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2
≤
μ
1
(
(
i
)
∀
(
e, f
)
∈ E
,e≤ f ⇒ μ
1
(
e
)
f
)
2
≺
μ
1
(
(
ii
)
∀
(
e, f
)
∈ E
,e ≺ f ⇒ μ
1
(
e
)
f
)
φ
◦ μ
1
(
iii
)
μ
0
◦ φ
=
α
◦ μ
1
(
iv
)
μ
2
◦ α
=
Fig. 7.
Illustration of the notion morphism
) mean that by a bSD morphism the order and
the type of the events are preserved (the type of an event is preserved means
that, for instance, a sending event of
Note that properties (
i
)and(
ii
M
will be always associated with a sending
event of
M
). Note that property (
iii
) also means that all events located on a
single lifeline of
M
are sent by
μ
1
on a single lifeline of
M
.Figure7showsa
bSD morphism
f
=
<f
0
,f
1
,f
2
>
:
pointcut → M
2 where only the morphism
f
1
associating the events is represented (for instance, the event
ep
1
which represents
the sending of the message
m
1 is associated with the event
em
2
). Note that since
M
each event of a bSD is unique, a bSD morphism
f
from a bSD
M
to a bSD
M
.
always defines a unique part of
Definition 8 (bSD isomorphism).
A bSD morphism
μ
μ
0
,μ
1
,μ
2
)
from a
=(
M
is an isomorphism if the three morphisms
bSD
M
to a bSD
μ
0
,
μ
1
,and
μ
2
μ
−
1
=(
μ
−
1
0
,μ
−
1
1
,μ
−
2
)
is also a
are isomorphic and if the converse morphism
bSD morphism.
With this definition of isomorphism, we can define the notion of join point in a
general way:
M
be a
Definition 9 (join point).
Let
M
be a bSD and
P
be a pointcut. Let
M
is a join point if and only if there exists a bSD
part of
M
. We will say that
M
where the morphisms
isomorphism
μ
=(
μ
0
,μ
1
,μ
2
)
from
P
to
μ
0
and
μ
2
are
M
have the same objects and action names).
identity morphisms (
P
and
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