Definitions for "Partial order"
A way of providing an order for a set of elements such that some elements may not be either greater or less than each other, hence the ordering is not total. Consider the trigrams ordered by degree of yangness. Clearly 101 has more yang than 100, but both 101 and 110 have the same amount of yang so neither is greater or less than the other.
Definition A partial order is a relation between elements of a set. That is, it defines relationships between all of the elements of a set, and these relationships obey three axioms (given below). Example For example, "≤" is a partial order over the natural numbers. For example: 2 ≤ 3 3 ≤ 4 3 ≤ 6 2 ≤ 2 Axioms Thus a partial order over a set S defines relationships between all of the elements of S, and these relationships all obey the following three axioms or rules: 1. Reflexivity For all elements A of S, it is true that: A ≤ A For example: 2 ≤ 2, 3 ≤ 3, 4 ≤ 4, ... etc. 2. Transitivity For all elements A, B, and C in S, this rule allows us to infer that: if A ≤ B and B ≤ C then A ≤ C For example, since 2 ≤ 3, and 3 ≤ 6, it must also be true that 2 ≤ 6. 3. Antisymmetry For all elements A and B of S, it is true that: if A ≤ B and B ≤ A then A = B For example, since "2 ≤ 2" and "2 ≥ 2", it must be true that "2=2".
An ordering relation on SxS, where S is some set.