A-2 Law and order: Fibering 503
Example A.5 (Fibering).
Let M = N
∗ , M 1 = {1}, M 2 = P (set of prime numbers) and M 3 =
x | ab = x for any a ∈ P and for
b ∈ N
∗
\ {1}
the set of compound numbers. Then
M = {M 1 , M 2 , M 3 }
(A.14)
is a partitioning or a fibering of M . For instance,
M 1 = {1} ,
M 2 = {2, 3, 5, 7, 11, 13, 17, . . .} ,
M 3 = {4, 6, 8, 9, 10, 12, 14, . . .}
(A.15)
fulfills
(i) M 0 = ∅ ,
(ii) M i ∩ M j = ∅ (i, j ∈ {1, 2, 3}, i = j) ,
(iii) M = M 1 ∪ M 2 ∪ M 3 = N
∗ .
(A.16)
End of Example.
Exercise A.3 (Fibering).
Find three fibres of the set of
M = {M 0 , M 1 , M 2 } .
(A.17)
Answer:
M 0 := 3Z :=
x | 3y = x and y ∈ Z
=
= {. . . , −6, −3, 0, 3, 6, . . .} ,
M 1 := 3Z + 1 :=
x | 3y + 1 = x and y ∈ Z
=
= {. . . , −5, −2, 1, 4, 7, . . .} ,
M 2 := 3Z + 2 :=
x | 3y + 2 = x and y ∈ Z
=
= {. . . , −4, −1, 2, 5, 8, . . .} .
(A.18)
End of Exercise.
Example A.6 (Hasse diagram, lexicographic order).
The Hasse diagram of a lexicographic order in a set of words which begin with the initial letter “a”
is called a chain. That is, all elements are ordered along a half line or line; there are no bifurcations:
compare with Fig. A.10.
End of Example.