Nonstandard Digraphs

TL;DR

This paper extends the concept of nonstandard graphs to nonstandard digraphs, exploring their structure, properties, and various constructions using ultrapower and transfer principles, including special cases like hyperfinite digraphs.

Contribution

It introduces the theory of nonstandard digraphs, providing definitions, constructions, and results that are specific to the directed case, expanding the nonstandard graph framework.

Findings

Nonstandard digraphs can be constructed via ultrapower and transfer methods.

Various properties like incidences, adjacencies, and connectedness are characterized.

Special cases include enlargements of infinite digraphs and hyperfinite digraphs.

Abstract

Nonstandard graphs have been defined and examined in prior works. The present work does the same for nonstandard digraphs. Since digraphs have more structure than do graphs, the present discussion requires more complicated definitions and yields a variety of results peculiar to nonstandard digraphs. A nonstandard digraph can be obtained by means of an ultrapower construction based on a sequence of digraphs or more elegantly by using the transfer principle. We use either or both techniques in particular circumstances. As special cases, we have the enlargement of a single infinite digraph and also hyperfinite digraphs based on sequences of finite digraphs. Also examined are such ideas as incidences and adjacencies for nonstandard arcs and vertices, connectedness, components, and galaxies in nonstandard digraphs.