User:Dfeuer/Definition:Lexicographic Ordering on Product
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Definition
Let $\struct {I, \preceq}$ be a well-ordered set.
For each $i \in I$, let $\struct {X_i, \le_i}$ be an ordered set.
Let $X = \ds \prod_{i \mathop \in I} X_i$ be the Cartesian product of the $X_i$.
The lexicographic ordering on $X$ is the binary relation $\le$, defined as follows:
For $a, b \in X$, let $M_{a, b} = \set {i \in I: a_i \ne b_i}$.
Then $a \le b$ if and only if either of the following holds:
Here, $\min M_{a, b}$ denotes the smallest element of $M_{a, b}$ under $\preceq$, which exists as $I$ is a woset.
Also see
User:Dfeuer/Lexicographic Ordering is Ordering
Lexicographic Order of Family of Totally Ordered Sets is Totally Ordered Set
User:Dfeuer/Lexicographic Ordering of Finite Product of Well-Orderings is Well-Ordering