Definitions 1: A binary relation on set is called a partial order / partial ordering on if it is:

1. REFLECTION:

if the relation R on A is called reflexive properties, if every element of A associated with him.

For each a∈A, (a, a) ∈A

or

For each a ∈A, aRa

2. ANTISIMETRY:

for each different assumptions a and b), the relation of this kind is called an anti-symmetric relation.

For each a, b∈A, a =/ b -> ((a, b) ∈R -> (b, a) ∈ / R)

or

For each a, b∈A, a =/ b -> (aRb -> ~ (bRa))

In most literature it is usually written as contraption as below. The advantage of this form is that it does not contain negation, and contains only one implication.

For each a, b∈A, (a, b) ∈R & (b, a) ∈R -> (a = b)

or

For every a, b∈A, aRb & bRa -> (a = b)

3. TRANSITIVE:

the relation is called transitive if it has properties, if a is related to b, and b associated with c, then a is related to c directly.

(a, b) ∈R & (b, c) ∈R -> (a, c) ∈R

or

for every a, b, c∈A, aRb & bRc -> aRc

Partial Order relation example

1. The relation “less than equal to” the Z
2. Relations “divide” on Z
3. Relation “subset” on Powerset (set power)
4. Relations “more than equal to” the Z

DEFINITION 2: Suppose (S, ≤) is POSET, a and b∈S, then:

a and b are comparable if a ≤b or b≤a.

a and b noncomparable if a≰ b and b≰ a.

DEFINITION 3: A partial sequence ≤ in a set is called a total order or linear order / linear order, if applicable: ∀ x, y ∈S | x ≤y or y ≤x, meaning that each pair is comparable. Pair (S, ≤) is called a linearly ordered set or a chain / CHAIN.

DEFINITION 4: If (S, ≤) is a POSET, A⊆S and A ≠ ∅, then:

a ∈A is called a minimal element of A if: there is no x∈A such that x≤a

a∈A is called the maximum element of A if: there is no x∈A such that a≤x

a∈A is called an element excluded from A if: a≤x, ∀x ∈A

a∈A is called the largest element of A if: x≤a, ∀x ∈A

b∈S is called the lower limit of A if: b≤x, ∀x ∈A

b∈S is called the upper limit of A if: x ≤b, ∀x ∈A

b∈S is called the biggest / infimum lower limit of A if: for every c which is another lower limit of A, applies c≤b.

b∈S is called the smallest / supremum upper limit of A if: for every c which is another upper limit of A, b≤c applies