Souslin Algebra Embeddings

TL;DR

This paper develops a representation theory for subalgebras of Souslin algebras and constructs examples with complex subalgebra structures using advanced set-theoretic methods.

Contribution

It introduces a new representation framework for subalgebras and constructs Souslin algebras with intricate embedding properties under the diamond principle.

Findings

Established a representation theory for subalgebras of Souslin algebras

Constructed Souslin algebras with non-isomorphic yet bi-embeddable subalgebras

Demonstrated the existence of complex subalgebra structures under set-theoretic assumptions

Abstract

A Souslin algebra is a complete Boolean algebra whose main features are ruled by a tight combination of an antichain condition with an infinite distributive law. The present article divides into two parts. In the first part a representation theory for the complete and atomless subalgebras of Souslin algebras is established (building on ideas of Jech and Jensen). With this we obtain some basic results on the possible types of subalgebras and their interrelation. The second part begins with a review of some generalizations of results from descriptive set theory concerning Baire category which are then used in non-trivial Souslin tree constructions that yield Souslin algebras with a remarkable subalgebra structure. In particular, we use this method to prove that under the diamond principle there is a bi-embeddable though not isomorphic pair of homogeneous Souslin algebras.