maximal element (original) (raw)

Let ≤ be an ordering on a set S, and let A⊆S. Then, with respect to the ordering ≤,

• •
  a∈A is the least element of A if a≤x, for all x∈A.
• •
  a∈A is a minimal element of A if there exists no x∈A such that x≤a and x≠a.
• •
  a∈A is the greatest element of A if x≤a for all x∈A.
• •
  a∈A is a maximal element of A if there exists no x∈A such that a≤x and x≠a.

Examples.

• •
• •
  The negative integers ordered by the standard definition of ≤ have a maximal element which is also the greatest element, -1. They have no minimal or least element.
• •
  The natural numbers ℕ ordered by the standard ≤ have a least element, 1, which is also a minimal element. They have no greatest or maximal element.
• •
  The rationals greater than zero with the standard ordering ≤ have no least element or minimal element, and no maximal or greatest element.