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Fig. 2.18
Schema of the
Cartesian product in the
category of algorithms
B=(
B
,
ʲ
)
˕
2
˕
i
˕
n
˕
1
A
1
A
2
A
i
=(
A
i
,
ʱ
i
)
n
A
=
(
A
,ʱ)
∈
obj
(
Al
g)
(2.47)
A
−
set
,
A
=∅
,ʱ
:
A
ₒ
A
The function
ʱ
corresponding to the function
F
is a partial function. However,
the function
ˆ
which is a morphism in the category
Al
g
(we denote as
ˆ
∈
˃(
Al
g)
)
may be defined as follows:
A
∈
obj
(
Al
g),
C
∈
obj
(
Al
g),
A
=
(
A
,ʱ),
C
=
(
C
,ʳ)
ˆ
:
A
ₒ
C
as that
ʳ(ˆ(
a
))
=
ˆ(ʱ(
a
))
and
a
,
∈
A
Summing up, the category
Al
g
consists of the set of objects
obj
(
Al
g)
and a set of
morphisms
as defined above.
The Cartesian product of objects that belongs to the category
Al
g
above the set of
indices
I
˃(
Al
g)
={
,
,...,
,...,
}
1
2
i
n
may be defined as follows (Fig.
2.18
):
B
=
ʠ
i
∈
I
Ai
(2.48)
where:
B
∈
obj
(
Al
g),
B
=
(
B
,ʲ), ʲ
:
B
ₒ
B
,
∀
i
∈
I
A
i
∈
obj
(
Al
g),
A
i
=
(
A
i
,ʱ
i
),
ʱ
i
:
A
i
ₒ
A
i
∀
i
∈
I
ˆ
i
∈
˃(
B
,
A
i
)
i.e.
ˆ
i
is a morphism from
B
to
A
i
and
∀
b
∈
B
as that
∀
i
∈
I
ˆ
i
(
b
)
=
a
i
∈
A
i
,ˆ
i
(ʲ(
b
))
=
ʱ
i
(ˆ
i
(
b
))
∈
A
i
The application of this Cartesian product to a given algorithm (in the category of
algorithms) enables decomposition of the algorithm
Al
g
into component algorithms
Al
g
i
. Each of the component algorithms (
Al
g
i
) obtained in this way
is autonomous towards another from the rest of component algorithms (
Al
g
j
for
i
=
(
A
i
,ʱ
i
)
j
), which results from the definition of the Cartesian product in the category of
algorithms.
=
