Limits of Mappings

TL;DR

This paper explores the concept of limits within the simple algebraic structure of sets equipped with a single endofunction, revealing complex behaviors and characterizing limit objects.

Contribution

It provides a comprehensive description of limit objects for sets with an endofunction and proves an inverse theorem for quantifier-free limits in this context.

Findings

Characterization of limit objects for sets with an endofunction

Proof of the inverse theorem for quantifier-free limits

Insights into the complexity of limits in simple algebraic structures

Abstract

In this paper we consider a simple algebraic structure --- sets with a single endofunction. We shall see that from the point of view of limits, even this simplest case is both interesting and difficult. Nevertheless we obtain the shape of limit objects in the full generality, and we prove the inverse theorem in the easiest case of quantifier-free limits.