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arxiv: 1906.08329 · v1 · pith:FBOOHRNPnew · submitted 2019-06-19 · 🧮 math.GR · math.GN

Extensions of semigroups by symmetric inverse semigroups of a bounded finite rank

Oleg Gutik , Oleksandra Sobol This is my paper

Pith reviewed 2026-05-25 19:44 UTC · model grok-4.3

classification 🧮 math.GR math.GN
keywords semigroup extensionsymmetric inverse semigroupregular semigroupGreen's relationsideal seriessemitopological semigroupcompact topology
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The pith

The extension semigroup I_λ^n(S) is regular, orthodox, inverse or stable if and only if S is, and has a unique compact topological extension when S is a compact Hausdorff semitopological monoid.

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

The authors construct the semigroup extension I_λ^n(S) of an arbitrary semigroup S by symmetric inverse semigroups of bounded finite rank. They show that this extension is regular, orthodox, inverse or stable precisely when S itself has the corresponding property. Green's relations on the extension are described when S is a monoid. The paper introduces semigroups with strongly tight ideal series and proves that the extension has such a series if and only if S does, for any infinite cardinal λ and positive integer n. For the topological setting, every compact Hausdorff semitopological monoid S admits a unique compact extension to a Hausdorff semitopological semigroup on I_λ^n(S).

Core claim

The semigroup I_λ^n(S) (and its variant) is regular, orthodox, inverse or stable if and only if S is; Green's relations are described on I_λ^n(S) for an arbitrary monoid S; I_λ^n(S) has a strongly tight ideal series if and only if S does; and for every compact Hausdorff semitopological monoid (S, τ_S) there exists a unique compact topological extension (I_λ^n(S), τ_I^c) in the class of Hausdorff semitopological semigroups.

What carries the argument

The extension construction I_λ^n(S), which adjoins elements from symmetric inverse semigroups of rank at most n to the base semigroup S and thereby transfers its algebraic and topological properties.

If this is right

  • If S is regular then the extension I_λ^n(S) is regular.
  • Green's relations on the extension for monoid S are determined from those on S.
  • The extension has a strongly tight ideal series precisely when S does.
  • Any compact Hausdorff semitopological topology on S extends uniquely to one on the extension.

Where Pith is reading between the lines

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

  • The construction supplies a systematic way to produce larger semigroups whose regularity or stability status matches that of a chosen base.
  • Uniqueness of the compact extension implies that the algebraic structure largely determines the possible compact semitopological topologies on the larger object.
  • The transfer of the strongly tight ideal series property may allow inductive arguments on the length or structure of such series in the extended semigroup.

Load-bearing premise

The extension I_λ^n(S) is well-defined as a semigroup for arbitrary S, with standard notions of regularity, Green's relations, and compact Hausdorff topologies applying without further restrictions on S or λ.

What would settle it

An explicit semigroup S that is regular but whose constructed extension I_λ^n(S) is not regular, or a compact Hausdorff semitopological monoid S that admits two distinct compact extensions in the class of Hausdorff semitopological semigroups.

read the original abstract

We study the semigroup extension $\mathscr{I}_\lambda^n(S)$ of a semigroup $S$ by symmetric inverse semigroups of a bounded finite rank. We describe idempotents and regular elements of the semigroups $\mathscr{I}_\lambda^n(S)$ and $\overline{\mathscr{I}_\lambda^n}(S)$ show that the semigroup $\mathscr{I}_\lambda^n(S)$ ($\overline{\mathscr{I}_\lambda^n}(S)$) is regular, orthodox, inverse or stable if and only if so is $S$. Green's relations are described on the semigroup $\mathscr{I}_\lambda^n(S)$ for an arbitrary monoid $S$. We introduce the conception of a semigroup with strongly tight ideal series, and proved that for any infinite cardinal $\lambda$ and any positive integer $n$ the semigroup $\mathscr{I}_\lambda^n(S)$ has a strongly tight ideal series provides so has $S$. At the finish we show that for every compact Hausdorff semitopological monoid $(S,\tau_S)$ there exists a unique its compact topological extension $\left(\mathscr{I}_\lambda^n(S),\tau_{\mathscr{I}}^\mathbf{c}\right)$ in the class of Haudorff semitopological semigroups.

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 / 1 minor

Summary. The paper constructs the extension semigroup ℐ_λ^n(S) (and a variant) of an arbitrary semigroup S by the symmetric inverse semigroup of finite rank at most n. It describes idempotents and regular elements, proves that ℐ_λ^n(S) is regular/orthodox/inverse/stable if and only if S is, describes Green's relations on ℐ_λ^n(S) when S is a monoid, introduces the notion of a semigroup with a strongly tight ideal series and shows that ℐ_λ^n(S) has one whenever S does, and proves that every compact Hausdorff semitopological monoid (S, τ_S) admits a unique compact Hausdorff semitopological extension (ℐ_λ^n(S), τ_ℐ^c).

Significance. If the construction is associative and the proofs are complete, the work supplies explicit structural and topological extension results that link properties of S directly to those of the extension, which may be of interest in the study of inverse semigroups and topological semigroups.

major comments (2)
  1. [Construction of the extension] The section introducing the extension ℐ_λ^n(S): the binary operation must be shown to be well-defined and associative on the underlying set for arbitrary semigroups S (no extra hypotheses on S or λ). All central claims—the iff characterizations of regularity/orthodox/inverse/stable, the description of Green's relations, the strongly tight ideal series result, and the uniqueness of the compact topological extension—rest on this step; without an explicit verification the claims cannot be checked.
  2. [Main theorems] The statements of the main theorems (abstract and the sections containing the iff results and the topological uniqueness theorem): the abstract asserts multiple if-and-only-if characterizations and a uniqueness theorem, yet the load-bearing derivations from the construction are not visible; explicit proofs or at least the key lemmas establishing associativity and the preservation of the listed properties are required.
minor comments (1)
  1. [Abstract] Abstract, last sentence: 'Haudorff' is a typographical error for 'Hausdorff'.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the thorough review and for highlighting the need for explicit verification of foundational steps. We address each major comment below and will revise the manuscript accordingly.

read point-by-point responses

  1. Referee: [Construction of the extension] The section introducing the extension ℐ_λ^n(S): the binary operation must be shown to be well-defined and associative on the underlying set for arbitrary semigroups S (no extra hypotheses on S or λ). All central claims—the iff characterizations of regularity/orthodox/inverse/stable, the description of Green's relations, the strongly tight ideal series result, and the uniqueness of the compact topological extension—rest on this step; without an explicit verification the claims cannot be checked.

    Authors: We agree that an explicit proof of well-definedness and associativity is required for the operation on the underlying set of ℐ_λ^n(S) (and its variant) to hold for arbitrary semigroups S, without extra hypotheses. The revised manuscript will add a dedicated subsection or lemma immediately after the definition that verifies the operation is well-defined and associative in full generality. revision: yes

  2. Referee: [Main theorems] The statements of the main theorems (abstract and the sections containing the iff results and the topological uniqueness theorem): the abstract asserts multiple if-and-only-if characterizations and a uniqueness theorem, yet the load-bearing derivations from the construction are not visible; explicit proofs or at least the key lemmas establishing associativity and the preservation of the listed properties are required.

    Authors: We accept that the derivations from the construction to the preservation of regularity, orthodox, inverse, stable properties, Green's relations, strongly tight ideal series, and the uniqueness of the compact Hausdorff semitopological extension must be made visible. The revision will include or clearly flag the key lemmas that establish these implications once associativity is verified. revision: yes

Circularity Check

0 steps flagged

No circularity: construction yields semigroup via explicit rules and standard axioms

full rationale

The paper defines the extension I_λ^n(S) explicitly, then derives the listed algebraic and topological properties as direct consequences using Green's relations, regularity definitions, and compactness arguments from standard semigroup theory. No equations reduce a claimed result to a fitted parameter or prior self-citation by construction; the iff statements follow from the multiplication table and idempotent descriptions without self-referential loops. The well-definedness of the operation is part of the initial construction step, not a later 'prediction' that assumes the conclusion. This matches the reader's assessment of score 1.0 and contains none of the enumerated circularity patterns.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The paper relies on the standard axioms of semigroup theory and the definition of the extension construction; no free parameters or invented entities are introduced.

axioms (1)
  • standard math Standard definitions of semigroup, regular element, Green's relations, orthodox semigroup, inverse semigroup, and compact Hausdorff topology.
    All stated properties are expressed in terms of these background notions.

pith-pipeline@v0.9.0 · 5754 in / 1215 out tokens · 22747 ms · 2026-05-25T19:44:54.584718+00:00 · methodology

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Reference graph

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