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chain complete
In order-theoretic mathematics, a partially ordered set is chain complete if every 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 poset is chain complete. Unlike complete posets, chain complete posets are relatively common. Examples include:
* The set of all linearly independent subsets of a vector space ''V'', ordered by inclusion.
* The set of all partial functions on a set, ordered by restriction.
* The set of all partial choice functions on a collection of non-empty sets, ordered by restriction.
* The set of all prime ideals of a ring, ordered by inclusion.
* The set of all consistent theories of a first-order language.
抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』
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