Order dependencies (ODs) describe the relationship among lexicographical orderings of sets of tuples. An order specification is a list of attributes marked as asc (ascending) or desc (descending). This is the notion of order used in SQL and within query optimization, as by the order-by operator (nested sort). Let X = [A |T] be a list of marked attributes, the marked attribute A is the head of the list, and the marked list T is the tail. For two tuples s and t, s ≼XtiffX = [ ]; or X = [Aasc|T] and sA < tA; or X = [Adesc|T] and sA > tA; or X = [Aasc|T] or X = [Adesc|T], sA = tA, and s ≼Tt. Let r be a table over a relation R that contains the attributes that appear in X and Y. Given two order specifications X and Y a table r satisfies an OD X ↦ Y, read as XordersY, iff for all s, t ∈r, s ≼Xt implies s ≼Yt . An OD is unidirectional when attributes within it are all marked as asc or desc [4, 6, 9, 7]. Unidirectional ODs are subsumed by ODs by definition.
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