Existence of almost free abelian groups and reflection of stationary set

TL;DR

This paper explores the existence of almost free abelian groups and the reflection properties of stationary sets, addressing open questions in set theory and group theory, and establishing new results on incompactness and non-transitivity of NPT.

Contribution

It provides answers to open questions about closure operations, stationary set reflection, and the non-transitivity of NPT, advancing understanding in set-theoretic and algebraic structures.

Findings

Answered Mekler and Eklof's question on closure operations.

Resolved Foreman and Magidor's question on stationary set reflection.

Proved NPT(lambda, mu) + NPT(mu, kappa) does not imply NPT(lambda, kappa).

Abstract

section 2: We answer a question of Mekler Eklof on the closure operations of the incompactness spectrum. We answer a question of Foreman and Magidor on reflection of stationary subsets of S_{< aleph_2}(lambda) = {a subseteq lambda : |a| < aleph_2}]. section 3 - NPT is not transitive. We prove NPT(lambda, mu) + NPT(mu, kappa) not => NPT(lambda, kappa)