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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 1 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
1
1
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 1 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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