Definition:Smallest/Ordered Set

Definition

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

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

Also see

• Smallest Element is Unique

• Greatest element

• Maximal element
• Minimal element

• Supremum of an Ordered Set
• Infimum of an Ordered Set

Sources

• Paul R. Halmos: Naive Set Theory (1960)... (previous)... (next): $\S 14$: Order
• Seth Warner: Modern Algebra (1965)... (previous)... (next): $\S 14$
• A.N. Kolmogorov and S.V. Fomin‎: Introductory Real Analysis (1968): $\S 3.5$
• T.S. Blyth: Set Theory and Abstract Algebra (1975): $\S 7$
• K.G. Binmore: Mathematical Analysis: A Straightforward Approach (1977)... (previous)... (next): $\S 2.7$