504 A Law and order
Example A.7 (Inverse relation).
The smaller-equal-relation ≤ is in the space of real numbers a reflexive partial order. Its inverse
relation is the inverse relation ≥, again a reflexive partial order.
End of Example.
Definition A.5 (Irreflexive partial order).
Let M be a non-empty set. The binary relation R 1 on M is called irreflexive partial order if for all
x, y, z ∈ M the two conditions
(i) −x < x (x < x is not true)
(irreflexivity) ,
(ii) if x < y and y < 2, then x < 2
(transitivity)
(A.19)
are fulfilled.
End of Definition.
Example A.7 (Inverse relation).
The smaller-equal-relation ≤ is in the space of real numbers a reflexive partial order. Its inverse
relation is the inverse relation ≥, again a reflexive partial order.
End of Example.
Definition A.5 (Irreflexive partial order).
Let M be a non-empty set. The binary relation R 1 on M is called irreflexive partial order if for all
x, y, z ∈ M the two conditions
(i) −x < x (x < x is not true)
(irreflexivity) ,
(ii) if x < y and y < 2, then x < 2
(transitivity)
(A.19)
are fulfilled.
End of Definition.