502 A Law and order
M 8
M 5
M 2
M 6
M 1
M 3
M 7
M 4
Fig. A.9. Hasse diagram for power(A), |A| = 3, |power(A)| = 8.
A-2 Law and order: Fibering
For a set system (which we ex perienced by power( A), for instance) M = {A 1 , A 2 , ..., A n }, we call
∩ M = ∩
n
i=1 A i and ∪ M = ∪
n
i=1 A i intersection and union of the set system. The inv erse operation
of the union of a set system, namely the partioning or fibering of a set system into specific subsets is
giv en by the following definition.
Definition A.4 (Fibering).
A set system M = {M 1 , M 2 , . . . , M n } (n ∈ N
∗ ) of subsets M 1 , . . . , M n is called a partioning or a
fibering of M if and only if
(i) M i = ∅ for any i ∈ {1, 2, ..., n} ,
(ii) M i ∩ M j = ∅ for any i, j ∈ {1, 2, ..., n} ,
(iii) M = M 1 ∪ M 2 ∪ . . . ∪ M n = ∪M = ∪
n
i=1 M i
holds. These subsets of M , the elements of M, are called fibres of M or of M, respectiv ely.
End of Definition.
I
n other words, M 1 , . . . , M n are non-empty subsets of M , their paired intersection M i ∩ M j is the
empty set and their ordered union is M again.
axiom
analysis
algorithm
algebra
absolute value
abacus
Fig. A.10. Hasse diagram, lexicographic order.