[Paper Review] Constructing a Lattice With the Cardinalities of the Sets of Congruences, Filters and Ideals Pairwise Distinct

Claudia Mureșan|arXiv (Cornell University)|Oct 27, 2017

Advanced Algebra and Logic4 citations

TL;DR

This paper constructs a bounded lattice in which the cardinalities of the sets of filters, ideals, and congruences are pairwise distinct, using horizontal sums of bounded lattices without assuming the Continuum Hypothesis. The key contribution is a novel lattice construction that resolves an open problem in universal algebra by ensuring all three sets have different infinite cardinalities.

ABSTRACT

In this paper, I am giving a solution to the problem I have proposed in [eucard]: finding a lattice with the cardinalities of the sets of filters, ideals and congruences pairwise distinct; I am constructing such a lattice by using horizontal sums, and without enforcing the Continuum Hypothesis. This research has also produced a set of results on (prime) filters, ideals and congruences in horizontal sums of bounded lattices.

Motivation & Objective

• To resolve an open problem posed in [eucard] concerning the existence of a lattice with pairwise distinct cardinalities for its sets of filters, ideals, and congruences.
• To construct such a lattice using horizontal sums of bounded lattices, avoiding reliance on the Continuum Hypothesis.
• To establish foundational results on (prime) filters, ideals, and congruences in horizontal sums of bounded lattices.

Proposed method

• The construction employs horizontal sums of bounded lattices to control the cardinalities of filters, ideals, and congruences independently.
• The method ensures that the resulting lattice's filter, ideal, and congruence sets have distinct infinite cardinalities.
• It leverages structural properties of horizontal sums to isolate and manipulate the sizes of these sets.
• The approach avoids set-theoretic assumptions like the Continuum Hypothesis, ensuring the result is valid in standard ZFC set theory.
• The analysis focuses on the interplay between lattice operations and the generation of filters, ideals, and congruences in the constructed sum.

Experimental results

Research questions

• RQ1 Can a bounded lattice be constructed such that the cardinalities of its sets of filters, ideals, and congruences are pairwise distinct?
• RQ2 Is such a construction possible without assuming the Continuum Hypothesis?
• RQ3 How do the properties of (prime) filters, ideals, and congruences behave in horizontal sums of bounded lattices?
• RQ4 What structural conditions ensure that the cardinalities of these three sets differ in a lattice?
• RQ5 Can the cardinality of the set of congruences be made strictly different from both the filter and ideal sets in a lattice?

Key findings

• A bounded lattice is explicitly constructed in which the sets of filters, ideals, and congruences have pairwise distinct infinite cardinalities.
• The construction relies solely on horizontal sums of bounded lattices and does not require the Continuum Hypothesis.
• The method produces a lattice where the cardinality of the set of congruences differs from both the filter and ideal sets.
• New structural results are established for (prime) filters, ideals, and congruences in horizontal sums of bounded lattices.
• The cardinalities of the three sets are shown to be independent in the constructed lattice, satisfying the original problem's conditions.