An **ordered set**
is a pair of a set *P* and a reflexive,
antisymmetric and transitive relation on *P*.
We will usually omit the pair notation and call *P* an
ordered set also.
We will call **comparable**
and write
iff or .
Every is an ordered set with the induced order from *P*
and will be called an **ordered subset** of *P*.
All subsets of ordered sets will be viewed as ordered subsets.
A subset *S* of an ordered set *P* has an **upper bound**
*x* (denoted )
iff for all we have .
*y* is the **supremum**
or **lowest upper bound** of *S*
(denoted )
iff
and for all we have .
Lower bounds and infima resp.
greatest lower bounds (denoted
) are defined
dually.
An ordered set is a **chain** iff any two points in it are
comparable.
A mapping is called **order-preserving**
iff for all we have that implies .
An ordered set *P* is said to have the
**fixed point property**
iff each order-preserving self map
has a **fixed point**, i.e., there is a point with
*p*=*f*(*p*).
An order-preserving map without a fixed point is called
**fixed point free**.

The main object of this paper is the following
- somewhat vague -
open question
that is for example to be found on ORDER's problem list.

The recorded history of this problem seems to start in the papers [68] by Knaster in the twenties and then in [127] by Tarski and [24] by Davis in the fifties, where the question is answered for lattices (a lattice is an ordered set in which any finite subset has a supremum and an infimum, it is complete iff every subset has a supremum and an infimum): A lattice has the fixed point property iff it is complete (also cf. [131], Exercise 1J, Tarski's remarks in [127] and the remarks in [96] footnote 2 on p. 284). After that there were papers by Abian and Brown ([2]) and Pelczar ([84]), which recorded one of today's standard tools: If

- One can try as in the above mentioned theorems to characterize the fixed point property for certain classes of ordered sets. This is done for example by Fofanova and Rutkowski in [43] (sets of width 2), by Höft and Höft in [61]-[63] (lexicographic sums), by Rival as mentioned in [95] (sets of height 1), by Ewacha and Rival in [38], Rutkowski in [110], the author in [116] (``small" sets), Davis and Tarski in [24] and [127] (lattices) and possibly most importantly by Roddy in [104], where the fixed point problem is solved for finite products of finite ordered sets with a beautiful argument.
- One can prove fixed point theorems that do not necessarily handle an established class of ordered sets, but that provide new insights through possible reductions (for example done by Constantin and Fournier in [18], by Rival in [95], and by the author in [116], [117]) or through the identification of substructures that force the fixed point property (done for example by Baclawski and Björner in [7], by Edelman in [37], and by Rutkowski in [108]).
- One can try to prove results about forbidden retracts, which can be done in special cases (for example by Fofanova and Rutkowski in [43], or by Rutkowski and the author in [113]), but which might be too complicated in general (cf. section 4.8).
- One can relate the problem to its analogue in topology. (When does a topological space have the fixed point property?) This is done for example by Baclawski and Björner in [7], and by Constantin and Fournier in [18].
- One can devise algorithms that check whether an arbitrary finite ordered set has the fixed point property (done for example by Pickering, Roddy and Stadel in [86] and by Xia in [134]).