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where the injective mapping
inr
X
represents the composition of terms with variables
from the more simple terms, for each two terms
t,t
∈
T
P
X
, as follows:
1.
inr
X
(t,i)
=
o
i
(t)
∈
T
P
X
, for each unary project/select operator
o
i
∈
Σ
R
,i
≥
1;
2.
inr
X
((t,t
),
t
TIMES
t
∈
T
P
X
, for Cartesian product;
−
1
)
=
3.
inr
X
((t,t
),
0
)
t
UNION
t
.
The mapping
h
A
:
Σ
P
(T A)
→
TA
is defined in Example
29
, for each unary (or
binary) operator
o
i
∈
=
Σ
P
=
Σ
R
and the relations
α
#
(t),α
#
(t
1
),α
#
(t
2
)
∈
TA
, by:
1.
h
A
(α
#
(t),i)
=
o
i
(t)
#
, where
o
i
(t)
∈
T
P
X
is a term and evaluation of terms
_
#
:
T
P
R→
Υ
in Definition
31
(R-algebra
α
is a restriction of
_
to
X
⊆R
).
−
=
TIMES
t
2
#
=
2.
h
A
((α
#
(t
1
),α
#
(t
2
)),
1
)
t
1
and
h
A
((α
#
(t
1
),α
#
(t
2
)),
0
)
otherwise.
Thus,
α
#
is the unique epic inductive extension of
h
A
along the assignment
t
1
UNION
t
2
#
if
α
#
(t
1
)
and
α
#
(t
2
)
are union compatible;
⊥
:
→
⊆
⊆
T
P
X
(t
he
α
X
A
TA
(for each
A
,
A
TA
) to all terms with variables in
⊆
T
P
X
) with
α
#
(r
WHERE 1
.
m
apping
α
#
is equal to
α
for the variables in
X
=
1
)
=
α(r)
∈
A
⊆
TA
(because we have the identity unary operation “_ WHERE
1
.
=
1” (point 4.1 in Sect.
5.1
)).
Notice that in the initial diagram above, in the case when
A
=
Υ
⊂
Υ
with
T
Υ
=
Υ
and
X
=R
, the R-algebra
α
corresponds to the evaluation
_
in Definition
31
.
α
∗
(
This initial semantics, which maps each pair
(
A
,α)
with
A
=
A
)
into the set
of views
TA
, algebraically defines the power-view operator
T
:
Ob
DB
→
Ob
DB
,
which is then extended to the fundamental power-view endofunctor
T
:
DB
→
DB
in Sect.
3.2.1
.
5.2
Action-Relational-Algebra RA Category
Action-based categories, for a given
Σ
-algebra, can be obtained if each arrow is a
unary operation of this algebra and the target object of such an arrow is equal to the
term of this algebra, obtained by applying this unary operation to the source object
(which is another term of the same algebra) of this arrow. Hence, the set of objects
in such a category is equal to the set of terms of this algebra, and for each arrow
its target object is obtained from its source object by applying the action (defined
by this arrow) to this source object. In this case, we have to transform the
n
-ary
operators,
n
2, of this algebra into the set of unary operators, as follows in this
simple example:
≥
