
4.
Γ
(Γ S
1
∩
ΓS
3
Γ
∪
(S
1
∩
S
2
,S
3
)
=
Γ
∪
(Γ S
1
∩
ΓS
2
,ΓS
3
)
=
ΓS
2
)
∪
(by the distributivity of
∩
and
∪
)
ΓS
3
)
(from the homomorphism property 1.)
Γ
(Γ S
1
∪
=
ΓS
3
)
∩
(Γ S
2
∪
=
Γ(ΓS
1
∪
∩
Γ(ΓS
2
∪
ΓS
3
)
ΓS
3
)
=
Γ(S
1
∪
S
3
)
∩
Γ(S
2
∪
S
3
)
=
Γ
∪
(S
1
,S
3
)
∩
Γ
∪
(S
2
,S
3
).
5.
Γ
∪
(S
1
,S
2
)
∩
S
3
=
Γ
∪
(S
1
,S
2
)
∩
ΓS
3
=
Γ(S
1
∪
S
2
)
∩
ΓS
3
(from the homomorphism property 1.)
=
Γ
(S
1
∪
S
2
)
∩
S
3
(by the distributivity of
∩
and
∪
)
Γ
(S
1
∩
S
3
)
=
∩
(S
1
∩
S
3
)
.
=
S
3
)
∪
(S
2
∩
Γ
S
3
),(S
2
∩
From the fact that the complete distributive lattice of hereditary subsets
H
(W)
sat-
isfies the infinite distributivity law (so it is a Heyting algebra), from this homomor-
phism (an analogous proof as for points 3 and 4), also the complete distributive
lattice
(
)
of closed hereditary subsets satisfies the infinite distributive law,
and, consequently, it is a Heyting algebra with:
6.
F
(W),
⊆
∪
↓
b
(
↓
a)
⇒
(
↓
b)
=
Γ
c
∈
F
(W)
|↓
c
∩↓
a
⊆↓
