Simple Order Product of Pair of Ordered Sets is Ordered Set
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 also an ordered set.
Proof
Let $\struct {T, \preccurlyeq_s} := \struct {S_1, \preccurlyeq_1} \otimes^s \struct {S_2, \preccurlyeq_2}$.
By definition of simple product:
- $T := S_1 \times S_2$ where $\times$ denotes Cartesian product
- $\forall \tuple {a, b}, \tuple {c, d} \in T: \tuple {a, b} \preccurlyeq_s \tuple {c, d} \iff a \preccurlyeq_1 c \text { and } b \preccurlyeq_2 d$
We check in turn each of the criteria for an ordering:
Reflexivity
We have by definition of Cartesian product:
- $\forall \tuple {a, b} \in S_1 \times S_2: a \in S_1 \land b \in S_2$
As $\preccurlyeq_1$ and $\preccurlyeq_2$ are both orderings, they are themselves both reflexive.
Hence both $a \preccurlyeq_1 a$ and $b \preccurlyeq_2 b$.
Hence by definition of simple product:
- $\forall \tuple {a, b} \in S: \tuple {a, b} \preccurlyeq_s \tuple {a, b}$
So $\preccurlyeq_s$ is reflexive.
$\Box$
Transitivity
Suppose $\tuple {a, b} \preccurlyeq_s \tuple {c, d}$ and $\tuple {c, d} \preccurlyeq_s \tuple {e, f}$.
Then by definition of simple product:
- $a \preccurlyeq_1 c$ and $b \preccurlyeq_2 d$
- $c \preccurlyeq_1 e$ and $d \preccurlyeq_2 f$
As $\preccurlyeq_1$ and $\preccurlyeq_2$ are both orderings, they are themselves both transitive.
Hence:
- $a \preccurlyeq_1 e$ and $b \preccurlyeq_2 f$
By definition of simple product:
- $\tuple {a, b} \preccurlyeq_s \tuple {e, f}$
So $\preccurlyeq_s$ is transitive.
$\Box$
Antisymmetry
Suppose $\tuple {a, b} \preccurlyeq_s \tuple {c, d}$ and $\tuple {c, d} \preccurlyeq_s \tuple {a, b}$.
Then by definition of simple product:
- $a \preccurlyeq_1 c$ and $b \preccurlyeq_2 d$
- $c \preccurlyeq_1 a$ and $d \preccurlyeq_2 b$
As $\preccurlyeq_1$ and $\preccurlyeq_2$ are both orderings, they are themselves both antisymmetric.
Hence:
- $a = c$ and $b = d$
So by Equality of Ordered Pairs:
- $\tuple {a, c} = \tuple {b, d}$
So $\preccurlyeq_s$ is antisymmetric.
$\Box$
So we have shown that $\preccurlyeq_s$ is reflexive, transitive and antisymmetric.
Thus by definition, $\preccurlyeq_s$ is an ordering and so $\struct {S_1, \preccurlyeq_1} \otimes^s \struct {S_2, \preccurlyeq_2}$ is an ordered set.
$\blacksquare$
Sources
- 1965: Seth Warner: Modern Algebra ... (previous) ... (next): Chapter $\text {III}$: The Natural Numbers: $\S 14$: Orderings: Exercise $14.18 \ \text {(a)}$
- 1980: G. Gierz, K.H. Hofmann, K. Keimel, J.D. Lawson, M.W. Mislove and D.S. Scott: A Compendium of Continuous Lattices
- 1996: Winfried Just and Martin Weese: Discovering Modern Set Theory. I: The Basics ... (previous) ... (next): Part $1$: Not Entirely Naive Set Theory: Chapter $2$: Partial Order Relations: Exercise $29 \ \text {(a)}$
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