Ordering is Equivalent to Subset Relation

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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.


Proof 1

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


For each $a \in S$, let $a^\prec$ be the lower closure of $a$:

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

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 $\struct {\mathbb S, \subseteq}$.

$\blacksquare$


Proof 2

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

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

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

By the Lemma, $\phi$ is order-preserving.


Suppose that ${a_1}^\preceq \subseteq {a_2}^\preceq$.

We have that:

$a_1 \in {a_1}^\preceq$

Thus by definition of subset:

$a_1 \in {a_2}^\preceq$

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

$a_1 \preceq a_2$

Thus $\phi$ is also order-reflecting.


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

$\blacksquare$