Some characterizations of pseudo-chains in pseudo-ordered sets

In this paper, we have proved that minimum condition in a pseudo-ordered set (psoset) is equivalent to the descending pseudo-chain condition. Characterization of a pseudo-chain in an acyclic psoset is obtained using graph theoretic approach as transitivity need not hold in psosets. Skala proved that every complete trellis has Fixed Point Property. A counterexample given in this paper shows that trellis having Fixed Point Property need not be complete. The notion of weakly isotone mapping and Strong Fixed Point Property is introduced in a psoset and the characterization of weakly isotone mapping is obtained. It is proved that a connected psoset containing a nontrivial cycle does not have Strong Fixed Point Property.

Automated summary

In this paper, the equivalence between the minimum condition and the descending pseudo-chain condition in pseudo-ordered sets is proved, using a graph-theoretic approach and a counterexample to show the relationship between complete trellises and Fixed Point Property. The concepts of weakly isotone mapping and Strong Fixed Point Property in psosets are introduced, with the result that connected psosets containing nontrivial cycles do not have the Strong Fixed Point Property.