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For (
J
P,M
, ∨, ∧
), the properties (1) and (2) of the lemma are verified (triv-
ial). Let
J
i
,
J
j
and
J
k
be three join points and
μ
i
=
<μ
i
0
,μ
i
1
,μ
i
2
>
,
μ
j
=
<
μ
j
0
,μ
j
1
,μ
j
2
>
and
μ
k
=
<μ
k
0
,μ
k
1
,μ
k
2
>
the three isomorphisms associating
P
J
i
,
J
j
and
J
k
.Let
e
f
M
e ≺ f
respectively
to
and
be two events of
such that
.
e
f
J
i
and (
J
i
∨ J
j
)
∨ J
k
,let
e
be the corresponding event
If
and
belong to
e
), then according to the definition of
P
e
μ
i
1
(
∨
e
in
such that
=
,
succeeds to
e
)and
e
). Therefore,
μ
j
1
(
μ
k
1
(
e
and
f
also belong to
J
i
∨
(
J
j
∨ J
k
). In this way,
we easily show that (
J
i
∨ J
j
)
∨ J
k
=
J
i
∨
(
J
j
∨ J
k
). In the same way, we also
show that (
J
i
∧ J
j
)
∧ J
k
=
J
i
∧
(
J
j
∧ J
k
). Finally, to prove the property (4),
let us consider the two join points
J
i
and
J
j
and their associated morphisms
μ
i
and
μ
j
.Let
e
2
and
f
2
be two events belonging to
J
j
and
J
i
∨
(
J
i
∧ J
j
)(and
e
the event belonging to
consequently to
J
i
∧ J
j
) but not to
J
i
. Let us note
P
e
). If
e
), then since
such that
e
2
=
μ
j
1
(
e
1
=
μ
i
1
(
e
2
belongs to
J
i
∧ J
j
,
e
2
≤ e
1
,
and since
e
2
belongs to
J
i
∨
(
J
i
∧J
j
),
e
1
≤ e
2
. Impossible, therefore all the events
of
J
i
∨
J
i
.
Accordingtothelemma,(
(
J
i
∧ J
j
)belongto
J
P,M
,
), with
defined by
J
i
J
j
if
J
i
∨ J
j
=
is equivalent to the order
J
j
, is a lattice. Moreover
of Definition 10. The
equivalence is easy to demonstrate. Let
J
i
and
J
j
be two join points, and
μ
i
and
J
i
J
j
, by definition
μ
j
their associated isomorphisms to
P
.If
J
i
∨ J
j
=
J
j
,
and thus all the message send events of
J
j
succeed those of
J
i
. The converse is
trivial.