411 REFERENCE · Wikibooks
Order Theory/Lexicographic order
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lexicographic order Let be preordered sets, where is well-ordered. Define an order on , the Cartesian product, by
.
lexicographic order induced by posets is poset Whenever
lexicographic order induced by total orders is total Whenever is a well-ordered set and are totally ordered sets, the lexicographic order on is total.
Let any two elements and of be given. Then either , or there exists a smallest so that . Since is total, either or , and thus either or .