A chain condition on a partially ordered set 
is a condition that requires sequences
in 
to become eventually constant. The condition that all increasing sequences in 
become eventually constant is called
the increasing chain condition, while
the dual condition that all decreasing sequences in 
become eventually constant is called the decreasing
chain condition.
Chain Condition
See also
Ascending Chain Condition, Descending Chain ConditionExplore with Wolfram|Alpha
Cite this as:
Weisstein, Eric W. "Chain Condition." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/ChainCondition.html