In this section we briefly indicate how to construct a large family of examples of finite ordered sets that have a fixed point free automorphism and are such that all retracts have the fixed point property. This shows that despite some nice results for special classes of ordered sets the approach 3 in the introduction might not lead to a resolution of the fixed point problem, as there are too many forbidden retracts. Theorem 4.51 was revealed to the author by an anonymous referee. It also gives a negative answer to Problem 2 in [96]. The author would like to hereby express his gratitude to this referee.

**Proof:**
Let be a retraction, i.e., a continuous
idempotent map with .
Then is isomorphic to a
retract of the *n*-dimensional unit ball which has the
topological fixed point property by Brouwer's fixed point theorem.
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**Proof:**
Let be the
truncated incidence lattice of the triangulation
*K* of .
Let be a nontrivial
retraction
with .
Then there is a minimal element *m* of
that is not in :
Indeed otherwise *r* fixes all minimal elements of , which
implies for all . Then there is a
such that *r*(*y*)>*y* and *r*(*p*)=*p* for all .
Then *y* has exactly one upper cover and
there is a maximal element such that
has exactly one point (otherwise *y* has more than
one upper cover).
Thus *M* is the only upper bound of *y*.
But then every point *x* that is in the interior of the
topological realization of *y* is such that
for all small enough we have that
is homeomorphic to
the upper half space
.
Since is an *n*-dimensional manifold
without boundary this is a contradiction
to |*K*| being homeomorphic to .

Let be an order-preserving map.
Let . For each minimal
element
choose a minimal element such that
.
For not minimal let

Then *G* is order-preserving on .
Moreover is a retraction
and induces a
simplicial map on *K* that is a
nontrivial
retraction
(we have shown above that the retract does not contain all minimal
elements), which can be
extended to a continuous retraction .
Now by Lemma 4.47
has the topological
fixed point property.
Thus the continuous map on *R*[|*K*|] induced by *G* has a fixed point *p*.
Let *S* be the smallest simplex in
such that .
Then *G* maps *S* to a sub-simplex of *S*, i.e.,
and thus *G* has a fixed point.
Thus has a fixed point, which must be a fixed point
of *f*.
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**Proof:**
For each simplex *S* of *K* let *A*(*S*):=-*S*.
By hypothesis this is well-defined.
Since no simplex is equal to its antipode
is a fixed point free
order-preserving automorphism of .
By Lemma 4.50 all nontrivial retracts of
have the (order-theoretical) fixed point property.
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