Definition B.1 (Partial Ordering) A binary relation ≼ on a nonempty set X is called partial ordering if the following conditions hold:

(i) For all a ∈ X, a ≼ a; (Reflexivity)

(ii) If a ≼ b and b ≼ a, then a = b; (Antisymmetry)

(iii) If a ≼ b and b ≼ c, then a ≼ c. (Transitivity)

Definition B.2 (Partially Ordered Set) A nonempty set X is called partially ordered if there is a partial ordering on X.

Remark B.1 The word ‘partially’ emphasizes that X may contain elements a and b for which neither a ≼ b nor b ≼ a holds. In this case, a and b are called incomparable elements. If a ≼ b or b ≼ a (or both), then a and b are called comparable elements.

Definition B.3 (Totally Ordered Set) A partially ordered set X is said to be totally ordered if every two elements of X are comparable.

In other word, partially ordered set X is totally ordered if it has no incomparable elements.

Definition B.4 Let M be a nonempty subset of a partially ordered set X.

(a) The element x ∈ X is called an upper bound of M if a ≼ x for all a ∈ M.

(b) The element x ∈ X is called a lower bound of M if x ≼ a for all a ∈ M.

(c) The element x ∈ M is called a maximal element of M if x ≼ a implies x = a.

Remark B.2 A subset of a partially ordered set M may or may not have an upper or lower bound. Also, M may or may not have maximal elements. Note that a maximal element need not be an upper bound.

Example B.1 (a) Let X be the set of all real numbers and let a ≤ b have its usual meaning. Then, X is totally ordered and it has no maximal element.