Conditions for Lexicographic Order on Pair of Ordered Sets to be Lattice

Theorem
Let $\struct {S_1, \preccurlyeq_1}$ and $\struct {S_2, \preccurlyeq_2}$ be ordered sets.

Let $\preccurlyeq_l$ denote the lexicographic order on $S_1 \times S_2$''':
 * $\tuple {x_1, x_2} \preccurlyeq_l \tuple {y_1, y_2} \iff \tuple {x_1 \prec_1 y_1} \lor \tuple {x_1 = y_1 \land x_2 \preccurlyeq_2 y_2}$

Then:
 * $\struct {S_1 \times S_2, \preccurlyeq_l}$ is a lattice

all of the following conditions hold:


 * $(1): \quad \struct {S_1, \preccurlyeq_1}$ is a lattice


 * $(2): \quad$ Either $\preccurlyeq_1$ is a total ordering, or $\struct {S_2, \preccurlyeq_2}$ has a greatest element and a smallest element


 * $(3): \quad$ Every doubleton subset of $S_2$ is either unbounded above or admits a supremum, and is also either unbounded below or admits an infimum


 * $(4): \quad$ Either every doubleton subset of $S_2$ admits a supremum, or every element of $S_1$ has an immediate successor and $\struct {S_2, \preccurlyeq_2}$ has a smallest element


 * $(5): \quad$ Either every doubleton subset of $S_2$ admits an infimum, or every element of $S_1$ has an immediate predecessor and $\struct {S_2, \preccurlyeq_2}$ has a greatest element.

Proof
Let $\struct {T, \preccurlyeq_l} := \struct {S_1, \preccurlyeq_1} \otimes^s \struct {S_2, \preccurlyeq_2}$.

From Lexicographic Order is Ordering we have that $\struct {T, \preccurlyeq_l}$ is an ordered set.

Recall the definition of lexicographic order:

Lemma $2$
Recall the definition of lattice:

Sufficient Condition
Let $\struct {T, \preccurlyeq_l}$ be a lattice.


 * Condition $(1)$:

Let $x_1, y_1 \in S_1$ and $x_2, y_2 \in S_2$ be arbitrary.

Then $\set {\tuple {x_1, x_2}, \tuple {y_1, y_2} }$ admits a supremum and admits an infimum in $T$.

Let $\tuple {c_1, c_2} \in T$ be a supremum of $\set {\tuple {x_1, x_2}, \tuple {y_1, y_2} }$.

Thus:
 * $(1): \quad \tuple {c_1, c_2}$ is an upper bound of $\set {\tuple {x_1, x_2}, \tuple {y_1, y_2} }$ in $T$
 * $(2): \quad \tuple {c_1, c_2} \preccurlyeq_l \tuple {d_1, d_2}$ 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:

and:

Thus:
 * $d_1$ is an upper bound of $\set {x_1, y_1}$

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 $\set {x_1, y_1}$

Then:

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 $\struct {S_1, \preccurlyeq_1}$.

Now let $\tuple {c_1, c_2} = \inf \set {\tuple {x_1, x_2}, \tuple {y_1, y_2} }$ be the infimum of $\set {\tuple {x_1, x_2}, \tuple {y_1, y_2} }$.

Mutatis mutandis, we use a similar argument to the above to show that:


 * $\set {x_1, y_1}$ admits an infimum $c_1$ in $\struct {S_1, \preccurlyeq_1}$

As $x_1$, $x_2$, $y_1$ and $y_2$ are arbitrary, it follows that $\struct {S_1, \preccurlyeq_1}$ is a lattice.


 * Condition $(2)$:

Suppose $\preccurlyeq_1$ is not a total ordering.

Then there exist non-comparable elements $x_1$ and $y_1$ in $S_1$:
 * $\lnot \paren {x_1 \preccurlyeq_1 y_1}$ and $\lnot \paren {y_1 \preccurlyeq_1 x_1}$

Hence, from Lemma $2$, $\tuple {x_1, x_2}$ and $\tuple {y_1, y_2}$ are non-comparable elements in $T$:
 * $\lnot \paren {\tuple {x_1, x_2} \preccurlyeq_l \tuple {y_1, y_2} }$ and $\lnot \paren {\tuple {y_1, y_2} \preccurlyeq_l \tuple {x_1, x_2} }$

Necessary Condition
Let $\struct {S_1, \preccurlyeq_1}$ and $\struct {S_2, \preccurlyeq_2}$ fulfil the conditions that:


 * $(1): \quad \struct {S_1, \preccurlyeq_1}$ is a lattice


 * $(2): \quad$ Either $\preccurlyeq_1$ is a total ordering, or $\struct {S_2, \preccurlyeq_2}$ has a greatest element and a smallest element


 * $(3): \quad$ Every doubleton subset of $S_2$ is either unbounded above or admits a supremum, and is also either unbounded below or admits an infimum


 * $(4): \quad$ Either every doubleton subset of $S_2$ admits a supremum, or every element of $S_1$ has an immediate successor and $\struct {S_2, \preccurlyeq_2}$ has a smallest element


 * $(5): \quad$ Either every doubleton subset of $S_2$ admits an infimum, or every element of $S_1$ has an immediate predecessor and $\struct {S_2, \preccurlyeq_2}$ has a greatest element.