Definition:Smallest Element

Definition
Let $\left({S, \preceq}\right)$ be an ordered set.

An element $x \in S$ is the smallest element :


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

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.

The smallest element of $S$ is denoted $\min S$.

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 defined as
Some sources do not bother to define this concept on a general ordered set and instead apply it directly to a totally ordered set or even a well-ordered set.

Also known as
The smallest element of a set is also called:
 * The least element
 * The lowest element (particularly with numbers)
 * The first element
 * The minimum element (but beware confusing with minimal - see above)
 * The null element

Also see

 * Smallest Element is Unique


 * Definition:Greatest Element


 * Definition:Maximal Element
 * Definition:Minimal Element


 * Definition:Supremum of Set
 * Definition:Infimum of Set