Ordering is Equivalent to Subset Relation/Proof 1

Theorem

Let $\struct {S, \preceq}$ be an ordered set.

Then there exists a set $\mathbb S$ of subsets of $S$ such that:

$\struct {S, \preceq} \cong \struct {\mathbb S, \subseteq}$

where:

$\struct {\mathbb S, \subseteq}$ is the relational structure consisting of $\mathbb S$ and the subset relation

$\cong$ denotes order isomorphism.

Hence any ordering on a set can be modelled uniquely by a set of subsets of that set under the subset relation.

Specifically:

Let

$\mathbb S := \set {a^\preceq: a \in S}$

where $a^\preceq$ is the lower closure of $a$.

That is:

$a^\preceq := \set {b \in S: b \preceq a}$

Let the mapping $\phi: S \to \mathbb S$ be defined as:

$\map \phi a = a^\preceq$

Then $\phi$ is an order isomorphism from $\struct {S, \preceq}$ to $\struct {\mathbb S, \subseteq}$.

Proof

From Subset Relation is Ordering, we have that $\struct {\mathbb S, \subseteq}$ is an ordered set.

Then let $T$ be defined as:

$T := \set {a^\prec: a \in S}$

Let the mapping $\phi: S \to T$ be defined as:

$\map \phi a = a^\prec$

We are to show that $\phi$ is an order isomorphism.

$\phi$ is clearly surjective, as every $a^\prec$ is defined from some $a \in S$.

Now suppose $x^\prec, y^\prec \in T: x^\prec = y^\prec$.

Then:

$\set {b \in S: b \preceq x} = \set {b \in S: b \preceq y}$

We have that:

$x \in x^\prec = y^\prec$ and $y \in y^\prec = x^\prec$

which means:

$x \preceq y$ and $y \preceq x$

So as an ordering is antisymmetric, we have $x = y$ and so $\phi$ is injective.

Hence by definition, $\phi$ is a bijection.

Now let $a_1 \preceq a_2$.

Then by definition:

$a_1 \in {a_2}^\prec$

Let $a_3 \in {a_1}^\prec$.

Then by definition:

$a_3 \preceq a_1$

As an ordering is transitive, it follows that:

$a_3 \preceq a_2$

and so:

$a_3 \in {a_2}^\prec$

So by definition of a subset:

${a_1}^\prec \subseteq {a_2}^\prec$

Therefore, $\phi$ is order-preserving

Conversely, suppose that ${a_1}^\prec \subseteq {a_2}^\prec$.

Then, since $a_1 \in {a_1}^\prec$, also $a_1 \in {a_2}^\prec$ by definition of subset.

By definition of ${a_2}^\prec$:

$a_1 \preceq a_2$

Hence it is seen that $\phi^{-1}$ is also order-preserving.

Thus it follows that $\phi$ is an order isomorphism between $\struct {S, \preceq}$ and $\struct {\mathbb S, \subseteq}$.

$\blacksquare$

Sources

• 1993: Keith Devlin: The Joy of Sets: Fundamentals of Contemporary Set Theory (2nd ed.) ... (previous) ... (next): $\S 1$: Naive Set Theory: $\S 1.5$: Relations: Theorem $1.5.2$