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.
=== Corollary ===
== 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 $1$ ===
=== 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 $\set {\tuple {x_1, x_2}, \tuple {y_1, y_2} }$.
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.
== Sources ==
* : Chapter $\text {III}$: The Natural Numbers: $\S 14$: Orderings: Exercise $14.19 \ \text{(c)}$
Category:Lexicographic Order
Category:Lattice Theory
Category:Conditions for Lexicographic Order on Pair of Ordered Sets to be Lattice