Definition:Minimal Element/Comparison with Smallest Element

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

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


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

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

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


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

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

If all elements are comparable to $x$, 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.