Definition:Minimal/Element

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

Let $T \subseteq S$ be a subset of $S$.

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


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

That is, the only element of $T$ that $x$ succeeds or is equal to is itself.

Alternatively, this can be put as:

$x \in T$ is a minimal element of $T$ iff:
 * $\neg \exists y \in T: y \prec x$

where $y \prec x$ denotes that $y \preceq x \land y \ne x$.

Comparison with Smallest Element
Compare this definition with that for a smallest element.

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


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

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

Note that when an ordered set is in fact a totally ordered set, the terms minimal element and smallest element are equivalent.

Also defined as
Some treatments of this subject do not restrict the relation $\preceq$ to the requirement that it be an ordering.

Also known as
Some sources refer to these as atoms.

Also see

 * Definition:Maximal Element