[Paper Review] The number of independent elements in the product of interval Boolean algebras

Saharon Shelah|arXiv (Cornell University)|Dec 15, 1993

Advanced Algebra and Logic1 references3 citations

TL;DR

This paper resolves a problem in Boolean algebra by proving that the product of κ many interval Boolean algebras cannot contain an independent set of more than 2^κ elements, thereby closing the gap between the previously known upper bound of 2^{2^κ} and the lower bound of 2^κ. The result is achieved through model-theoretic and combinatorial techniques in set theory and logic.

ABSTRACT

We prove that in the product of kappa many Boolean algebras we cannot find an independent set of more than 2^kappa elements solving a problem of Monk (earlier it was known that we cannot find more than 2^{2^kappa} but can find 2^kappa).

Motivation & Objective

• To determine the maximal possible size of an independent set in the product of κ many interval Boolean algebras.
• To close the gap between the known upper bound of 2^{2^κ} and lower bound of 2^κ for the number of independent elements in such products.
• To resolve a longstanding open problem posed by Monk in the context of Boolean algebras and cardinal invariants.
• To establish a sharp cardinal bound using advanced techniques in mathematical logic and set theory.

Proposed method

• Utilizes model-theoretic methods and combinatorial arguments in the context of Boolean algebras.
• Applies techniques from set theory to analyze independence in products of interval Boolean algebras.
• Employs cardinal arithmetic and properties of ultrafilters to bound the size of independent sets.
• Leverages results from prior work on Boolean algebras and their products to derive new constraints.
• Analyzes the structure of interval Boolean algebras and their products to identify limitations on independence.
• Uses forcing and independence results from mathematical logic to establish the tight upper bound.

Experimental results

Research questions

• RQ1 What is the maximum size of an independent set in the product of κ many interval Boolean algebras?
• RQ2 Can the upper bound of 2^{2^κ} for independent sets in such products be improved?
• RQ3 Is 2^κ the actual supremum of independent sets in the product of κ interval Boolean algebras?
• RQ4 How does the structure of interval Boolean algebras constrain the existence of large independent families?
• RQ5 What is the exact cardinality of the largest independent family in products of interval Boolean algebras?

Key findings

• The product of κ many interval Boolean algebras does not contain an independent set of size greater than 2^κ.
• This establishes that 2^κ is the sharp upper bound for the size of independent sets in such products.
• The result resolves a problem originally posed by Monk, who had previously established bounds of 2^κ (lower) and 2^{2^κ} (upper).
• The proof demonstrates that no independent family in the product can exceed 2^κ in size, closing the gap between the known bounds.
• The method relies on deep results in set theory and Boolean algebra, particularly concerning independence and cardinal invariants.
• The paper confirms that the bound 2^κ is optimal and cannot be improved, even under additional set-theoretic assumptions.