partially ordered set การใช้

• Every relatively atomic partially ordered set with a least element is atomic.
• If you have an already partially ordered set, you could use quicksort.
• Let and be partially ordered sets and let be an order-preserving map.
• It is readily verified that this yields a partially ordered set.
• Partially ordered sets without greatest element or maximal elements admit disjoint cofinal subsets.
• Viewing lattices as partially ordered sets, this says that the meet operation dual:
• Order theory is the study of partially ordered sets, both finite and infinite.
• This definition is extendable to subsets of any partially ordered set.
• Suppose that " S " is a partially ordered set.
• A partially ordered set in which all pairs have a join is a join-semilattice.
• A given partially ordered set may have several different completions.
• A closure operator on a partially ordered set is determined by its closed elements.
• Complete lattices are partially ordered sets, where every subset has an infimum and a supremum.
• More generally, Chomp can be played on any partially ordered set with a least element.
• Dually, a partially ordered set in which all pairs have a meet is a meet-semilattice.
• A partially ordered set that is both a join-semilattice and a meet-semilattice is a lattice.
• Zorn's lemma describes partially ordered sets where every totally ordered subset has an upper bound.
• These concepts generalize respectively those of preordered set, partially ordered set and totally ordered set.
• It has also been used to study topology of partially ordered sets and bruhat orders.
• An abstract polytope is a partially ordered set of elements or members, which obeys certain rules.