Natural extensions of embeddable semigroup actions

\'Alvaro Bustos-Gajardo; Miguel Donoso-Echenique; Raimundo Brice\~no

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Natural extensions of embeddable semigroup actions

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Pith's one-line read This paper proves that the free group on an embeddable semigroup is the unique receiving group that always converts a surjective semigroup action into an invertible group action.

desk verdict A genuinely new and mostly sound theory of natural extensions for semigroup actions, with one repairable gap in Proposition 4.6 that should be fixed before publication. read the letter →

arxiv 2501.05536 v2 pith:VYKQ66ZL submitted 2025-01-09 math.DS math.GR

Raimundo Briceño , Álvaro Bustos-Gajardo , Miguel Donoso-Echenique This is my paper · ORCID

classification math.DSmath.GR MSC 20M3037B0220M0537B1020F0520M50

keywords countablesemigroupactionnaturalextensionfreeS-groupleftreversibletopologicalentropytransitivitysubshift

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

This paper develops a theory of natural extensions for continuous actions of countable embeddable semigroups: given a semigroup S embedded in a group G, an S-action extends to G by recording, at each group element, the future of the orbit. The central result, Theorem B, says that a receiving group G extends every surjective continuous S-action—equivalently, every surjective S-subshift—if and only if G is a realization of the free S-group, the group generated by S with no relations beyond those already in S. This shows that both the choice of the group and the choice of the embedding are essential, and that a surjective action can fail to have any invertible extension to a particular generating group precisely when the semigroup is not left reversible. In the left-reversible case the paper proves that the natural extension is the smallest compact extension, and in the amenable case topological entropy, ergodicity, transitivity, and minimality lift from S to the group action.

What carries the argument

The central object is the free S-group (Γ,γ): the group presented by the generators and relations of S, existing for every semigroup and characterized as the initial object among S-groups. The concrete construction is the space $X_G = \{ (x_h)_{h\in G} \in X^G : s\cdot x_h = x_{\eta(s)h} \text{ for all } s\in S,\, h\in G \}$, equipped with the shift action of G and the projection π(x)=x_{1_G}; Proposition 2.4 identifies X_G with the natural G-extension exactly when π is surjective, and Proposition 2.10 rewrites it as a G-subshift by reading the same forbidden patterns over G instead of S. The proof of Theorem B separates into a feasibility direction, in which every surjective S-action is shown to extend to the free S-group by lifting through the free semigroup on a presentation, and an infeasibility direction, in which any proper quotient G of Γ is shown to admit a surjective S-subshift with X_G empty. For Theorem D the engine is the cited Lemma 4.4, which equates the property that G is the group of right fractions of S with downward directedness of the pre-order $g \leq_S h \iff hg^{-1} \in \eta(S)$ and with thickness of η(S) in G; this provides the translates of finite sets into η(S) that carry the compact-extension factor maps and the dynamical lifting arguments.

What would settle it

Take the subshift $X = \{ x \in \Gamma^S : x(s) = \gamma(s)x(1_S) \}$ constructed in Proposition 3.8 for a proper quotient G of the free S-group Γ, and compute whether its G-extension space $X_G$ is empty. The theorem predicts it is empty for every nontrivial element of the kernel of the quotient map Γ → G; one nonempty $X_G$ would disprove Theorem B.

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Extended reading notes

Core claim

The load-bearing claim is Theorem B: for an embeddable monoid S and a realization (Γ,γ) of the free S-group, a receiving S-group G has the property that every surjective continuous S-action (equivalently, every surjective S-subshift) is G-extensible if and only if G is a realization of the free S-group. When Γ is residually finite, the characterization reduces to finite surjective S-subshifts: any proper quotient of Γ is betrayed by a finite symbolic subshift whose G-extension space is empty. The paper also establishes Theorem C, that the natural Γ-extension is the terminal object in the category of all invertible extensions, so it encodes every other extension; and Theorem D, that the classical demand that every compact G-extension factor onto the natural extension holds exactly when S is left reversible and G is its group of right fractions. For left reversible bicancellative monoids, the natural extension to the group of right fractions preserves amenable topological entropy and lifts topological ergodicity, transitivity, and minimality.

Load-bearing premise

The left-reversible half of the paper rests on a cited algebraic equivalence: for a left reversible semigroup, a receiving group being the group of right fractions is the same as being able to move any finite set of group elements into the semigroup by one right translation; if that equivalence fails, the compact-extension characterization and the entropy and dynamical lifting theorems no longer follow.

Editorial extensions

If this is right

Every surjective continuous action of an embeddable monoid extends to the free S-group, while no other receiving S-group can play this role for all actions simultaneously.
Failure of extensibility to a given generating group is an algebraic obstruction: the group must be a proper quotient of the free S-group, and when the free S-group is residually finite a finite symbolic subshift certifies the failure.
For a left reversible bicancellative semigroup the receiving group is unique up to isomorphism—the group of right fractions—and the natural extension is the smallest compact extension, with factor maps lifting to it.
Amenable topological entropy is preserved by the natural extension: $h(X,S)$ equals $h(X_{\Gamma_S}, \Gamma_S)$, so entropy can be computed in the invertible group setting.
Topological ergodicity, topological transitivity, and minimality all lift from the semigroup action to the group action on its natural extension.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

The paper leaves implicit that Theorem B supplies a dynamical certificate of non-universality: to show a proposed receiving pair (G,η) is not the free S-group, it suffices to exhibit any surjective S-subshift whose G-extension space is empty, a check that may be easier than comparing algebraic presentations.
In the non-left-reversible case, natural extensions are only unique relative to a fixed receiving group; different embeddings of the same semigroup into the same non-Hopfian group can therefore produce genuinely different invertible envelopes of the same dynamics, and the paper's examples suggest this is a real phenomenon rather than a pathology.
The residual-finiteness criterion suggests a computational route: for semigroups whose free S-group is residually finite, verifying universality of a receiving group can be reduced to testing finite surjective subshifts, which is a finite-state search in practice.
The entropy equality raises the question, not addressed here, of whether finer invariants—measure-theoretic entropy, periodic orbit data, or the maximal equicontinuous factor—also lift through the natural extension in the amenable left-reversible case.

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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the letter

Short version: this is a real contribution. The paper gives the first general treatment of natural extensions for countable embeddable semigroup actions, and the main new results—Theorem B (free S-group is the unique receiving pair that always admits extensions), Theorem C (terminal object), Theorem D (left reversibility characterized by factor maps)—are genuinely new, not in the cited literature. The symbolic and algebraic arguments are careful; the free S-group construction and the transfer to subshifts are sound.

The strongest part is the free S-group characterization. The proof of Proposition 3.6 is detailed and checks out. I also like the concrete examples (BS(1,2), BS(2,3) non-Hopfian) showing that group choice and embedding both matter.

Soft spots:

1. Proposition 4.6 has a real gap as printed. The proof picks a cofinal sequence (g_k) and defines y^k by choosing arbitrary preimages outside γ(S)g_k. But the coherence computation for the subnet limit assumes that for each h, both h and γ(s)h lie in γ(S)g_{k(λ)}—not just that each h is covered eventually. The proof never ensures the right ideals are increasing, so the equality y^k_{γ(s)h} = (s s_k)·y^k_{g_k} is unjustified for those λ where γ(s)h falls outside γ(S)g_k. The fix is straightforward: choose g_{k+1} ≤_S g_k with g_k ≤_S h_i for the first k enumerated elements, which Lemma 4.4's downward directedness allows. Then for each h, h ∈ γ(S)g_k eventually, and the subnet computation goes through. But as printed, Theorem D's first direction is not fully proved. This is repairable, not fatal.

2. Minor: the definition of arrows in Extα is confusing but actually consistent—for an arrow (G′,X̂′)→(G,X̂), θ maps G→G′, contrary to what one might first guess. The proof of Theorem C matches this. So I'd call it a presentation issue, not an error.

3. Example 4.8 cites "Proposition 4.4" for the cofinal sequence; should be Lemma 4.4. Also the expression "ss−1U" in Proposition 5.4(i) is garbled, though the intended inequality is clear given surjectivity.

The cited Lemma 4.4 is the engine for several later results. It is from Hilgert–Neeb and standard; I don't see a problem with relying on it. The paper cites the sparse prior work honestly, including the folklore status of some results.

Who this is for: people working on semigroup actions, symbolic dynamics over monoids, or topological dynamics of non-invertible maps. It deserves a serious referee. Send it to review; the referee will ask for the Proposition 4.6 fix and some cleaning up, but the core is solid.

Referee Report

1 major / 5 minor

Summary. The paper develops a theory of natural extensions for continuous actions of countable embeddable monoids. The natural extension is defined by a universal property (Definition A) and is shown, in Proposition 2.4, to be concretely realizable as the space X^G = {x ∈ X^G : s·x_h = x_{η(s)h} for all s,h} whenever that space projects onto X. The central characterization is Theorem B: for an embeddable monoid S, a receiving S-group G has the property that every surjective continuous S-action, equivalently every surjective S-subshift, is G-extensible if and only if G is a realization of the free S-group. The paper also proves a joint universality property for the natural extension over the free S-group (Theorem C), a characterization of left reversibility through factor maps of compact extensions (Theorem D), and, in the left reversible bicancellative case, equality of amenable topological entropy and lifting of topological ergodicity, transitivity, and minimality to the natural extension (Propositions 5.3 and 5.4).

Significance. If the results are correct, this gives a clean and quite general framework for natural extensions beyond N^d-actions. Theorem B is a sharp, falsifiable statement: both the receiving group and the embedding matter, and in the residually finite case the characterization reduces to finite subshifts. The paper has several strengths: the universal-property definition is independent of the concrete construction, the proofs of Proposition 2.4 and Proposition 3.6 are detailed and the main algebraic steps check out, and the reliance on standard tools (Ore–Dubreil, Følner sequences, Mal'cev's embeddability theorem) is explicit and appropriate. I found no circularity in the main line of argument. The main issue identified below is a genuine but local gap in the proof of Proposition 4.6, which affects Theorem D as printed; it appears repairable without changing the paper's scope.

major comments (1)

[§4, Proposition 4.6] The proof does not justify that the subnet limit y belongs to Y_{ΓS}. The sequence (g_k) is chosen only to be cofinal in the sense that for every h ∈ ΓS there is some k with h ∈ γ(S)g_k. However, the displayed computation y^{k(λ)}_{γ(s)h} = (s s_{k(λ)})·y^{k(λ)}_{g_{k(λ)}} = s·y^{k(λ)}_h is valid only when h ∈ γ(S)g_{k(λ)}; for indices with h outside γ(S)g_{k(λ)} the coordinates y^{k(λ)}_h were chosen arbitrarily, so no coherence is forced. Cofinality alone does not imply that h ∈ γ(S)g_{k(λ)} eventually along the monotone final subnet, and therefore the limit y may fail the defining relation of Y_{ΓS}. This affects the surjectivity claim for φ_{ΓS}, the 'in particular' part of Proposition 4.6, and consequently Theorem D. The gap is local and repairable: one should choose the sequence additionally with g_{k+1} ≤_S g_k, so that the right ideals γ(S)g_k are increasing as well as cofinal; then for each h one has h ∈ γ(S)g_{k(λ)} eventually, and the displayed computation does establish y ∈ Y_{ΓS}.

minor comments (5)

[§3.3, Theorem C proof] The equality 'ϕ′ ◦ π = τ' is a typo in the uniqueness argument; the arrow condition is π ◦ ϕ′ = τ.
[§3.3, category Ext_α] The direction of θ in the definition of arrows is easy to misread. For an arrow (G′, X̂′) → (G, X̂), θ is a morphism of S-groups from G to G′ and φ : Y′ → Y; writing the equivariance as φ(θ(g)·y′) = g·φ(y′) and displaying the composition rule explicitly would remove the apparent inconsistency.
[Example 4.8] The sentence justifying that {n : h ≤_S g_n} is finite appears to derive the conclusion from the opposite inequality. The conclusion is true, but it needs the missing argument that if h ≤_S g_n for infinitely many n, then h is ≤_S-equivalent to all sufficiently large g_n, which forces the connecting semigroup elements s_n to be units and contradicts the strict choice g_{n+1} <_S g_n.
[Example 4.8] The reference to 'Proposition 4.4' should be to Lemma 4.4.
[§5.2, Proposition 5.4(i)] The expression '∅ ≠ s s^{-1}U ∩ s^{-1}V' is confusing; the argument only needs U ∩ s^{-1}V ≠ ∅, which is exactly what topological ergodicity provides.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: the free S-group characterization and extensibility results are derived from algebraic universal properties and explicit constructions, not from fitted or self-referential assumptions.

full rationale

The paper's central claims are obtained by direct mathematical construction and standard external results, not by defining outputs into inputs. Definition A introduces natural extensions via a categorical terminal-object universal property; Proposition 2.4 then characterizes existence of the natural extension by surjectivity of the projection π, which is an independent criterion. The free S-group is defined algebraically, via the initial object in the category of S-groups (Definition 3.1) and via the presentation F(B)/R, before it is characterized in Theorem B. Corollary 3.7 builds on Proposition 3.6, which explicitly constructs the Γ-extension from surjectivity of the S-action and the universal property of the free group, while Proposition 3.8 constructs a concrete counterexample subshift from a non-trivial kernel element; no fitted parameter is being renamed as a prediction. Lemma 4.4, cited from Hilgert–Neeb [13], is external background mathematics and does not assume Theorem D or the results it supports. The only self-citation, reference [4], appears in Remark 4.10 as a contextual pointer and is not load-bearing. The skeptical note about Proposition 4.6 concerns a possible gap in the cofinality/subnet argument; that is a proof-correctness issue, not circularity, because it does not involve reducing a claimed result to its own assumptions by definition or by self-citation. Overall, the derivation chain is self-contained and non-circular.

Assumptions & free parameters 0 free parameters · 6 assumptions · 0 invented entities

The central theorems rely on standard algebraic and dynamical background theorems, plus the Axiom of Choice for choosing right inverses in Proposition 3.6. There are no free numerical parameters and no invented entities such as new particles, forces, or conserved quantities.

assumptions (6)

standard math Free S-group exists for every semigroup and the embedding criterion: S embeds in a group iff γ: S → Γ is injective (Theorem 3.2, cited to [7, Theorem 12.4]).
Used to identify the free S-group as the canonical receiving group and to justify that all other receiving groups are quotients of it.
standard math Ore-Dubreil theorem: a bicancellative semigroup S is left reversible iff its group of right fractions ΓS exists (Theorem 4.1, cited to [10,18]).
Basis for the left-reversibility characterization and for uniqueness of the receiving group in Section 4.
standard math Lemma 4.4: for left reversible S, G is the group of right fractions of S iff the pre-order ≤S is downward directed iff η(S) is thick in G (cited to [13, Section 3.6]).
Engine behind Proposition 4.6, Theorem D, and the entropy and transference results in Section 5.
standard math A left cancellative semigroup is left amenable iff it admits a left Følner sequence ([17, Corollary 3.6]).
Needed for Proposition 5.1 and for the definition of amenable topological entropy used in Proposition 5.3.
standard math For actions of left amenable semigroups on compact spaces, the topological entropy limit h_U(X,S) exists and is independent of the Følner sequence ([5]).
Foundation for the entropy equality in Proposition 5.3.
standard math Axiom of choice is used to select right inverses f_b^{-1}: Y → Y for each generator b in Proposition 3.6.
The construction of the free group extension requires choosing a right inverse for each surjective map y ↦ b·y; this is a standard set-theoretic input.

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Pith. "Pith review of Natural extensions of embeddable semigroup actions." pith.science (2026). https://pith.science/paper/VYKQ66ZL

@misc{pith2026250105536,
  author       = {Pith},
  title        = {Pith review of: Natural extensions of embeddable semigroup actions},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/VYKQ66ZL}},
  note         = {Machine review of arXiv:2501.05536}
}

read the original abstract

Semigroup actions and their invertible extensions are discussed. First, we develop a theory of natural extensions for continuous actions of countable, embeddable semigroups. Second, we demonstrate that not every surjective such action of a semigroup, which embeds into a group and generates it, can be extended to an action of said group, and that this phenomenon is specific to non-reversible semigroups. Furthermore, we characterize the free group on a semigroup (the group together with the embedding) as the unique pair that always admits such an extension, showing that both the choice of the receiving group and the embedding are crucial for this construction. Next, we prove that the classical notion of a natural extension -- requiring all other invertible extensions to factor through it -- only works in the context of compact extensions of left reversible semigroup actions and fails outside of it, thus providing a characterization of left reversibility. We finish by briefly studying topological dynamical properties of the natural extension in the amenable case.

Discussion (0). Continue with ORCID to comment.

Forward citations

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Pith tools

Reviewed August 10, 2026 · model on record in the stance chip above.

[1] [1]

L. N. Argabright and C. O. Wilde. Semigroups satisfying a strong Følner condition. Proc. Amer. Math. Soc., 18:587–591, 1967

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work page Pith review arXiv

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arXiv

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work page Pith review arXiv

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