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where
Φ
is equal to SOtgd
Ψ
{∃
and
f
r
i
is the characteristic function of the relation
α(r
i
)
. Consequently, from the fact that
f
r
i
∀
x
i
((f
r
i
(
x
i
)
=
1
)
⇒
r
i
(
x
i
))
|
r
i
/
∈
S
}
the information flux of the component
{∃
f
r
i
∀
x
i
((f
r
i
(
x
i
)
=
1
)
⇒
r
i
(
x
i
))
|
r
i
/
∈
S
}
is empty, the information fluxes of
f
and
Tf
are equal, that is,
Tf
=
f
.
Thus,
Tf
=
f
in the observational sense of Definition
23
, while in the set-
theoretic sense we have that
f
⊆
Tf
.
Example 21
Let us consider the definition of this endofunctor for simple isomor-
phic arrows
is
A
:
TA
,
is
−
A
:
A
→
TA
→
A
where, from the proof of Proposition
8
,
is
−
A
=
is
A
=S
A
={
id
R
:
R
→
R
|
R
∈
A
}
.
Let us show that
T
1
(is
−
A
)
T
1
(is
A
)
=
=
is
TA
=
id
TA
:
TA
→
TA
.
In fact, from the definition of the isomorphic arrow
is
TA
:
TA
→
T(TA)
and
from the idempotence
TTA
=
TA
, we obtain
is
TA
= S
TA
={
id
R
:
R
→
R
|
R
∈
TA
. Analogously,
is
−
1
TA
}:
TA
→
TA
, that is,
is
TA
=
id
TA
:
TA
→
TA
=S
TA
=
{
id
R
:
R
→
R
|
R
∈
TA
}:
T(TA)
→
TA
, that is,
is
−
1
(a)
TA
=
is
TA
=
id
TA
:
TA
→
TA
.
=
From Theorem
2
above (here
B
TA
)wehave
(b)
T
1
(is
A
)
is
−
1
A
=
is
B
◦
is
A
◦
from
is
A
◦
id
TA
is
−
A
=
=
is
B
◦
id
TA
=
is
TA
◦
id
TA
=
is
TA
,
and (here we replace
A
by
TA
and
B
by
A
)
(c)
T
1
is
−
A
=
is
−
A
◦
is
−
1
TA
is
A
◦
from
is
A
◦
id
TA
is
−
A
=
is
−
1
is
−
1
=
id
TA
◦
TA
=
TA
.
Thus, from (a), (b), and (c) we obtain
(d)
T
1
(is
−
A
)
=
T
1
(is
A
)
=
is
−
1
TA
=
is
TA
=
id
TA
:
TA
→
TA
, where
id
TA
=
T
1
(id
A
)
.
The endofunctor
T
is a right and left adjoint to the identity functor
I
DB
, i.e.,
T
T,I
DB
,η
C
,η
where the unit
η
C
I
DB
. Thus, we have the equivalence adjunction
:
I
DB
(such that for any object
A
the arrow
η
A
η
C
(A)
is
−
A
:
T
=
TA
−→
A
) and
the counit
η
:
I
DB
T
(such that for any
A
the arrow
η
A
η(A)
=
is
A
:
A
−→
TA
)
are isomorphic arrows in
DB
(by duality Theorem
3
it holds that
η
C
η
OP
).
We have already explained that the views of a database may be seen as its
ob-
servable computations
: what we need, in order to obtain an expressive power of
computations in the category
DB
, are the categorial computational properties which
are, as it is well known [
17
], based on monads:
=
