Definition:Minimal Element/Comparison with Smallest Element

Definition
Compare the definition of minimal element with that of a smallest element.

Consider the ordered set $\struct {S, \preceq}$ such that $T \subseteq S$.

An element $x \in T$ is the smallest element of $T$ :


 * $\forall y \in T: x \preceq y$

That is, $x$ is comparable with, and precedes, or is equal to, every $y \in T$.

An element $x \in T$ is a minimal element of $T$ :


 * $y \preceq x \implies x = y$

That is, $x$ precedes, or is equal to, every $y \in T$ which is comparable with $x$.

If all elements are comparable with $x$, then such a minimal element is indeed the smallest element.

Note that when an ordered set is in fact a totally ordered set, the terms minimal element and smallest element are equivalent.