REVIEW 3 major objections 6 minor 1 cited by
On the dichotomy of a locally compact semitopological monoid of order isomorphisms between principal filters of $\mathbb{N}^n$ with adjoined zero
T0 review · 3 major / 6 minor · reviewed 2026-08-14 · deepseek-v4-flash
Pith's one-line read The monoid IPF(N^n)^0 admits only two locally compact semitopological topologies: discrete or compact.
desk verdict Clean dichotomy for IPF(N^n)^0 that extends the bicyclic monoid result; the theorem is believable and the strategy is sound, but the printed typos and a heavy reliance on [14] mean it needs a careful referee before publication. read the letter →
The pith
A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.
The reading
What carries the argument
The load-bearing machinery is the presentation of $\mathscr{I\!P\!F}(\mathbb{N}^n)$ as a semidirect product $S_n \ltimes C(p,q)^n$, where $C(p,q)$ is the bicyclic monoid; elements are written as $(\sigma,[x,y])$ with $\sigma\in S_n$ and $x,y\in\mathbb{N}^n$. The argument first uses the prior theorem that every shift-continuous topology on $\mathscr{I\!P\!F}(\mathbb{N}^n)$ is discrete, so in a non-discrete locally compact semitopological topology all non-zero points are isolated and a zero-neighbourhood can be chosen open and compact (Lemma 1). A combinatorial path lemma (Lemma 4) shows that when $\mathbb{N}^n$ is split into two infinite disjoint sets, some infinite subfamily of one side is sent entirely into the other side by a unit coordinate shift $g_k$ or its inverse $g_k^{-1}$. In the monoid, left and right translations by elements such as $(1,[1,\mathbf{2}_k])$ and $(1,[\mathbf{2}_k,1])$, where $\mathbf{2}_k$ has a $2$ in the $k$-th coordinate and $1$ elsewhere, realize these shifts on the fibres $L^a_\sigma=\{(\sigma,[a,x]):x\in\mathbb{N}^n\}$, and separate continuity lets the proof transfer a compact open zero-neighbourhood into any given zero-neighbourhood. Iterating this forces the complement of every zero-neighbourhood to be finite, which is exactly the cofinite topology of $\tau_{\mathrm{Ac}}$.
What would settle it
Construct a Hausdorff locally compact semitopological semigroup topology on $\mathscr{I\!P\!F}(\mathbb{N}^n)^0$ in which some neighbourhood of zero has infinite complement; the theorem's Corollary 8 predicts that in any non-discrete such topology the complement of every zero-neighbourhood is finite, so even one example with a co-infinite zero-neighbourhood, with separate continuity verified, would settle the dichotomy false.
Extended reading notes
Core claim
The central claim is Theorem 11: for every positive integer $n$, if $\mathscr{I\!P\!F}(\mathbb{N}^n)^0$ is a Hausdorff locally compact semitopological semigroup, then either it is discrete or it is topologically isomorphic to $(\mathscr{I\!P\!F}(\mathbb{N}^n)^0,\tau_{\mathrm{Ac}})$, where $\tau_{\mathrm{Ac}}$ is the topology of the Alexandroff one-point compactification of the discrete space $\mathscr{I\!P\!F}(\mathbb{N}^n)$ with the zero as the point at infinity. Equivalently, the dichotomy is between two extremes: isolated points everywhere, or isolated points except for one compactifying zero whose neighbourhoods are cofinite. The proof also yields Corollary 12: if the operation is jointly continuous rather than only separately continuous, then the locally compact topology must be discrete, because $\mathscr{I\!P\!F}(\mathbb{N}^n)$ does not embed into a compact Hausdorff topological semigroup.
Load-bearing premise
The proof assumes as a starting point the prior theorem that every topology on $\mathscr{I\!P\!F}(\mathbb{N}^n)$ making shifts continuous is discrete, and this is what forces every non-zero element to be an isolated point; without it, Lemma 1's compact-open zero-neighbourhood and finite-complement conclusions do not follow.
Editorial extensions
If this is right
- For each $n$, the monoid $\mathscr{I\!P\!F}(\mathbb{N}^n)^0$ has no Hausdorff locally compact semitopological semigroup topology between the discrete topology and the Alexandroff one-point compactification.
- Corollary 12: in the jointly continuous case, the only locally compact semitopological topology is the discrete one, since the underlying monoid cannot be embedded in a compact Hausdorff topological semigroup.
- The topology $\tau_{\mathrm{Ac}}$ is the unique compact semitopological topology on this monoid, so compactness determines the topology uniquely.
- The dichotomy holds uniformly across all dimensions $n$, reducing to the bicyclic-monoid dichotomy when $n=1$ and extending it to every higher power of $\mathbb{N}$.
Reading between the lines
- Lemma 4 is a self-contained combinatorial statement about partitions of $\mathbb{N}^n$ that could be applied to other semigroups with coordinate-shift translations; the topological part of the proof would then transfer unchanged.
- A plausible testable extension is that any inverse monoid with a similar direct-product-of-bicyclic structure, finitely many solutions to first-order equations, and only discrete shift-continuous topologies should satisfy the same compact-versus-discrete dichotomy.
- The theorem illustrates a general mechanism: algebraic finiteness of solution sets plus a discreteness theorem for the monoid without zero forces locally compact semitopological topologies to collapse to two extremes, suggesting the same pair of extremes for monoids built from other partial order isomorphisms.
Signed reviews
Editorial analysis
A structured set of objections, weighed in public.
Referee Report
Summary. The paper studies the monoid IPF(N^n) of all order isomorphisms between principal filters of N^n with the product order, with a zero adjoined, and proves a dichotomy theorem: every Hausdorff locally compact semitopological semigroup topology on this monoid is either discrete or topologically isomorphic to the Alexandroff one-point compactification of the discrete topology (Theorem 11). The proof combines structure theory of IPF(N^n) imported from [14] with a sequence of lemmas (Lemmas 1, 3, 4, 5, 6, 7, Corollary 8) showing that every open neighborhood of zero is cofinite under the locally compact semitopological assumption. Example 9 verifies that the Alexandroff compactification topology indeed makes the monoid a compact semitopological semigroup, and Corollary 12 draws the topological-semigroup consequence that the space must be discrete.
Significance. If the dichotomy is correct, it gives a complete classification of locally compact semitopological topologies on IPF(N^n)^0, extending the known dichotomy for the bicyclic monoid with adjoined zero and for several related classes. The result is a natural and worthwhile contribution to topological semigroup theory. The proof is not a reduction by definition to prior results; the finite-difference propagation lemmas constitute substantive new work, and the paper contains no free parameters or data fitting. The main strength is the clarity of the overall strategy; the main weakness is the heavy reliance on the external paper [14], which is listed as to-appear and is partly by the same author, for the crucial fact that non-zero elements are isolated.
major comments (3)
- [Lemma 1(1)] The proof asserts that 'all non-zero elements of the semigroup IPF(N^n)^0 are isolated points' without proof. This assertion is imported from [14], where it is stated that every shift-continuous topology on IPF(N^n) is discrete. The manuscript neither states the precise theorem from [14] nor proves that the topology induced on IPF(N^n) by a semitopological topology on IPF(N^n)^0 is shift-continuous. Since Lemma 1(2) and every subsequent lemma depend on this isolation property, Theorem 11 is conditional on an unverified external result. The authors should state the relevant theorem from [14] explicitly and prove the reduction to it, or provide a self-contained proof of the isolation of non-zero elements.
- [Lemma 6] The displayed definition of q and p is incorrect: the claim q - p = a - b is false as printed. For example, with n = 1, a = 3, b = 5, the definitions give p = 2, q = 1, so q - p = -1, whereas a - b = -2. Consequently the computation (σ,[q,p] ∗ [b,x]) = (σ,[a,x]) does not follow. In addition, the proof concludes that 'the set L^b_σ \ V(0) is finite' after V(0) is chosen using separate continuity; however, Lemma 5 was applied to U(0), not to V(0), so this finiteness is not available for the V(0) constructed later in the proof. Both flaws are repairable: choose the compact open neighborhood V0 first, apply Lemma 5 to V0 to obtain b, then choose a compact open V′ ⊇ V0 with h·V′ ⊂ U for the translation h; and define q,p so that q - p = a - b with p_i,q_i ≥ 1 and p ≤ b. As printed, however, the proof of Lemma 6 has a load-bearing gap.
- [Lemma 5, case (ii)] The displayed identity for ρ(1,[2_k,1]) states that it equals g_k^{-1} on L^a_σ \ {x_k = 2}, but the right translation by (1,[2_k,1]) agrees with g_k^{-1} on L^a_σ \ {x_k = 1}: for x_k = 2 it sends x_k to 1, and for x_k = 1 it fixes the coordinate. The set C used in case (ii) satisfies x_k ≥ 2 because g_k^{-1} is applied to it, so the intended argument works with the correct restriction, but the printed formula is wrong and should be corrected.
minor comments (6)
- [Lemma 4] The statement contains the typo 'there xists' instead of 'there exists', and the proof writes 'A ⊔ B = Nn and A ∩ B = ∅', where the disjoint union notation already implies the intersection condition.
- [Example 9] The text contains 'followinh' instead of 'following', and the parenthetical 'first order equation in I PF(Nn' is missing a closing parenthesis; it should read 'I PF(Nn)'.
- [Theorem 11] Theorem 11 says 'Lemma 8 and Remark 10 implies the following dichotomy', but the reference should be to Corollary 8, not Lemma 8.
- [Remark 10] The sentence 'In [14] showed that the discrete topology τd is a unique topology...' is grammatically incomplete; it should read '[14] showed that the discrete topology τd is the unique topology...'. The precise formulation of the result used from [14] should also be given.
- [Introduction] The definition 'a semigroup S with the an adjoined by S^0' is garbled; it should read 'a semigroup S with an adjoined zero is denoted by S^0'.
- [References] Reference [14] is listed as 'to appear'; if it has appeared by the time of publication, the full publication data should be supplied, and the specific theorems from [14] used in this paper (isolation of non-zero elements, Proposition 2.26, and the algebraic structure of IPF(N^n)) should be cited with precise theorem numbers.
Circularity Check
No circularity: the dichotomy is a new compactness argument built on a legitimate external theorem from [14], not a restatement of its inputs.
full rationale
The claimed dichotomy does not reduce by construction to any fitted parameter or definitional identity. Lemma 1(1) invokes the prior result from [14] that every shift-continuous topology on IPF(N^n) is discrete to conclude that non-zero points are isolated; this is a real external premise about IPF(N^n) without the adjoined zero, and it does not assert the target dichotomy for locally compact semitopological IPF(N^n)^0. The new content lies in Lemmas 3, 5, 6, 7 and Corollary 8, which use local compactness and separate continuity to force cofinite neighbourhoods of zero; these arguments are not restatements of [14]. Example 9 checks separate continuity of tau_Ac using Proposition 2.26 of [14], an independent finiteness-of-solutions counting fact. The printed mistakes in Lemmas 5-6 (e.g., q-p=a-b is false for positive integers; finiteness of L_b^sigma \ V(0) is not justified) are local correctness defects with an available repair, not a circular step: selecting a compact open V0 first and then a smaller V' with h·V' subset U restores the argument. Because the load-bearing citation is a prior parameter-free theorem whose assumptions do not include the conclusion of this paper, the reliance on [14] is legitimate evidence rather than circularity.
Assumptions & free parameters
assumptions (4)
- domain assumption IPF(N^n) is isomorphic to the semidirect product S_n ⋉ C(p,q)^n, where C(p,q) is the bicyclic monoid.
- domain assumption Every shift-continuous (semitopological) topology on IPF(N^n) is discrete, so non-zero elements of IPF(N^n)^0 are isolated points.
- domain assumption IPF(N^n) does not embed into a compact Hausdorff topological semigroup.
- domain assumption Every equation of the form x·s=t or s·x=t in IPF(N^n) has finitely many solutions for fixed s and t.
Cite this review
Pith. "Pith review of On the dichotomy of a locally compact semitopological monoid of order isomorphisms between principal filters of $\mathbb{N}^n$ with adjoined zero." pith.science (2026). https://pith.science/paper/6OQGQILQ
@misc{pith2026190808521,
author = {Pith},
title = {Pith review of: On the dichotomy of a locally compact semitopological monoid of order isomorphisms between principal filters of $\mathbbN^n$ with adjoined zero},
year = {2026},
howpublished = {\url{https://pith.science/paper/6OQGQILQ}},
note = {Machine review of arXiv:1908.08521}
}
abstract
Let $n$ be any positive integer and $\mathscr{I\!P\!F}(\mathbb{N}^n)$ be the semigroup of all order isomorphisms between principal filters of the $n$-th power of the set of positive integers $\mathbb{N}$ with the product order. We prove that a Hausdorff locally compact semitopological semigroup ${\mathscr{I\!P\!F}(\mathbb{N}^n)}$ with an adjoined zero is either compact or discrete.
Forward citations
Cited by 1 Pith paper
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On the lattice of weak topologies on the bicyclic monoid with adjoined zero
The lattice of weak shift-continuous topologies on C^0 is isomorphic to the product of two copies of the poset of shift-invariant filters with a top element attached, and it contains antichains of size 2^c and chains ...
Reference graph
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