totally ordered set การใช้

• Directed sets are a generalization of nonempty totally ordered sets.
• The maximal totally ordered sets are horizontal lines in  ! 2.
• A totally ordered set without infinite descending chains is called well-ordered.
• Every non-empty totally ordered set is directed.
• Unlike the latter, the product order of two totally ordered sets is not total.
• Join and meet are symmetric totally ordered set is simply its maximal / minimal element.
• From any totally ordered set of representations, we can construct a representation as follows.
• A sequence is usually indexed by the natural numbers, which are a totally ordered set.
• The lexicographic order of totally ordered sets is however a linear extension of their product order.
• These concepts generalize respectively those of preordered set, partially ordered set and totally ordered set.
• A totally ordered set ( with its order topology ) which is a complete lattice is compact.
• The lexicographical order of two totally ordered sets is thus a linear extension of their product order.
• This covers series, which is PM's term for what is now called a totally ordered set.
• That is, all totally ordered sets are directed sets ( contrast lattices are directed sets both upward and downward.
• For totally ordered sets, the notions of maximal element and maximum coincide, and the notions of minimal element and minimum coincide.
• Weak orders are a generalization of totally ordered sets ( rankings without ties ) and are in turn generalized by partially ordered sets and preorders.
• For any point x we take the union of all the sets that have x as a member from the representations in the totally ordered set.
• Typical examples of distributive lattice are totally ordered sets, isomorphic to a lattice of sets, ordered by inclusion ( Birkhoff's representation theorem ).
• In particular, totally ordered sets can also be referred to as " ordered sets ", especially in areas where these structures are more common than posets.
• The affinely extended real number system turns into a totally ordered set by defining-\ infty \ leq a \ leq + \ infty for all a.