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 a₀∈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 a₀ (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).

This note summarizes known results about this problem and provides a slight generalization of some of them.

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