\begin{definition}[Definition:Minimal Element/Comparison with Smallest Element]
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 $T$ {{iff}}:
:$\forall y \in T: x \preceq y$
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 $T$ {{iff}}:
:$y \preceq x \implies x = y$
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.
\end{definition}