Definition:Minimal/Ordered Set

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Definition

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

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


Definition 1

An element $x \in T$ is a minimal element of $T$ if and only if:

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


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


Definition 2

An element $x \in T$ is a minimal element of $T$ if and only if:

$\neg \exists y \in T: y \prec x$

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


That is, if and only if $x$ has no strict predecessor.


Comparison with Smallest Element

Compare the definition of minimal element with that of a smallest element.


An element $x \in T$ is the smallest element of $T$ if and only if:

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

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


An element $x \in S$ is a minimal element of $T$ if and only if:

$y \preceq x \implies x = y$

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 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