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In , which depicts the ordering relation between certain pairs of elements and allows one to reconstruct the whole partial order structure. FORMAL DEFINITION A partial order is a Binary Relation "≤" over a Set ''P'' which is Reflexive , Antisymmetric , and Transitive , I.e. , for all ''a'', ''b'', and ''c'' in ''P'', we have that:
In other words, a partial order is an antisymmetric Preorder . A set with a partial order is called a partially ordered set (also called a '''poset'''). The term ''ordered set'' is sometimes also used for posets, as long as it is clear from the context that no other kinds of orders are meant. In particular, Totally Ordered Set s can also be referred to as "ordered sets", especially in areas where these structures are more common than posets. EXAMPLES Standard examples of posets arising in mathematics include:
ORDERS ON THE CARTESIAN PRODUCT OF PARTIALLY ORDERED SETS In order of increasing strength, i.e., decreasing sets of pairs, three of the possible partial orders on the Cartesian Product of two partially ordered sets are:
All three can similarly be defined for the Cartesian product of more than two sets. Applied to Ordered Vector Space s over the same Field , the result is in each case also an ordered vector space. See also . STRICT AND NON-STRICT PARTIAL ORDERS In some contexts, the partial order defined above is called a non-strict (or '''reflexive''') '''partial order'''. In these contexts a '''strict''' (or '''irreflexive''') '''partial order''' "<" is a binary relation that is Irreflexive and Transitive , and therefore Asymmetric . In other words, asymmetric (hence irreflexive) and transitive. Thus, for all ''a'', ''b'', and ''c'' in ''P'', we have that:
There is a 1-to-1 correspondence between all non-strict and strict partial orders. If "≤" is a non-strict partial order, then the corresponding strict partial order "<" is the reflexive reduction given by: a Conversely, if "<" is a strict partial order, then the corresponding non-strict partial order "<" is the "≤" given by: : ''a'' ≤ ''b'' if and only if ''a'' < ''b'' or ''a'' = ''b''. This is the reason for using the notation "≤". Strict partial orders are useful because they correspond more directly to of a dag is both a strict partial order and also a dag itself. INVERSE AND ORDER DUAL The inverse or converse ≥ of a partial order relation ≤ satisfies ''x''≥''y'' if and only iff ''y''≤''x''. The inverse of a partial order relation is reflexive, transitive, and antisymmetric, and hence itself a partial order relation. The ''order dual'' of a partially ordered set is the same set with the partial order relation replaced by its inverse. The irreflexive relation > is to ≥ as < is to ≤. Any of these four relations ≤, <, ≥, and > on a given set uniquely determine the other three. In general two elements ''x'' and ''y'' of a partial order may stand in any of four mutually exclusive relationships to each other: either ''x'' < ''y'', or ''x'' = ''y'', or ''x'' > ''y'', or ''x'' and ''y'' are ''incomparable'' (none of the other three). A holds. The Natural Number s, the Integer s, the Rational s, and the Real s are all totally ordered by their algebraic (signed) magnitude whereas the Complex Number s are not. This is not to say that the complex numbers cannot be totally ordered; we could for example order them lexicographically via ''x''+i''y'' < ''u''+i''v'' if and only if ''x'' < ''u'' or (''x'' = ''u'' and ''y'' < ''v''), but this is not ordering by magnitude in any reasonable sense as it makes 1 greater than 100i. Ordering them by absolute magnitude yields a preorder in which all pairs are comparable, but this is not a partial order since 1 and i have the same absolute magnitude but are not equal, violating antisymmetry. NUMBER OF PARTIAL ORDERS Sequence A001035 in OEIS gives the number of partial orders on a set of ''n'' elements: The number of strict partial orders is the same as that of partial orders. LINEAR EXTENSION A Total Order ''T'' is a Linear Extension of a partial order ''P'' if, whenever ''x'' ≤ ''y'' in ''P'' it also holds that ''x'' ≤ ''y'' in ''T''. In Computer Science , algorithms for finding linear extensions of partial orders are called Topological Sorting . CATEGORY THEORY |
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