Abstract
Let
(X,≤)
be a {\em 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 $f:X\a X$, 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
(implying that
a
k
is a fixpoint of
f
):
\begin{itemize}\itemsep=0mm \item $a_{k+1}=f(a_k)$ \item $a_l=\lub\{a_k\mid k
\textless{} l\}$ if $l$ is a limit ordinal, i.e. $l=lub(l)$ \end{itemize}
This note summarizes known results about this problem and provides a slight generalization of some of them.