Comparing Classes of Finite Structures

TL;DR

This paper introduces a new reducibility concept for classes of structures, compares various classes, and establishes their relative complexity, revealing maximal and minimal classes among finite structures.

Contribution

It defines a uniform enumeration reducibility for classes of structures and characterizes their relative complexity, including the maximal and minimal classes.

Findings

Class of cyclic graphs and finite prime fields are equivalent.

Finite graphs and finite linear orders are maximal among finite structures.

Constructed large chains and antichains of classes.

Abstract

We introduce a reducibility on classes of structures, essentially a uniform enumeration reducibility. This reducibility is inspired by the Friedman-Stanley paper on using Borel reductions to compare classes of countable structures. This reducibility is calibrated by comparing several classes of structures. The class of cyclic graphs and the class of finite prime fields are equivalent, and are properly below the class of arbitrary finite graphs. The class of finite graphs and the class of finite linear orders are maximal among all classes of finite structures. We also prove some general characterizations of reducibility to certain classes. Examples of large chains and antichains of classes are constructed.