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arxiv: 2303.08491 · v4 · submitted 2023-03-15 · 🧮 math.GN · math.LO

Many subalgebras of mathcal{P}(ω)/fin

Klaas Pieter Hart This is my paper

Pith reviewed 2026-05-24 09:53 UTC · model grok-4.3

classification 🧮 math.GN math.LO
keywords Boolean algebrasP(ω)/finsubalgebrasStone dualityzero-dimensional compact spacescontinuumembeddings
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The pith

The Boolean algebra P(ω)/fin contains a family of subalgebras {B_X : X ⊆ c} such that X ⊆ Y implies B_Y is a subalgebra of B_X while X notsubseteq Y implies B_Y does not embed into B_X.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper shows that P(ω)/fin admits a family of subalgebras indexed by subsets of the continuum. When one index set sits inside another, the algebra attached to the larger index is contained as a subalgebra in the algebra for the smaller index. When two index sets are incomparable under inclusion, neither algebra embeds into the other. The construction proceeds by producing a parallel family of separable zero-dimensional compact spaces and transferring the inclusion and non-embeddability relations across Stone duality.

Core claim

There exists a family {B_X : X ⊆ c} of subalgebras of P(ω)/fin such that X ⊆ Y implies B_Y is a subalgebra of B_X, and if X ⊈ Y then B_Y is not embeddable into B_X.

What carries the argument

Stone duality applied to an auxiliary family of separable zero-dimensional compact spaces indexed by subsets of the continuum.

If this is right

  • The poset of subalgebras of P(ω)/fin under inclusion contains an order-isomorphic copy of the power set of the continuum ordered by reverse inclusion.
  • Embeddings between these subalgebras are possible only when the index sets stand in the inclusion relation.
  • The algebra P(ω)/fin therefore realizes at least continuum many distinct subalgebras whose mutual embeddability is rigidly controlled by set-theoretic inclusion.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The same indexing technique could be attempted inside other quotients or Boolean algebras that admit sufficiently many zero-dimensional compact representations.
  • The construction supplies a concrete lower bound on the width and height of the subalgebra lattice of P(ω)/fin.
  • One could test whether the spaces can be chosen metrizable or with additional separation properties without losing the non-embeddability.

Load-bearing premise

There exists a family of separable zero-dimensional compact spaces, indexed by subsets of the continuum, whose Stone spaces yield Boolean algebras satisfying the stated subalgebra and non-embeddability relations.

What would settle it

An explicit counter-example would be a pair of subsets X and Y with X notsubseteq Y together with an embedding of the corresponding algebra B_Y into B_X; failure to construct any such family of spaces at all would also falsify the existence claim.

read the original abstract

In answer to a question on Mathoverflow we show that the Boolean algebra $\mathcal{P}(\omega)/\mathit{fin}$ contains a family $\{\mathcal{B}_X:X\subseteq\mathfrak{c}\}$ of subalgebras with the property that $X\subseteq Y$ implies $\mathcal{B}_Y$ is a subalgebra of $\mathcal{B}_X$ and if $X\not\subseteq Y$ then $\mathcal{B}_Y$ is not embeddable into~$\mathcal{B}_X$. The proof proceeds by Stone duality and the construction of a suitable family of separable zero-dimensional compact spaces.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

0 major / 2 minor

Summary. The manuscript proves that the Boolean algebra P(ω)/fin contains a family {B_X : X ⊆ c} of subalgebras such that X ⊆ Y implies B_Y is a subalgebra of B_X, and if X ⊈ Y then B_Y does not embed into B_X. The argument proceeds by Stone duality, via an explicit construction of a family of separable zero-dimensional compact spaces indexed by subsets of the continuum whose Stone spaces realize the desired algebraic relations.

Significance. This supplies an affirmative answer to a MathOverflow question by realizing a large indexed family of subalgebras inside P(ω)/fin whose inclusion and embeddability relations are controlled by the index sets. The construction via Stone duality is a standard and direct method in this area; when the spaces are shown to exist and to produce the stated non-embeddability via topological invariants, the result is a concrete contribution to the structure theory of this Boolean algebra.

minor comments (2)
  1. The abstract states the claim and method but does not sketch the key properties of the constructed spaces; a single sentence indicating how the spaces enforce non-embeddability when X ⊈ Y would improve readability.
  2. Notation for the continuum alternates between c and frak{c}; uniform use throughout the text would reduce minor confusion.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their positive assessment of the manuscript and for recommending acceptance.

Circularity Check

0 steps flagged

No significant circularity; direct existence construction via Stone duality

full rationale

The paper establishes an existence result by explicitly constructing a family of separable zero-dimensional compact spaces indexed by subsets of the continuum. These spaces are used with Stone duality to produce the required subalgebras B_X of P(ω)/fin satisfying the inclusion and non-embeddability conditions. No equations reduce outputs to fitted inputs, no self-definitional loops appear, and no load-bearing self-citations or imported uniqueness theorems are invoked. The derivation is self-contained as a topological construction independent of the target Boolean-algebra properties.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The proof relies on the standard theorem of Stone duality between Boolean algebras and compact zero-dimensional spaces; no free parameters, ad-hoc axioms, or new entities are introduced in the abstract.

axioms (1)
  • standard math Stone duality: every Boolean algebra is isomorphic to the clopen algebra of its Stone space (compact zero-dimensional Hausdorff space).
    Invoked explicitly as the bridge between the algebraic statement and the topological construction.

pith-pipeline@v0.9.0 · 5618 in / 1268 out tokens · 24140 ms · 2026-05-24T09:53:04.624238+00:00 · methodology

discussion (0)

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Reference graph

Works this paper leans on

8 extracted references · 8 canonical work pages

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