Simple Order Product of Pair of Ordered Sets is Lattice iff Factors are Lattices

== Theorem ==
Let $\struct {S_1, \preccurlyeq_1}$ and $\struct {S_2, \preccurlyeq_2}$ be ordered sets.
Let $\struct {S_1, \preccurlyeq_1} \otimes^s \struct {S_2, \preccurlyeq_2}$ denote the simple (order) product of $\struct {S_1, \preccurlyeq_1}$ and $\struct {S_2, \preccurlyeq_2}$.
Then $\struct {S_1, \preccurlyeq_1} \otimes^s \struct {S_2, \preccurlyeq_2}$ is a lattice both $\struct {S_1, \preccurlyeq_1}$ and $\struct {S_2, \preccurlyeq_2}$ are lattices.
== Proof ==
Let $\struct {T, \preccurlyeq_s} := \struct {S_1, \preccurlyeq_1} \otimes^s \struct {S_2, \preccurlyeq_2}$.
First we note that from Simple Order Product of Pair of Ordered Sets is Ordered Set, $\struct {T, \preccurlyeq_s}$ is an ordered set.
=== Sufficient Condition ===
Let $\struct {T, \preccurlyeq_s}$ be a lattice.
Let $x_1$ and $y_1$ be arbitrary elements of $S_1$.
Let $x_2$ and $y_2$ be arbitrary elements of $S_2$.
Then we have that $\tuple {x_1, x_2}$ and $\tuple {y_1, y_2}$ are elements of $T$.
By definition of lattice, $\set {\tuple {x_1, x_2}, \tuple {y_1, y_2} }$ admits both a supremum and an infimum.
Let $\tuple {c_1, c_2} = \sup \set {\tuple {x_1, x_2}, \tuple {y_1, y_2} }$ be the supremum of $\set {\tuple {x_1, x_2}, \tuple {y_1, y_2} }$.
Then by definition:
:$\tuple {c_1, c_2}$ is an upper bound of $\set {\tuple {x_1, x_2}, \tuple {y_1, y_2} }$ in $T$
:$\tuple {c_1, c_2} \preccurlyeq_s \tuple {d_1, d_2}$ for all upper bounds $\tuple {d_1, d_2}$ of $\set {\tuple {x_1, x_2}, \tuple {y_1, y_2} }$ in $T$.
Let $\tuple {d_1, d_2}$ be an arbitrary upper bound of $\set {\tuple {x_1, x_2}, \tuple {y_1, y_2} }$ in $T$.
Then:
Similarly:
Thus:
:$d_1$ is an upper bound of $\set {x_1, y_1}$
and:
:$d_2$ is an upper bound of $\set {x_2, y_2}$.
As $\tuple {c_1, c_2}$ is also an upper bound of $\set {\tuple {x_1, x_2}, \tuple {y_1, y_2} }$, it similarly follows that:
:$c_1$ is an upper bound of $\set {x_1, y_1}$
and:
:$c_2$ is an upper bound of $\set {x_2, y_2}$.
Now we have:
Thus:
:$c_1$ is an upper bound of $\set {x_1, y_1}$
and:
:if $d_1$ is an upper bound of $\set {x_1, y_1}$, then $c_1 \preccurlyeq_1 d_1$
so $\set {x_1, y_1}$ admits a supremum $c_1$ in $\struct {S_1, \preccurlyeq_1}$.
Also:
:$c_2$ is an upper bound of $\set {x_2, y_2}$
and:
:if $d_2$ is an upper bound of $\set {x_2, y_2}$, then $c_2 \preccurlyeq_1 d_2$
so $\set {x_2, y_2}$ admits a supremum $c_2$ in $\struct {S_2, \preccurlyeq_2}$.
Now let $\tuple {c_1, c_2} = \inf \set {\tuple {x_1, x_2}, \tuple {y_1, y_2} }$ be the infimum of $\set {\tuple {x_1, x_2}, \tuple {y_1, y_2} }$.
We use a similar argument to the above, , to show that:
:$\set {x_1, y_1}$ admits an infimum $c_1$ in $\struct {S_1, \preccurlyeq_1}$
and:
:$\set {x_2, y_2}$ admits an infimum $c_2$ in $\struct {S_2, \preccurlyeq_2}$.
As $x_1$, $x_2$, $y_1$ and $y_2$ are arbitrary, it follows that both $\struct {S_1, \preccurlyeq_1}$ and $\struct {S_2, \preccurlyeq_2}$ are lattices.
=== Necessary Condition ===
Let $\struct {S_1, \preccurlyeq_1}$ and $\struct {S_2, \preccurlyeq_2}$ be lattices.
Let $\tuple {x_1, x_2}$ and $\tuple {y_1, y_2}$ be arbitrary elements of $T$.
Then:
:$x_1$ and $y_1$ are elements of $S_1$
:$x_2$ and $y_2$ are elements of $S_2$.
By definition of lattice, both $\set {x_1, y_1}$ and $\set {x_2, y_2}$ admit both a supremum and an infimum.
Let $c_1 = \sup \set {x_1, y_1}$ and $c_2 = \sup \set {x_2, y_2}$ be the suprema of $\set {x_1, y_1}$ and $\set {x_2, y_2}$ respectively.
Then by definition:
:$c_1$ is an upper bound of $\set {x_1, y_1}$ in $S_1$
:$c_1 \preccurlyeq_1 d_1$ for all upper bounds $d_1$ of $\set {x_1, y_1}$ in $S_1$
and:
:$c_2$ is an upper bound of $\set {x_2, y_2}$ in $S_2$
:$c_2 \preccurlyeq_2 d_2$ for all upper bounds $d_2$ of $\set {x_2, y_2}$ in $S_2$
Let:
:$d_1$ be an arbitrary upper bound of $\set {x_1, y_1}$ in $S_1$
:$d_2$ be an arbitrary upper bound of $\set {x_2, y_2}$ in $S_2$.
Similarly:
Thus:
:$\tuple {d_1, d_2}$ is an upper bound of $\set {\tuple {x_1, x_2}, \tuple {y_1, y_2} }$
As $c_1$ and $c_2$ are also upper bounds of of $\set {x_1, y_1}$ and $\set {x_2, y_2}$ respectively, it similarly follows that:
:$\tuple {c_1, c_2}$ is an upper bound of $\set {\tuple {x_1, x_2}, \tuple {y_1, y_2} }$.
Now we have:
Thus:
:$\tuple {c_1, c_2}$ is an upper bound of $\set {\tuple {x_1, x_2}, \tuple {y_1, y_2} }$
and:
:if $\tuple {d_1, d_2}$ is an upper bound of $\set {\tuple {x_1, x_2}, \tuple {y_1, y_2} }$, then $\tuple {c_1, c_2} \preccurlyeq_s \tuple {d_1, d_2}$
so $\set {\tuple {x_1, x_2}, \tuple {y_1, y_2} }$ admits a supremum $\tuple {c_1, c_2}$ in $\struct {T, \preccurlyeq_s}$.
Now let:
:$c_1 = \inf \set {x_1, y_1}$ be the infimum of $\set {x_1, y_1}$
and:
:$c_2 = \inf \set {x_2, y_2}$ be the infimum of $\set {x_2, y_2}$.
We use a similar argument to the above, , to show that:
:$\set {\tuple {x_1, x_2}, \tuple {y_1, y_2} }$ admits an infimum $\tuple {c_1, c_2}$ in $\struct {T, \preccurlyeq_s}$.
As $\tuple {x_1, x_2}$ and $\tuple {y_1, y_2}$ are arbitrary, it follows that $\struct {T, \preccurlyeq_s}$ is a lattice.
== Sources ==
* : Chapter $\text {III}$: The Natural Numbers: $\S 14$: Orderings: Exercise $14.18 \ \text {(b)}$
Category:Simple Order Product
Category:Lattice Theory