{"id":"e24c23e1-2fc5-4766-a3bb-4f4aa8c2ed9f","arxiv_id":"2411.14027","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":5.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"Every self-similar k-graph C*-algebra, for finitely aligned k-graphs, is the tight C*-algebra of an explicitly constructed inverse semigroup, and simplicity can be read off from graphical conditions.","lead":"The paper builds a new algebraic object, an inverse semigroup, that completely describes the C*-algebra associated to a self-similar higher-rank graph, even when the graph has sources or fails the usual freeness conditions. This gives a unified way to tell when such C*-algebras are simple, using only combinatorial properties of the graph.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Proposition 6.3's claim that every boundary path yields an ultra-filter is false; the proof identifying the tight spectrum with ∂Λ is therefore invalid.","rationale":"The reader's weakest_assumption correctly identifies the identification of the tight spectrum with ∂Λ as load-bearing. I refine this: the specific flaw is in Proposition 6.3, which claims the ultra-filter space is homeomorphic to ∂Λ. The proof that F_x is an ultra-filter for every boundary path x is incorrect; a counterexample shows that non-generic boundary paths (like an empty path at an infinitely branching vertex) yield non-ultra-filters. This invalidates the paper's derivation of the tight-spectrum identification, even though the identification itself may be recovered from the known results for the inverse semigroup S_Λ. Since the paper repeatedly uses the equality of the tight and ultra-filter spaces (for example in Proposition 11.3 to apply [7, Lemma 5.6]), the proof of the simplicity criteria is compromised. The reader's verdict of CONDITIONAL remains appropriate: the gaps are specific and potentially fixable, but as written the arguments do not go through. The other gaps noted by the reader (the reverse implication in Corollary 10.2 and the unverified hypothesis in Proposition 11.3) are also present, but the Prop 6.3 issue is more fundamental because it affects the groupoid model used throughout. My concrete test settles the concern by exhibiting a finitely aligned k-graph where the claimed homeomorphism fails, so the proof cannot be patched without changing the statement of Prop 6.3.","tokens_in":186,"tokens_out":30056,"duration_ms":481048,"concrete_test":"Let Λ be the 1-graph with one vertex v, countably many edges e_n : v→v_n, and each v_n a sink. Let G be trivial. Compute RE_∞(S_Λ) and RE_tight(S_Λ) using the filter topology. The boundary path space ∂Λ contains v, e_n, and v_n. Verify that F_v = {ι_v} is a tight filter but not an ultra-filter (because ι_{e_n} ⋓ ι_v for all n, yet ι_{e_n} ∉ F_v). This disproves the surjectivity claim in Proposition 6.3 and shows that RE_∞ ≠ ∂Λ. The correct statement is RE_tight ≅ ∂Λ, which must be proven via [14] for S_Λ, not via the flawed Prop 6.3.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The paper's groupoid model and all later simplicity criteria rest on the identification of the tight spectrum of S_{G,Λ} with the boundary path space ∂Λ. The proof of Proposition 6.3 asserts that for every x∈∂Λ, the filter F_x = ⋃_{n≤d(x)} ι_{x(0,n)}↑ is an ultra-filter. This is false: for a boundary path x that is 'non-generic' (e.g., an empty path at a vertex with infinitely many pairwise-incomparable outgoing edges), F_x need not be maximal. In the star graph with one vertex v and countably many edges e_n to distinct sinks v_n, the empty path v is a boundary path vacuously (no finite exhaustive set exists at v), and F_v = {ι_v}. But ι_v is not an ultra-filter: the idempotent ι_{e_n} intersects ι_v for every n, yet ι_{e_n} ∉ F_v, so the maximality condition of Remark 6.2(1) fails. Thus the map in Proposition 6.3 is not surjective onto the ultra-filter space. Consequently, the deduction that RE_tight(S_{G,Λ}) = RE_∞(S_{G,Λ}) = ∂Λ is unjustified; in fact, the tight spectrum is strictly larger than the ultra-filter space in this example. Although the tight spectrum of S_{G,Λ} is still homeomorphic to ∂Λ via the known identification for S_Λ (since E(S_{G,Λ})=E(S_Λ)), the paper's proof of this key fact is wrong, and the later use of the equality RE_tight = RE_∞ (e.g., in Proposition 11.3) is unsupported.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper generalizes the Li-Yang notion of self-similar k-graphs and their C*-algebras to the setting of finitely aligned k-graphs, and introduces an inverse semigroup S_{G,Λ} whose tight C*-algebra is claimed to be canonically isomorphic to O_{G,Λ}. The main structural result, Theorem 4.7, asserts O_{G,Λ} ≅ C*_tight(S_{G,Λ}) ≅ C*(G_tight(S_{G,Λ})). The paper then characterizes Hausdorffness, minimality, effectiveness, and simplicity of the tight groupoid in terms of graphical conditions such as local exhaustion of strongly fixed paths, G-cofinality, and a G-aperiodicity condition, and applies these to obtain simplicity criteria for O_{G,Λ} in both Hausdorff and non-Hausdorff settings.","tokens_in":34734,"tokens_out":12083,"duration_ms":124013,"significance":"If the main structural theorem and the spectral identifications hold, the paper provides a genuinely useful framework: it extends self-similar k-graph C*-algebras to finitely aligned graphs with sources, constructs the first inverse semigroup model for this class, and connects the tight groupoid to both the earlier path-like groupoid of Li-Yang and to Spielberg's LCSC groupoids. The proof of Theorem 4.7 is detailed and has a clear universal-property strategy: an explicit representation π is shown to be cover-to-join, and universality is verified by deriving the Cuntz-Krieger and self-similarity relations. The paper also explicitly acknowledges the relationship with the more general LCSC framework, which is a useful contextualization. However, the later simplicity results rest on an identification of the tight spectrum with the boundary path space, and the proof of that identification contains a substantive error; consequently the non-Hausdorff simplicity theorem and the claimed iff in the Hausdorff simplicity corollary are not established as written.","major_comments":[{"comment":"The assertion that every boundary path x gives an ultrafilter F_x is false. Consider the finitely aligned 1-graph with one vertex v and countably many edges e_n to distinct sinks v_n. The empty path v is a boundary path vacuously, but F_v = {E ∈ E(S_G,Λ) : (v,e,v) ∈ E} is not an ultrafilter: for every n, ι_{e_n} intersects ι_v, yet ι_{e_n} ∉ F_v, contradicting Remark 6.2(1). Thus the map x ↦ F_x is not onto the ultrafilter space, and the conclusion after Proposition 6.3 that Ê_tight(S_G,Λ) = Ê_∞(S_G,Λ) = ∂Λ is unjustified. This equality is used in load-bearing ways: the explicit description of G_tight(S_G,Λ) in (6.1)-(6.2) and the proof of Proposition 11.3 both rely on it. The tight-spectrum homeomorphism might be recoverable from E(S_G,Λ) = E(S_Λ) and the known results of [14], but it is not proved by the argument given in the paper.","section":"Section 6, Proposition 6.3"},{"comment":"Corollary 10.2 states an iff criterion for simplicity of O_{G,Λ}, but Proposition 10.1 only proves the forward direction. The converse direction—that simplicity of O_{G,Λ} forces G-cofinality, the G-aperiodicity condition (A), and the strongly-fixed exhaustive condition—is not argued. To make the claimed equivalence valid, the author needs to invoke the converse of the Brown-Clark-Farthing-Sims theorem and the converses of Theorems 8.3 and 9.6, and to spell out how those converses apply under the standing assumptions. As written, the corollary exceeds what is proved.","section":"Section 10, Corollary 10.2"},{"comment":"The proof asserts that G_tight(S_G,Λ) is minimal and effective by Theorems 8.3 and 9.6, but Theorem 9.6(3) requires condition (3)(b): if g fixes every x ∈ v∂Λ, then there exists a finite exhaustive set X such that every τ ∈ X is strongly fixed by g. Proposition 11.3's hypothesis (3) only gives, for each individually fixed path x, a prefix with trivial cocycle; it does not, as stated, supply the uniform finite exhaustive set required by Theorem 9.6(3)(b). Even if row-finiteness and the source-free assumption can be used to extract such an X by compactness, the argument is not given. Additionally, the proof uses the equality Ê_tight(S_G,Λ) = Ê_∞(S_G,Λ), which is false by the counterexample in the first major comment.","section":"Section 11, Proposition 11.3"}],"minor_comments":[{"comment":"The converse implication in the proof of (7.2) is left to the reader, but injectivity of the proposed groupoid isomorphism is central to the comparison with Li-Yang's groupoid; it should be written out.","section":"Section 7, Proposition 7.2"},{"comment":"The decomposition of a cover C for J_F into covers C_i for J_{ι_{μ_i}} is stated very tersely and would benefit from a sentence explaining how the outer-cover and cover conditions interact under the union.","section":"Section 4, Remark 4.5"},{"comment":"There are several typographical issues, e.g., 'beacuse' in the proof of Lemma 3.3 and inconsistent spacing in 'C ∗-algebras'; these do not affect the mathematics but should be cleaned up.","section":"General"}],"recommendation":"major_revision","confidential_remarks":"The main obstruction is the false Proposition 6.3. I view this as repairable in principle: the tight-spectrum identification with ∂Λ may be recoverable from E(S_G,Λ) = E(S_Λ) together with the known results of [14], and then the Hausdorff-case simplicity results could be placed on firmer ground. The non-Hausdorff result in Proposition 11.3 needs more substantial repair because it explicitly relies on the false equality Ê_tight = Ê_∞. I do not see evidence of novelty or attribution problems: the relationship to the LCSC framework is acknowledged. The paper has a valuable central construction, but the spectral gap affects the later simplicity theorems, so major revision is appropriate rather than rejection."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"The thing to know: the inverse semigroup S_{G,Λ} for finitely aligned self-similar k-graphs is a real contribution, and the main isomorphism theorem (4.7) is argued in enough detail that I find it convincing. The simplicity results, though, are not ready as written. Three specific gaps need work.\n\nWhat is genuinely new: the S_{G,Λ} construction, the cover-to-join proof that its tight C*-algebra is O_{G,Λ}, the Hausdorffness criterion via SF_g locally exhausted, the G-cofinality minimality criterion, and the G-aperiodicity condition. These extend Li-Yang's row-finite pseudo-free theory to the non-pseudo-free case. The author also honestly connects to the Ortega-Pardo/Spielberg LCSC framework via Proposition 6.4, and that comparison is useful.\n\nNow the soft spots.\n\n1. Proposition 6.3 is false as stated. The claimed homeomorphism from ∂Λ onto the ultra-filter space fails. Take a vertex v with infinitely many edges e_n to distinct sinks. The empty path v is a boundary path (no finite exhaustive sets exist), and F_v = {E : ι_v ∈ E} is not an ultra-filter: the idempotent {ι_{e_n}} intersects every element of F_v (since {ι_{e_n}}ι_v = {ι_{e_n}}), yet {ι_{e_n}} is not in F_v. The proof's step where finiteness of E forces some x(0,n) into E breaks down at n=0. The tight spectrum may still equal ∂Λ via the known identification for S_Λ, since E(S_{G,Λ})=E(S_Λ), but the paper's argument—and the later equality E_tight = E_∞ used in Proposition 11.3—is unsupported.\n\n2. Corollary 10.2 asserts an \"if and only if\" for simplicity, but only the forward direction (Proposition 10.1) is proven. The reverse implication is a missing theorem, not a minor detail.\n\n3. Proposition 11.3 invokes Theorem 9.6 to get effectiveness but does not verify condition 9.6(3)(b). Theorem 9.6(3)(b) requires a finite exhaustive set of strongly fixed paths when g fixes all boundary paths at a vertex; the paper's condition (3) is a pointwise statement about individual infinite paths. I don't see how the latter implies the former.\n\nThese are addressable. The central construction is sound, and the Hausdorffness and minimality criteria look right. A serious revision could fix the gaps. As it stands, I would not cite the simplicity claims, but I would send this to a referee. The referee should insist on a corrected Proposition 6.3 (or a direct appeal to the S_Λ tight spectrum result), a proof of the converse in 10.2, and a repair of 11.3.\n\nOverall: deserves peer review, yes. I'd cite it for the inverse semigroup model once cleaned up.","headline":"A genuinely useful inverse semigroup model with a convincing main isomorphism, but the simplicity theorems rest on a false ultra-filter claim and a missing converse; needs revision and refereeing.","tokens_in":35311,"tokens_out":10138,"would_cite":true,"duration_ms":92763,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["46L05"],"pacs":[],"model":"deepseek-v4-flash","headline":"Every self-similar k-graph C*-algebra is the tight C*-algebra of an inverse semigroup.","keywords":["finitely aligned k-graph","self-similar k-graph","inverse semigroup","tight groupoid of germs","C*-algebra","simplicity","boundary path space","G-cofinality"],"falsifier":"Look for a self-similar k-graph over a finitely aligned k-graph for which two different boundary paths produce the same filter $\\mathcal{F}_x$, or for which a tight filter contains no idempotent $\\iota_{x(0,n)}$; either would disprove Proposition 6.3 and void the simplicity theorems. Concretely, testing a small example with a non-pseudo-free cocycle and comparing the tight-spectrum topology with the cylinder-set topology on $\\partial\\Lambda$ would settle it.","tokens_in":34201,"feed_emoji":"🔗","tokens_out":6522,"duration_ms":57376,"temperature":0.7,"pith_summary":"This paper shows that every self-similar k-graph C*-algebra, defined for a group action on a finitely aligned higher-rank graph (sources allowed), is really the tight C*-algebra of a single inverse semigroup built from the graph and the action. The payoff is that the algebra's groupoid, its tight groupoid of germs, can be read off from graphical data: its unit space is the boundary path space of the graph, Hausdorffness is controlled by strongly fixed paths, minimality by G-cofinality, and effectiveness by an aperiodicity condition. From these, the paper derives concrete necessary and sufficient conditions for simplicity of the algebra, covering both Hausdorff and non-Hausdorff cases. A sympathetic reader should care because this unifies and extends earlier row-finite, source-free, pseudo-free results to a substantially larger class, and gives an inverse-semigroup machine for future questions.","feed_headline":"One inverse semigroup encodes every self-similar k-graph C*-algebra","feed_subtitle":"Tight groupoid methods now apply to finitely aligned k-graphs with sources, yielding simplicity criteria.","key_machinery":"The central object is the inverse semigroup $S_{G,\\Lambda}$: its elements are finite sets of pairwise orthogonal triples $(\\mu,g,\\nu)$ of paths and group elements with $s(\\mu)=g\\cdot s(\\nu)$, the product is defined by splicing minimal common extensions, and the inverse reverses triples. It carries the argument because its tight C*-algebra is the universal algebra $\\mathcal{O}_{G,\\Lambda}$ (Theorem 4.7), its tight groupoid of germs is the groupoid whose C*-algebra is the same algebra, and its idempotent semilattice is exactly that of the underlying k-graph's inverse semigroup. That last fact, via a homeomorphism between ultrafilters in that semilattice and boundary paths, identifies the tight spectrum with $\\partial\\Lambda$, letting every later graphical criterion be read off from the k-graph itself.","core_discovery":"The paper's central claim is that the universal C*-algebra $\\mathcal{O}_{G,\\Lambda}$ of a self-similar k-graph $(G,\\Lambda)$ over a finitely aligned k-graph $\\Lambda$ admits a canonical inverse semigroup model: there is an inverse semigroup $S_{G,\\Lambda}$ whose tight C*-algebra is canonically isomorphic to $\\mathcal{O}_{G,\\Lambda}$, and whose tight groupoid of germs $\\mathcal{G}_{\\mathrm{tight}}(S_{G,\\Lambda})$ satisfies $C^*(\\mathcal{G}_{\\mathrm{tight}}(S_{G,\\Lambda})) \\cong C^*_{\\mathrm{tight}}(S_{G,\\Lambda}) \\cong \\mathcal{O}_{G,\\Lambda}$. The elements of $S_{G,\\Lambda}$ are finite sets of pairwise orthogonal triples $(\\mu,g,\\nu)$ with $s(\\mu)=g\\cdot s(\\nu)$, with multiplication computed through minimal common extensions; its idempotents coincide with those of the inverse semigroup of the underlying k-graph, so the tight spectrum is homeomorphic to the boundary path space $\\partial\\Lambda$. On this model, Hausdorffness of the groupoid is equivalent to the set of strongly fixed paths by each nontrivial group element being locally exhausted; minimality is equivalent to G-cofinality; effectiveness is equivalent, under that Hausdorff condition, to a G-aperiodicity condition. These criteria combine into simplicity theorems for $\\mathcal{O}_{G,\\Lambda}$, including a non-Hausdorff case handled by a stronger condition on the semigroup action.","pith_inferences":["Because the idempotent semilattice is independent of the group, the unit space of the tight groupoid is the same boundary-path space as for the underlying k-graph; the action's dynamics live entirely in the germs. A natural extension would be to classify ideals or K-theory of $\\mathcal{O}_{G,\\Lambda}$ through invariant subsets of $\\partial\\Lambda$, a route the paper does not take.","The non-Hausdorff simplicity result via Condition (S) suggests that the Zappa-Szep product picture for left-cancellative small categories should admit analogous simplicity criteria under weaker hypotheses than right cancelation, using the same strongly-fixed-path analysis.","One could test whether the G-cofinality criterion alone characterizes simplicity of the full algebra when the tight groupoid is amenable but non-Hausdorff; the paper only states this under additional Condition (S) hypotheses.","The finite-set form of $S_{G,\\Lambda}$, rather than single triples as in the one-graph case, may be the right model for handling sources, and it invites explicit computation for examples where $\\Lambda$ is not row-finite by replacing covers with finite exhaustive sets."],"forward_implications":["For every finitely aligned self-similar k-graph, $\\mathcal{O}_{G,\\Lambda}$ is canonically isomorphic to the tight C*-algebra of $S_{G,\\Lambda}$ and to $C^*(\\mathcal{G}_{\\mathrm{tight}}(S_{G,\\Lambda}))$, so inverse semigroup machinery applies even when $\\Lambda$ has sources and the action is not pseudo-free.","The tight groupoid is Hausdorff exactly when strongly fixed paths are locally exhausted for every nontrivial group element; in particular every pseudo-free self-similar k-graph has a Hausdorff tight groupoid.","The tight spectrum is $\\partial\\Lambda$, so the groupoid is minimal exactly when $(G,\\Lambda)$ is G-cofinal, and effective exactly under the stated G-aperiodicity conditions, with equivalence when Hausdorffness holds.","Under Hausdorffness and amenability of the groupoid, $\\mathcal{O}_{G,\\Lambda}$ is simple if and only if it is G-cofinal, G-aperiodic in the stated sense, and satisfies the local strong-fixed-point condition; a separate non-Hausdorff theorem gives simplicity of the reduced algebra from G-cofinality, aperiodicity, and eventual triviality of the cocycle along infinite paths.","These results extend earlier simplicity criteria that were restricted to row-finite, source-free k-graphs and pseudo-free actions."],"supporting_citations":[{"why":"Supplies the tight C*-algebra of an inverse semigroup, universal tight representations, and the isomorphism $C^*(\\mathcal{G}_{\\mathrm{tight}}(S))\\cong C^*_{\\mathrm{tight}}(S)$.","marker":"[9]"},{"why":"Provides the tight groupoid of an inverse semigroup, the cover-to-join tightness criterion, and the Hausdorffness, minimality, and effectiveness criteria used throughout.","marker":"[12]"},{"why":"Supplies the inverse semigroup $S_\\Lambda$ of a finitely aligned k-graph, the boundary path space $\\partial\\Lambda$, and its topology, used to identify the tight spectrum.","marker":"[14]"},{"why":"Original construction of self-similar k-graph C*-algebras and the path-like groupoid $\\mathcal{G}_{G,\\Lambda}$ for row-finite source-free pseudo-free cases, which this paper extends.","marker":"[27]"},{"why":"Defines the row-finite unital version $A_{G,\\Lambda}$ and proves nuclearity and simplicity results that the paper generalizes to the Hausdorff case.","marker":"[28]"},{"why":"Provides the tight groupoid of inverse semigroups of left-cancellative small categories and the Zappa-Szep product version used for the comparison in Proposition 6.4.","marker":"[34]"},{"why":"Introduces self-similar graph C*-algebras and their inverse semigroup; the k-graph semigroup is modeled on this, with pseudo-freeness inherited.","marker":"[11]"},{"why":"Defines finitely aligned k-graphs and Cuntz-Krieger families; its (CK4) relation is exactly what the tightness proof verifies.","marker":"[37]"},{"why":"Supplies the groupoid simplicity theorem used in the Hausdorff case: minimal plus effective plus Hausdorff implies the reduced C*-algebra is simple.","marker":"[5]"},{"why":"Supplies Condition (S) and the non-Hausdorff simplicity criterion used in Section 11 for $C^*_{\\mathrm{red}}(\\mathcal{G}_{\\mathrm{tight}}(S_{G,\\Lambda}))$.","marker":"[7]"}],"fun_headline_variants":["Inverse semigroup encodes self-similar k-graph C*-algebras","Finitely aligned k-graphs get inverse semigroup C*-algebras","Simplicity via tight groupoids in self-similar k-graphs","Tight groupoid criteria for simple self-similar k-graph algebras","Self-similar k-graph algebras via inverse semigroups"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The whole bridge from algebras to graph conditions rests on the claim that the tight filters of $S_{G,\\Lambda}$ correspond exactly to boundary paths of $\\Lambda$ (Proposition 6.3); if that homeomorphism failed, the simplicity and minimality criteria would not transfer to the algebra.","fun_headline_variants_meta":{"raw":{"variants":["Inverse semigroup encodes self-similar k-graph C*-algebras","Finitely aligned k-graphs get inverse semigroup C*-algebras","Simplicity via tight groupoids in self-similar k-graphs","Tight groupoid criteria for simple self-similar k-graph algebras","Self-similar k-graph algebras via inverse semigroups"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000629,"raw_usage":{"total_tokens":2898,"prompt_tokens":927,"completion_tokens":1971,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":543,"completion_tokens_details":{"reasoning_tokens":1875}},"tokens_in":543,"tokens_out":1971,"duration_ms":13896,"temperature":1.0,"reasoning_tokens":1875,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-12T15:36:47.990466+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Look for a self-similar k-graph over a finitely aligned k-graph for which two different boundary paths produce the same filter $\\mathcal{F}_x$, or for which a tight filter contains no idempotent $\\iota_{x(0,n)}$; either would disprove Proposition 6.3 and void the simplicity theorems. Concretely, testing a small example with a non-pseudo-free cocycle and comparing the tight-spectrum topology with the cylinder-set topology on $\\partial\\Lambda$ would settle it.","supporting_citations":[{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Supplies the tight C*-algebra of an inverse semigroup, universal tight representations, and the isomorphism $C^*(\\mathcal{G}_{\\mathrm{tight}}(S))\\cong C^*_{\\mathrm{tight}}(S)$."},{"cited_title":"Exel and E","cited_arxiv_id":null,"evidence_quote":"Provides the tight groupoid of an inverse semigroup, the cover-to-join tightness criterion, and the Hausdorffness, minimality, and effectiveness criteria used throughout."},{"cited_title":"Farthing, P.S","cited_arxiv_id":null,"evidence_quote":"Supplies the inverse semigroup $S_\\Lambda$ of a finitely aligned k-graph, the boundary path space $\\partial\\Lambda$, and its topology, used to identify the tight spectrum."},{"cited_title":"Li and D","cited_arxiv_id":null,"evidence_quote":"Original construction of self-similar k-graph C*-algebras and the path-like groupoid $\\mathcal{G}_{G,\\Lambda}$ for row-finite source-free pseudo-free cases, which this paper extends."},{"cited_title":"Li and D","cited_arxiv_id":null,"evidence_quote":"Defines the row-finite unital version $A_{G,\\Lambda}$ and proves nuclearity and simplicity results that the paper generalizes to the Hausdorff case."},{"cited_title":"Ortega and E","cited_arxiv_id":null,"evidence_quote":"Provides the tight groupoid of inverse semigroups of left-cancellative small categories and the Zappa-Szep product version used for the comparison in Proposition 6.4."},{"cited_title":"Exel and E","cited_arxiv_id":null,"evidence_quote":"Introduces self-similar graph C*-algebras and their inverse semigroup; the k-graph semigroup is modeled on this, with pseudo-freeness inherited."},{"cited_title":"Raeburn, A","cited_arxiv_id":null,"evidence_quote":"Defines finitely aligned k-graphs and Cuntz-Krieger families; its (CK4) relation is exactly what the tightness proof verifies."},{"cited_title":"Brown, L.O","cited_arxiv_id":null,"evidence_quote":"Supplies the groupoid simplicity theorem used in the Hausdorff case: minimal plus effective plus Hausdorff implies the reduced C*-algebra is simple."},{"cited_title":"Clark, R","cited_arxiv_id":null,"evidence_quote":"Supplies Condition (S) and the non-Hausdorff simplicity criterion used in Section 11 for $C^*_{\\mathrm{red}}(\\mathcal{G}_{\\mathrm{tight}}(S_{G,\\Lambda}))$."}],"review_version":1}