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

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

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

Let $\struct {S_2, \preccurlyeq_2}$ have neither a greatest element nor a smallest element.

Then:
 * $\preccurlyeq_l$ is a lattice ordering




 * $\preccurlyeq_1$ is a total ordering

and:
 * $\preccurlyeq_2$ is a lattice ordering.

Sufficient Condition
Let $\preccurlyeq_l$ be a lattice ordering.

We are given that $\struct {S_2, \preccurlyeq_2}$ has neither a greatest element nor a smallest element.

From Condition $(2)$ of Conditions for Lexicographic Order on Pair of Ordered Sets to be Lattice, it follows that $\preccurlyeq_1$ is a total ordering.

Because $\struct {S_2, \preccurlyeq_2}$ has no smallest element, it follows from Condition $(4)$ of Conditions for Lexicographic Order on Pair of Ordered Sets to be Lattice that every doubleton subset of $S_2$ admits a supremum.

Because $\struct {S_2, \preccurlyeq_2}$ has no greatest element, it follows from Condition $(5)$ of Conditions for Lexicographic Order on Pair of Ordered Sets to be Lattice that every doubleton subset of $S_2$ admits an infimum.

Thus every doubleton subset of $S_2$ admits a supremum and an infimum.

That is, $\preccurlyeq_2$ is a lattice ordering.

Necessary Condition
Let:
 * $\preccurlyeq_1$ be a total ordering

and:
 * $\preccurlyeq_2$ be a lattice ordering.

From Totally Ordered Set is Lattice, $\preccurlyeq_1$ is a lattice ordering.

Thus Condition $(1)$ of Conditions for Lexicographic Order on Pair of Ordered Sets to be Lattice holds.

We are given that $\preccurlyeq_1$ be a total ordering.

Thus Condition $(2)$ of Conditions for Lexicographic Order on Pair of Ordered Sets to be Lattice holds.

We are given that $\preccurlyeq_2$ is a lattice ordering.

Hence every doubleton subset of $S_2$ admits a supremum and an infimum.

Thus Conditions $(3)$, $(4)$ and $(5)$ of Conditions for Lexicographic Order on Pair of Ordered Sets to be Lattice hold.

Hence from Conditions for Lexicographic Order on Pair of Ordered Sets to be Lattice:
 * $\preccurlyeq_l$ is a lattice ordering.