# On double-membership graphs of models of Anti-Foundation

## TL;DR

This paper investigates the structure of graphs derived from models of Anti-Foundation by examining the double-membership relation, revealing a vast diversity of such graphs and their model-theoretic properties.

## Contribution

It demonstrates the existence of continuum-many double-membership graphs and characterizes their theories and models, including non-reducts of Anti-Foundation models.

## Key findings

- There are continuum-many such graphs.
- Each graph has continuum-many countable models.
- Some models are not reducts of Anti-Foundation models.

## Abstract

We answer some questions about graphs which are reducts of countable models of Anti-Foundation, obtained by considering the binary relation of double-membership $x\in y\in x$. We show that there are continuum-many such graphs, and study their connected components. We describe their complete theories and prove that each has continuum-many countable models, some of which are not reducts of models of Anti-Foundation.