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Chain Complete
In mathematics, specifically order theory, a partially ordered set is chain-complete if every chain (order theory), chain in it has a least upper bound. It is ω-complete when every increasing sequence of elements (a type of countable chain) has a least upper bound; the same notion can be extended to other cardinalities of chains.. Examples Every complete lattice, complete lattice (order), lattice is chain-complete. Unlike complete lattices, chain-complete posets are relatively common. Examples include: * The set of all linearly independent subsets of a vector space ''V'', ordered by inclusion (set theory), inclusion. * The set of all partial functions on a set, ordered by Restriction (mathematics), restriction. * The set of all partial choice functions on a collection of empty set, non-empty sets, ordered by restriction. * The set of all prime ideals of a ring (mathematics), ring, ordered by inclusion. * The set of all consistent theory (mathematical logic), theories of a first-o ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics with the major subdisciplines of number theory, algebra, geometry, and analysis, respectively. There is no general consensus among mathematicians about a common definition for their academic discipline. Most mathematical activity involves the discovery of properties of abstract objects and the use of pure reason to prove them. These objects consist of either abstractions from nature orin modern mathematicsentities that are stipulated to have certain properties, called axioms. A ''proof'' consists of a succession of applications of deductive rules to already established results. These results include previously proved theorems, axioms, andin case of abstraction from naturesome basic properties that are considered true starting points of ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
