# Definition:Minimal/Element

## Definition

Let $\struct {S, \RR}$ be a relational structure.

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

### Definition 1

An element $x \in T$ is a **minimal element (under $\RR$) of $T$** if and only if:

- $\forall y \in T: y \mathrel \RR x \implies x = y$

### Definition 2

An element $x \in T$ is a **minimal element (under $\RR$) of $T$** if and only if:

- $\neg \exists y \in T: y \mathrel {\RR^\ne} x$

where $y \mathrel {\RR^\ne} x$ denotes that $y \mathrel \RR x$ but $y \ne x$.

## Comparison with Smallest Element

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

Consider the ordered set $\struct {S, \preceq}$ such that $T \subseteq S$.

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 with, and precedes, or is equal to, every $y \in T$.

An element $x \in T$ 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 $y \in T$ which *is* comparable with $x$.

If *all* elements are comparable with $x$, 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

Most treatments of the concept of a **minimal element** restrict the definition of the relation $\RR$ to the requirement that it be an ordering.

However, this is not strictly required, and this more general definition as used on $\mathsf{Pr} \infty \mathsf{fWiki}$ is of far more use.

## Also known as

Some sources refer to a **minimal element** as an **atom**.

However, the latter term has a meaning in several different contexts, so will not be used like this on $\mathsf{Pr} \infty \mathsf{fWiki}$.

## Examples

### Finite Subsets of Natural Numbers

Let $\FF$ denote the set of finite subsets of the natural numbers $\N$.

Consider the ordered set $\struct {\FF, \subseteq}$.

There is one minimal element of $\struct {\FF, \subseteq}$, and that is the empty set $\O$.

### Finite Subsets of Natural Numbers less Empty Set

Let $\FF$ denote the set of finite subsets of the natural numbers $\N$.

Let $\GG$ denote the set $\FF \setminus \set \O$, that is, $\FF$ with the empty set excluded.

Consider the ordered set $\struct {\GG, \subseteq}$.

The minimal elements of $\struct {\GG, \subseteq}$ are the sets of the form $\set n$, for $n \in \N$.

## Also see

- Results about
**minimal elements**can be found**here**.