Database Reference
In-Depth Information
r
1
[
S
1
]
WHERE
C
1
TIMES
r
2
[
S
2
]
WHERE
C
2
REDUCE
S
2
TO
S
1
WHERE
C
5
[
S
5
]
TIMES
r
3
[
S
3
]
TIMES
(r
4
WHERE
C
4
)
S
4
]
[
REDUCE
S
4
TO
S
3
WHERE
(name
e)
[
S
6
]
a,name,e
=
DISJOINT
S
6
FROM
S
5
represented by the tree above (notice that with this reduction of the term-trees of the
Σ
RE
-algebra to the equivalent terms in
T
RA
X
, we replace the original binary nodes
with the 'TIMES' nodes and we introduce the new unary operations, and hence the
obtained tree is more complex than original tree).
Let us now define the category for this new action-relational algebra:
Proposition 21
We define the action-relational category
RA
as follows
:
1.
The objects of this category are
(
here '
\
' is the set difference
):
T
RA
X
⊂
1.1.
Ob
RA
,
for X
=R
;
Σ
RA
,
the
1.2.
For each object t
RA
∈
Ob
RA
and each unary operator o
i
∈
Σ
RA
\
Ob
RA
.
2.
The
atomic
arrows of this category are the set of all unary operators in
Σ
RA
\
term o
i
(t
RA
) is an object
,
i
.
e
.,
o
i
(t
RA
)
∈
Σ
RA
.
The category RA is skeletal
.
Proof
Notice that this is really an action-category, and hence for each unary opera-
tion
o
i
and term
t
RA
we have the arrow
o
i
:
t
RA
→
o
i
(t
RA
)
because
o
i
(t
RA
)
is a also
term in
T
RA
X
and hence an object in
RA
.
For any object
t
RA
∈
T
RA
X
) we have that its identity ar-
row
id
t
RA
is equa
l
to
t
he identity unary operation _ WHERE
C
w
he
n
C
is the
Ob
RA
(a term in
atomic conditi
on
1
.
(
_ WHERE 1
.
=
1 (a tautology), so that
id
t
RA
=
=
1
)
:
t
R
A
→
(t
RA
WHERE 1
.
1
)
, form the fact that the term (i.e., object)
(t
RA
WHERE 1
.
=
=
1
)
is equal to
t
RA
.
Given any two arrows
o
i
:
o
j
(o
i
(t
RA
))
, their
composition is still a unary operation
o
j
o
i
obtained by their composition, and
hence the composition of these two arrows is an arrow equal to
o
j
◦
t
RA
→
o
i
(t
RA
)
and
o
j
:
o
i
(t
RA
)
→
o
i
:
t
RA
→
o
j
(o
i
(t
RA
))
.
From the fact that the composition of unary operations is associative, the compo-
sition of arrows in
RA
satisfies the associativity property.
Two objects (terms)
t
RA
,t
RA
are isomorphic,
t
RA
t
RA
, if they are equal
t
RA
#
) and hence this cate-
(if for all assignments
_
:R →
Υ
,
t
RA
#
=
t
RA
t
RA
, an
d
th
e
iso arrows are
is
gory is sk
el
eta
l.
Thus,
t
RA
iff
t
RA
≈
=
(
_WHERE1
.
=
(
_ WHERE 1
.
1
)
:
t
RA
→
t
R
A
and
is
−
1
1
)
:
t
RA
→
t
RA
, so that
=
=
(
_ WHERE
(
1
.
(
1
.
(
_ WHERE 1
.
is
−
1
◦
is
=
=
1
)
∧
=
1
))
=
=
1
)
=
id
:
t
RA
→
t
RA
t
RA
→
t
RA
). Notice that
is
is well defined, that is,
is
−
1
(analogously,
is
◦
=
id
:
