Presentation of power-ordered sets
by
Dietmar Schweigert
University of Kaiserslautern

We like to generalize the Boolean algebra in the sight of ordered sets. Especially we like to bridge the set theory and ordered sets. We endow a power set with an order in the following way. {a₁, ..., a_n} <= {b₁, ..., b_m} if and only if there exists an injective mapping \pi such that a_i <= \pi (a_i) for i=1, ..., n.
In our approach the role of the injective mapping is essentially. One can take for example graphs, equivalences and other relations instead of orders if injectivity is preserve. This concept of power-ordered sets has many applications in decision theory and in voting a game theory.
We will show some results on power-ordered sets. If (E; <= ) is a chain then (P(E); <= ) is a distributive lattice. Even more we can show same for disjoint union of chains. We point out that there are orders which are not distributive nor a lattice and we derive the general result that (P(E); <= ) is a polarity. Finally we introduct the leveled lattices and we show that some important examples are simple and have only trivial tolerances. Especially we present a list of Hassediagrams.

Date received: May 8, 2000

Copyright © 2000 by the author(s). The author(s) of this document and the organizers of the conference have granted their consent to include this abstract in Atlas Conferences Inc. Document # caee-26.