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member-
collective
This is a parthood relation between a functional complex or a collective (as a
part) and a collective (as a whole). Examples include: (a) a tree is part of
forest; (b) a card is part of a deck of ca rds; (c) a club mem ber is part of a
club. We use the symbols
to represent the
shareable and non-shareable versions of this relation, respectively.
M
and
M
General Constraints
1. Weak Supplementation: Let T be a type whose instances are wholes and let {T 1 …T 2 }
be a set of types related to T via the subcollective-collective or member-collective
relations. Let lower Ci be the value of the minimum cardinality constraint of the
association end connected to C i in the aggregation relation. Then, we have that
n
(
=
lower Ci ) ≥ 2;
i 1
Constraints applied to the subcollective-collective relation
1. This relation only holds between collectives, i.e., they must be either stereotyped as
«collective» or be a subtype of a type stereotyped as «collective»;
2. Collectives are maximal entities. For this reason, it is not the case that a collective can
have as a part another collective of the same type (i.e., unified by the same relation). As a
consequence, these relations are irreflexive at the type level. In UML terms, the two
association ends of this relation must be connected to classes of different types (albeit
both stereotyped as «collective»);
3. Also because collectives are maximal entities, a collective can have at maximum one
subcollective of a given type. For this reason, the maximum cardinality constraint in the
association end connected to the part in this relation must be one (in UML terms,
self.target.upper = 1);
4. All subcollective-collective relations are relations of inseparable parthood . These
relations are marked with a tagged value {insperable} and the association end connected
to the whole must be immutable (in UML terms, self.source.readOnly = true);
5. This relation conforms to the axiomatization of Minimum Mereology (MM), i.e., it is an
Irreflexive, Asymmetric and Transitivity relation which obeys the Weak Supplementation
axiom. Moreover, if a collective W has one single direct subcollective W', then it must
have members which are disjoint from W';
Constraints applied to the member-collective relation
1. This relation can only represent essential parthood if the object representing the whole
on this relation is an extensional individual. In this case, all parthood relations in which
this individual participates as a whole are essential parthood relations. These relations
are marked with tagged value {essential} and the association end connected to the part
must be immutable (in UML terms, self.target.readOnly = true);
2. The class connected to association end relative to the whole individual must be a type
whose instances are collectives, i.e., they must be either stereotyped as «collective» or
be a subtype of a type stereotyped as «collective»;
3. This is an Irreflexive and Asymmetric relation which obeys the Weak Supplementation
axiom. However, it is also an Intransitive relation. Although transitivity does not hold
across two member-collective relations, a member-collective relation followed by
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