A CHARACTERIZATION OF INDUCTIVE POSETS

Abstract. In this paper some aspects of the fixedpoint theory of posets are studied. A new type of selfmapping on posets so called ascending mappings is defined. This new concept enables to prove a fixedpoint theorem which is a generalization of the Bourbaki's fixedpoint theorem. This result is applied to two characterizations of inductiveness for posets. The first one shows that inductiveness in semilattices is equivalent to the existence of fixedpoints of extensive mappings The second characterization proves the inductiveness property for general posets.| Key words: partially ordered sets, fixedpoints, isotone mappings. The aim of this short paper is to give some characterizations of inductiveness for posets in a way similar to that of A. C. Davis [4]. First we derive our basic tool, the fixed point theorem for ascending mappings of an inductive poset into itself. Further we discuss the relationship between the inductiveness in posets and the existence of fixed points. We also present a related characterization of an inductiveness for semilattices in terms of the fixed points. Throughout this paper a poset is a partially ordered set and a chain will mean a totally ordered set (it may be empty). A lower directed set will mean an ordere