Ordinal operations on graph representations of sets



Any set x is uniquely specified by the graph of the membership relation on the set obtained by adjoining x to the transitive closure of x. Thus any operation on sets can be looked at as an operation on these graphs. We look at the operations of ordinal arithmetic of sets in this light. This turns out to be simplest for a modified ordinal arithmetic based on the Zermelo ordinals, instead of the usual von Neumann ordinals. In this arithmetic, addition of sets corresponds to concatenating graphs, and multiplication corresponds to replacing each edge of a graph by a copy of another graph. Characterizations for the von Neumann case are also given.