Ordinal optimization: Difference between revisions

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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 sets can also be referred to as "ordered sets", especially in areas where these structures are more common than posets.

For distinct elements ''a, b'' distinct elements of a partially ordered set ''P'', if ''a ≤ b'' or ''b ≤ a'', then ''a'' and ''b'' are '''comparable'''. Otherwise they are '''incomparable'''. If every two elements of a poset are comparable, the poset is called a [[totally ordered set]] or '''chain''' (e.g. the natural numbers under order). A poset in which every two elements are incomparable is called an [[antichain]].

=== Examples ===