Definition
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[071] Given two ordered sets \((X,≤_ X)\) and \((Y,≤_ Y)\), setting \(Z=X× Y\), we define the lexicographic order \(≤_ Z\) on \(Z\); let \(z_ 1=⦇x_ 1,y_ 1⦈\in Z\) and \(z_ 2=⦇x_ 2,y_ 2⦈\in Z\), then:
in the case \(x_ 1≠ x_ 2\) , then \(z_ 1≤_ Z z_ 2\) if and only if \(x_ 1≤_ X x_ 2\);
in the case \(x_ 1= x_ 2\) , then \(z_ 1≤_ Z z_ 2\) if and only if \(y_ 1≤_ Y y_ 2\).
This definition is then extended to products of more than two sets: given two vectors, if the first elements are different then we compare them, if they are equal we compare the second elements, if they are equal the thirds, etc.