On projections of the tails of a power

TL;DR

This paper explores the structure of endomorphisms on tail projections of power sets in set theory, revealing new constructions that challenge previous assumptions about their origins and supports.

Contribution

It introduces general methods for constructing idempotent endomorphisms on tail quotients that are not derived from small support endomorphisms, expanding understanding of their complexity.

Findings

Existence of non-derivable tail endomorphisms under mild assumptions

Construction of idempotent endomorphisms not from small support endomorphisms

Application to endomorphisms of integer power sets

Abstract

Let $\kappa$ be an inaccessible cardinal, $\mathfrak{U}$ be a universal algebra, and $\sim$ be the equivalence relation on $\mathfrak{U}^{\kappa}$ of eventual equality. From mild assumptions on $\kappa$ we give general constructions of $\mathcal{E} \in End(\mathfrak{U}^{\kappa}/\sim)$ satisfying $\mathcal{E} \circ \mathcal{E} = \mathcal{E}$ which do not descend from $\Delta \in End(\mathfrak{U}^{\kappa})$ having small strong supports. As an application there exists an $\mathcal{E} \in End(\mathbb{Z}^{\kappa}/\sim)$ which does not come from a $\Delta \in End(\mathbb{Z}^{\kappa})$.