Recall that a relation ≥ defined on a set X is called pre-order if (i) x ≥ x, for all x ∈ X, and (ii) x ≥ y and y ≥ z imply x ≥z. If x ≥ y and y ≥ x, then x and y are called equivalent elements. A pre-order relation is called complete if, for any two elements x and y, either x ≥ y or y ≥ x.
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