Existence of Endo-Rigid Boolean Algebras

TL;DR

This paper constructs Boolean algebras of certain infinite sizes that have only the trivial endomorphisms, demonstrating the existence of endo-rigid structures in this context.

Contribution

It proves the existence of endo-rigid Boolean algebras of cardinality lambda where lambda equals lambda^{aleph_0}, using a novel application of black box techniques.

Findings

Existence of Boolean algebras with only trivial endomorphisms at certain infinite cardinalities

Use of black box methods to construct endo-rigid Boolean algebras

Commentary on the necessity of the cardinality restriction

Abstract

How many endomorphisms does a Boolean algebra have? Can we find Boolean algebras with as few endomorphisms as possible? Of course from any ultrafilter of the Boolean algebra we can define an endomorphism, and we can combine finitely many such endomorphisms in some reasonable ways. We prove that in any cardinality lambda=lambda^ {aleph_0} there is a Boolean algebra with no other endomorphisms. For this we use the so called "black boxes", but in a self contained way. We comment on how necessary the restriction on the cardinal is.