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Minimal and Maximal Element

Definition: If x is an element of S such that no element of S is smaller than x, then x is called a minimal element of S.

x is minimal element of S w.r.t. <= : <=>

x in S /\  forall y in S: y <= x => y = x.

If x is an element of S such that no element of S is greater than x, then x is called a maximal element of S.

x is maximal element of S w.r.t. <= : <=>

x in S /\  forall y in S: x <= y => x = y.

Minimal and maximal elements are not necessarily unique.