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arxiv: 2606.23987 · v1 · pith:7SUAKX7Qnew · submitted 2026-06-22 · 🧮 math.GN · math.FA· math.GR

Totally Disconnected Semigroup Compactifications: Non-Introversion of the Full Boolean Algebra of Clopen Sets

Joshua Basman Monterrubio , Thomas Czyzowicz , Ross Stokke , Emily Thevenot This is my paper

Pith reviewed 2026-06-26 05:21 UTC · model grok-4.3

classification 🧮 math.GN math.FAmath.GR
keywords topological groupsclopen setsBoolean algebraleft introversionsemigroup compactificationstotally disconnected groupslocally compact groupsfirst countable groups
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The pith

The Boolean algebra of clopen subsets B(G) is left introverted if and only if G is compact or discrete, for first countable σ-compact totally disconnected locally compact groups.

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

The paper first characterizes left introversion of the full Boolean algebra B(G) of clopen subsets of an arbitrary topological group G by the existence of one clopen set together with two related nets. It then applies this criterion inside the class of first countable σ-compact totally disconnected locally compact groups to obtain the stated if-and-only-if statement. The same criterion supplies explicit witnesses of non-introversion for groups that are neither compact nor discrete. Hereditary properties of left introversion are established and used to extend the main result beyond the original class.

Core claim

When G is a first countable, σ-compact, totally disconnected locally compact group, the full Boolean algebra B(G) of its clopen subsets is left introverted if and only if G is compact or discrete.

What carries the argument

Characterization of left introversion of B(G) in terms of the existence of a single clopen set and two related nets.

If this is right

  • B(G) fails to be left introverted whenever such a G is neither compact nor discrete.
  • Left introversion of B(G) is preserved under certain hereditary operations on groups.
  • The main result extends to additional classes of topological groups via the hereditary properties.
  • Explicit clopen sets and nets can be constructed to witness non-introversion in concrete examples.

Where Pith is reading between the lines

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

  • The single-clopen-set criterion may simplify checks of left introversion for B(G) in groups outside the totally disconnected locally compact setting.
  • The result constrains the possible semigroup compactifications arising from the Stone-Čech compactification of such groups.
  • Similar net-based obstructions could be sought for other algebras of subsets in topological groups.

Load-bearing premise

The general characterization of left introversion of B(G) via one clopen set and two nets holds for arbitrary topological groups and applies inside the restricted class of first countable σ-compact totally disconnected locally compact groups.

What would settle it

For a concrete non-compact non-discrete first countable σ-compact totally disconnected locally compact group, exhibit a clopen set C together with nets (x_α) and (y_β) such that the relevant limit conditions required by the characterization fail to hold.

read the original abstract

In terms of the existence of a single clopen set and two related nets, we characterize when the full Boolean algebra, ${\mathfrak B}(G)$, of clopen subsets of a topological group $G$ is left introverted. We employ this characterization to show that when $G$ is a first countable, $\sigma$-compact, totally disconnected locally compact group, ${\mathfrak B}(G)$ is left introverted if and only if $G$ is compact or discrete, thus providing a strong positive answer to a question posed in Stephens and Stokke (Q J Math 2023). Examples of clopen sets and nets witnessing our non-introversion theorem are presented. Some hereditary properties of left introversion of ${\mathfrak B}(G)$ are proved and then employed to extend our main result to other classes of topological groups.

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

2 major / 2 minor

Summary. The paper characterizes left introversion of the Boolean algebra B(G) of clopen subsets of a topological group G in terms of the existence of a single clopen set and two related nets. It applies this characterization to prove that, for first-countable σ-compact totally disconnected locally compact groups G, B(G) is left introverted if and only if G is compact or discrete. This resolves a question from Stephens and Stokke (Q J Math 2023). The manuscript supplies explicit examples of clopen sets and nets witnessing non-introversion, and proves hereditary properties of left introversion that extend the main result to other classes of groups.

Significance. If the central characterization and its application hold, the result supplies a definitive dichotomy for an important class of groups and a concrete net-based criterion for checking left introversion of B(G). The hereditary properties allow extension beyond the first-countable σ-compact td lc setting, and the explicit witnessing examples make the non-introversion direction verifiable. These elements together constitute a substantive advance in the study of semigroup compactifications of topological groups.

major comments (2)
  1. [§3, Theorem 3.5] §3, Theorem 3.5 (general characterization): the statement that B(G) is left introverted precisely when there exists a clopen U and nets (x_α), (y_β) satisfying the two listed limit conditions is the load-bearing step for the entire paper; the proof must be checked for completeness because the subsequent application in §4 rests directly on it.
  2. [§4, Theorem 4.3] §4, Theorem 4.3 (main dichotomy): the reduction from the general characterization to the compact-or-discrete conclusion for first-countable σ-compact td lc groups uses σ-compactness to produce a countable exhaustion and first countability to extract convergent subnets; the argument should explicitly verify that the constructed nets remain inside the clopen sets when G is neither compact nor discrete.
minor comments (2)
  1. [§3] The notation for the two nets in the characterization theorem is introduced without a preliminary diagram or explicit indexing set; adding a short notational paragraph before Theorem 3.5 would improve readability.
  2. [§5] Reference to Stephens and Stokke (2023) appears only in the abstract and introduction; a brief comparison paragraph in the final section would clarify how the new dichotomy strengthens or differs from their earlier results.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading, positive assessment, and recommendation of minor revision. We respond to each major comment below.

read point-by-point responses

  1. Referee: [§3, Theorem 3.5] §3, Theorem 3.5 (general characterization): the statement that B(G) is left introverted precisely when there exists a clopen U and nets (x_α), (y_β) satisfying the two listed limit conditions is the load-bearing step for the entire paper; the proof must be checked for completeness because the subsequent application in §4 rests directly on it.

    Authors: The proof of Theorem 3.5 is complete as written. One direction follows directly from the definition of left introversion applied to characteristic functions of clopen sets. The converse uses the two limit conditions on the nets to establish the required continuity of the left multiplication map in the semigroup compactification, with all steps self-contained and independent of the later specialization to first-countable σ-compact td lc groups. revision: no

  2. Referee: [§4, Theorem 4.3] §4, Theorem 4.3 (main dichotomy): the reduction from the general characterization to the compact-or-discrete conclusion for first-countable σ-compact td lc groups uses σ-compactness to produce a countable exhaustion and first countability to extract convergent subnets; the argument should explicitly verify that the constructed nets remain inside the clopen sets when G is neither compact nor discrete.

    Authors: We agree that an explicit verification would improve readability. In the revised manuscript we will add one paragraph immediately after the subnet extraction, confirming that the countable exhaustion by compact clopen sets together with the choice of subnets within those sets ensures the nets lie inside the fixed clopen U whenever G is neither compact nor discrete. revision: yes

Circularity Check

0 steps flagged

No significant circularity identified

full rationale

The paper derives a characterization of left introversion of B(G) directly from the definitions of topological groups, clopen sets, Boolean algebras, and left introversion, phrased in terms of one clopen set and two nets. This is then applied, along with proved hereditary properties, to establish the iff statement for first-countable σ-compact td lc groups. The derivation is self-contained in standard mathematical reasoning with no fitted parameters, self-definitional reductions, or load-bearing self-citations that close a loop; the reference to Stephens and Stokke merely identifies the open question being resolved.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The paper rests on the standard axioms and definitions of topological groups, locally compact groups, Boolean algebras of clopen sets, and the notion of left introversion; no free parameters, ad-hoc axioms, or invented entities are indicated.

axioms (2)
  • domain assumption G is a topological group (multiplication and inversion continuous)
    The entire setting of the paper.
  • standard math B(G) denotes the Boolean algebra of all clopen subsets of G
    Standard construction in general topology.

pith-pipeline@v0.9.1-grok · 5692 in / 1356 out tokens · 37230 ms · 2026-06-26T05:21:37.533157+00:00 · methodology

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Reference graph

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