The relation
is said to be a partial ordering on the set A iff it has the following
properties

1.

2.

The partial ordering is said to be a linear ordering iff either
or
for any two elements
of the set A.

Any linearly ordered subset C of a partially ordered set A is called a chain.
The chain C is said to be maximal if it is not a nontrivial subset of any
other chain.

For a chain C we define the upper bound
as an element with the property
for any
.

The element
of the partially ordered set A is called maximal iff

Axiom

(Axiom of choice). Let
be an arbitrary index set
.
Suppose that for any
a set
is given. Then there exists a map
acting on
such that
.

Lemma

(Hausdorff maximal principle). Every chain of a partially ordered set is
contained in some maximal chain.

Lemma

(Zorn maximal principle). If every chain of
a partially ordered set has a upper bound then there exists a maximal element.

Both versions of the maximal principle are consequences of the axiom of
choice.