Database Reference
In-Depth Information
Example 31
∗}
for addition, substraction, and multiplication, respectively, and the unary operator
|
Consider the set of integers
I
with the set of binary operators
{+
,
−
,
_
|
(finding absolute value of integers).
The we define the set of only unary operators
Σ
={
+
−
∗
|
_
n,
_
n,
_
n
for all
≥
}∪{I
|
|}
finite positive integers
n
, as we have shown in Example
30
.
Hence, we define the set of ground terms
1
,
_
T
I
of this algebra as follows:
1. The constants in
are terms.
2. If
t
I
is a term and _
I
∈
Σ
a unary operator then
t
I
is a term.
So, for instance, when _
is equal to the unary operation _
+
4 and for a
T
I
.
Consequently, we can define an action-category for this
Σ
-algebra as follows:
1. The set of objects of this category is equal to the set of terms
constant
−
1, the expression
−
1
+
4 is a term in
T
I
.
2. The set of arrows of this category is equal to the set of unary operators in
Σ
.The
identity arrows are just the identity operators (_
+
0
,
_
−
0
,
_
∗
1).
Consequently, for a source object
t
I
∈
T
I
and an operation _
∈
Σ
, we have the ar-
row _
:
t
I
→
t
I
in this action-category as, for example, the arrow _
+
4
:
(
−
1
)
→
(
4
)
. Notice that in such an action-category the
nidification
of (unary) operators
in a given term can be simply represented as a
composition of arrows
. For instance,
the term
(((
−
1
+
6, can be equivalently represented as the following
composition of three arrows: _
−
1
)
+
4
)
−
3
)
∗
∗
6
◦
_
−
3
◦
_
+
4
:
(
−
1
)
→
(((
−
1
)
+
4
)
−
3
)
∗
6.
For a given term
t
I
∈
T
I
we denote by
t
I
the evaluated value of this term: for
6. We say that two terms
t
I
,t
I
are
instance, let
t
I
be the term 4
−
5
+
7, then
t
I
=
t
I
equal if
t
I
=
, and hence the terms 4
−
5
+
7 and 9
−
3 are equal.
As we may see from this example, in an action-category the nidification of
(unary) operators of a given finite term of an algebra can be represented as a com-
position of finite number of arrows in the
same ordering
of the nidification of the
atomic operators of this algebra. Or, equivalently, the composition of the algebraic
operators is traduced into the composition of the arrows in such an action-category,
with the property that each target object of a given arrow is equal to the application
of the operator of this arrow to the source object of this arrow: the target object is
exactly the result of application of this unary operator to the source object of this
arrow.
Consequently, what we need in order to define an action-category
RA
for the
Σ
RE
-algebra introduced in Sect.
5.1
is to reduce all its binary operators to a unique
binary operator and by introducing the new auxiliary unary operators to replace the
eliminated binary operators. The natural choice for the unique binary operator is the
Cartesian product _ TIMES _, with the additional requirement that the updates in
any given term
t
R
are unfolded directly to the real relational tables.
The necessity to transform each term
t
R
of the
Σ
RE
-algebra into a composed
arrow (by using unary operators only) in
RA
means that we must unfold the
_ TIMES _ binary operations toward the source object of this composed object:
it means that at the end of these transformations, the operators _ TIMES _ appear
only at the most nidified level of each term, and consequently, the Cartesian products
will be executed only over (eventually updated) real relational tables. Consequently,
