An alternative way of doing this is to use the transitive reduction of the partial ordering; in general, this produces DAGs with fewer edges, but the reachability relation in these DAGs is still the same partial order.
22.
Under the natural partial ordering on the set \ mathcal { C } ( X ) of equivalence classes of compactifications, any minimal element is equivalent to the Alexandroff extension ( Engelking, Theorem 3.5 . 12 ).
23.
Tsort reads its input ( from the given FILE, or standard input if no input file is given or for a FILE of'-') as pairs of strings, separated by blanks, indicating a partial ordering.
24.
The concept of a mean-preserving spread provides a partial ordering of probability distributions according to their dispersions : of two probability distributions, one may be ranked as having more dispersion than the other, or alternatively neither may be ranked as having more dispersion.
25.
We can define a partial ordering on the set of all such representations of f by declaring that the representation S1 is larger than or equal than represenation S2, if all the intervals in S1 are unions of those of S2, or closure of unions.
26.
If we take a semigroup and define " a " d " " b " if and only if " ab = b ", we obtain a partial ordering if and only if this semigroup is a left-regular band.
27.
Finally let " B " be a Gr�bner basis of " I " for a monomial ordering refining the total degree partial ordering and " G " the ( homogeneous ) ideal generated by the leading monomials of the elements of " B ".
28.
Together these two assumptions imply that given a set of exhaustive and exclusive actions to choose from, an individual can " rank " the elements of this set in terms of his preferences in an internally consistent way ( the ranking constitutes a partial ordering ), and the set has at least one maximal element.
29.
Such constraints define a partial ordering or total ordering and can be represented as a directed graph G = ( N, E ), where N is the set of variables ( observed values ) involved, and E is the set of pairs ( i, j ) for each constraint x _ i \ le x _ j.
30.
Conversely, any partial ordering on a finite set may be defined as the reachability relation in a DAG . One way of doing this is to define a DAG that has a vertex for every object in the partially ordered set, and an edge " xy " for every pair of objects for which " x " d " " y ".
