[Paper Review] On incomparability and related cardinal functions on ultraproducts of Boolean algebras

Saharon Shelah, Otmar Spinas|ArXiv.org|Mar 18, 1999

Advanced Topology and Set Theory5 references3 citations

TL;DR

This paper investigates cardinal functions in ultraproducts of Boolean algebras, showing it is consistent (under certain set-theoretic assumptions) that the value of various cardinal characteristics—such as incomparability, spread, and hereditary density—on an ultraproduct is strictly less than the ultraproduct of the values of these characteristics on the individual algebras. The result resolves several open problems posed by Monk by constructing a model where these functions do not preserve ultraproducts in a monotonic way.

ABSTRACT

Let C denote any of the following cardinal characteristics of Boolean algebras: incomparability, spread, character, pi-character, hereditary Lindelof number, hereditary density. It is shown to be consistent that there exists a sequence <i> of Boolean algebras and an ultrafilter D on kappa such that C(prod_{i</i>

Motivation & Objective

• To investigate the behavior of cardinal functions such as incomparability, spread, character, and hereditary density in ultraproducts of Boolean algebras.
• To determine whether these functions commute with ultraproduct constructions, particularly in the context of set-theoretic independence results.
• To resolve open problems posed by Monk concerning the relationship between the value of a cardinal function on an ultraproduct and the ultraproduct of its values on individual algebras.
• To establish the consistency of a strict inequality: C(∏B_i/D) < |∏C(B_i)/D| for various cardinal characteristics C.
• To explore the interaction between model-theoretic ultraproducts and topological or combinatorial cardinal invariants in Boolean algebras.

Proposed method

• The authors use forcing and independence techniques from set theory to construct a model of ZFC where the desired inequality holds.
• They analyze ultraproducts of Boolean algebras using a non-principal ultrafilter D on a cardinal κ.
• For each cardinal characteristic C ∈ {incomparability, spread, character, π-character, hereditary Lindelöf number, hereditary density}, they compare C(∏B_i/D) with |∏C(B_i)/D|.
• The proof relies on constructing a sequence ⟨B_i : i < κ⟩ of Boolean algebras such that the ultraproduct of their C-values is strictly larger than C of the ultraproduct.
• They employ techniques from Boolean algebra and general topology, particularly those related to cardinal invariants of Boolean algebras and their ultraproducts.
• The argument is set in a model where the Generalized Continuum Hypothesis fails in a controlled way, enabling the required inequalities to be forced.

Experimental results

Research questions

• RQ1 Is it consistent that the incomparability of an ultraproduct of Boolean algebras is strictly less than the ultraproduct of the incomparabilities of the individual algebras?
• RQ2 Can the spread of an ultraproduct of Boolean algebras be strictly smaller than the ultraproduct of the spreads of the components?
• RQ3 Does the hereditary density of an ultraproduct of Boolean algebras fail to equal the ultraproduct of the hereditary densities in some models of ZFC?
• RQ4 To what extent do cardinal functions on Boolean algebras commute with ultraproduct constructions?
• RQ5 Is there a model of ZFC in which C(∏B_i/D) < |∏C(B_i)/D| for multiple cardinal characteristics C?

Key findings

• It is consistent that the incomparability of an ultraproduct of Boolean algebras is strictly less than the ultraproduct of the incomparabilities of the individual algebras.
• For the spread, character, π-character, hereditary Lindelöf number, and hereditary density, the paper establishes the consistency of C(∏B_i/D) < |∏C(B_i)/D|.
• The result shows that these cardinal functions do not preserve ultraproducts in a monotonic way, contradicting a natural expectation.
• The construction relies on a forcing extension where the Generalized Continuum Hypothesis fails at certain cardinals, enabling the required inequalities.
• The paper resolves multiple open problems originally posed by Monk regarding the behavior of cardinal functions under ultraproducts.
• The findings demonstrate that ultraproducts can significantly reduce the values of certain cardinal invariants, even when the component algebras have large values.