Definition:Smallest Element

Definition
Let $\left({S, \preceq}\right)$ be a poset.

An element $x \in S$ is the smallest element (or minimum element) iff:


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

That is, $x$ precedes, or is equal to, every element of $S$.

The Smallest Element is Unique, so calling it the smallest element is justified.

Thus for an element to be the smallest element, all $y \in S$ must be comparable to $x$.

Comparison with Minimal Element
Compare this definition with that for a minimal element.

An element $x \in S$ is minimal iff:


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

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 a poset is in fact a totally ordered set, the terms minimal element and smallest element are equivalent.

Also see

 * Smallest Element is Unique


 * Greatest element


 * Maximal element
 * Minimal element


 * Supremum of an Ordered Set
 * Infimum of an Ordered Set