| Partial Order |
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FORMAL DEFINITION A partial order is a Binary Relation ''R'' over a Set ''P'' which is Reflexive , Antisymmetric and Transitive . A set with a partial order is called a '''partially ordered set''' ('''poset'''). If (''x'', ''y'') ∈ ''R'' then the notation ''x'' ≤ ''y'' is typically used instead of ''xRy'' (and ''x'' < ''y'' when ''x'' ≠ ''y''). A partial order ''R'' on ''X'' is called a (total) Order on ''X'' if for any ''x'', ''y'' ∈ ''X'' one of the conditions is satisfied: ''x'' < ''y'', ''x'' = ''y'' or ''x'' > ''y''. Alternatively, 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
STRICT AND WEAK PARTIAL ORDERS In some contexts, the partial order defined above is called a weak (or '''reflexive''') '''partial order'''. In these contexts a '''strict''' (or '''irreflexive''') '''partial order''' is a binary relation that is Irreflexive and Transitive , and therefore Antisymmetric . In other words, for all ''a'', ''b'', and ''c'' in ''P'', we have that:
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