Aleksander Rutkowski

The fixed point property for small sets

There exist exactly eleven (up to isomorphism and duality) ordered sets of size ≤10 with the fixed point property and containing no irreducible elements.

The fixed point property in ordered sets of width two

We consider the fixed point property (FPP) in an ordered set of width two (every antichain contains at most two elements). The necessary condition of the FPP and a number of equivalent conditions to the FPP in such sets is established. The product theorem is proved, as well.

Cores, cutsets and the fixed point property

AbstractThe purpose of this paper is the analysis and application of the concepts of a core (a pair of chains) and cutset in the fixed point theory for posets. The main results are: (1)(Theorem 3) If P is chain-complete and (*), it contains a cutset S such that every nonempty subset of S has a join or a meet in P, then P has the fixed point property (FPP),(2)(Theorem 5) If P or Q is chain-complete, Q satisfies (*) and both P and Q have the FPP, then P x Q has the FPP.(3)(Theorem 6) Let P or Q be chain-complete and there exist p∈P and a finite sequence f1, f2, ..., fn of order-preserving mappings of P into P such that

Retractability and the fixed point property for products

\left( {\forall x\varepsilon P} \right)x \leqslant f_1 \left( x \right) \geqslant f_2 \left( x \right) \leqslant \cdots \geqslant f_n \left( x \right) \leqslant p

Dimension two, fixed points and dismantlable ordered sets

If P and Q have the FPP then P x Q has the FPP.(4)(Theorem 7) If T is an ordered set with the FPP and {Pt:t∈T} is a disjoint family of ordered sets with the FPP then its ordered sum ∪{Pt:t∈T} has the FPP.

On Operations and Linear Extensions of Well Partially Ordered Sets

Sufficient conditions for the fixed point property for products of two partially ordered sets are proved. These conditions are formulated in terms of multifunctions (functions with non-empty sets as values).

The number of strictly increasing mappings of fences

LetP, Q be ordered sets and leta∈P. IfP \ {a} is a retract ofP and setsP and {x∈P:x>p} (or its dual) have the fixed point property then, for each chain complete setP,P×Q has the fixed point property if and only if (P\{a})×Q has this property. This establishes the fixed point property for some products of ordered sets which are beyond the reach of all known product theorems.

Counting the number of isotone selfmappings of crowns

AbstractLetY be a fence of sizem andr=⌊m−1/2⌊. The numberb(m) of order-preserving selfmappings ofY is equal toAr-Br-Cr-Dr, where, ifm is odd,