A point on fixpoints in posets. F. Blanqui. Note. 2014.

Let (X,≤) be a non-empty strictly inductive poset, that is, a
non-empty partially ordered set such that every non-empty chain Y has
a least upper bound lub(Y)∈X, a chain being a subset of X totally
ordered by ≤. We are interested in sufficient conditions such that,
given an element a_{0}∈X and a function f:X→X, there
is some ordinal k such that a_{k+1}=a_{k}, where
a_{k} is the transfinite sequence of iterates of f starting
from a_{0} (a_{k} is then a fixpoint of f):

- a
_{k+1}=f(a_{k}), - a
_{l}=lub{a_{k}|k<l} if l is a limit ordinal, i.e. l=lub(l).

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