{"id":"bb09af0a-e989-4817-ab47-e8cb978ad71f","arxiv_id":"1908.02471","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":7.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"Two new theorems obtained by Boolean valued transfer: a decomposition of universally complete vector lattices free of locally one-dimensional bands, and an Ando-type projection characterization for B-cyclic Banach lattices.","lead":"This paper uses Boolean valued analysis, a logical model-building technique, to prove two new theorems about vector lattices: a splitting theorem for universally complete spaces and an extension of the classical Ando characterization to cyclic Banach lattices. A smart generalist should read it to see how a transfer from a fictional universe of sets can turn elementary real-number facts into structural theorems about infinite-dimensional function spaces.","discovery_kind":"new_application","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Lemma 3.3's equivalence forces order completeness of the descended X1, X2 that Theorem 3.5(4) asserts are not order complete; the proof is internally inconsistent unless Lemma 3.3 is misquoted.","rationale":"The reader's weakest_assumption already targets Lemma 3.3 and order-completeness under descent, and my stress-test sharpens it into a near-contradiction inside the proof. The very application of Lemma 3.3, together with X_k⊥⊥=X and the asserted band preservation of the canonical projections, forces X1 and X2 to be order complete, which Theorem 3.5(4) denies. This is load-bearing: the theorem's novelty is precisely the existence of fragment-closed, laterally complete sublattices that are not order complete. If Lemma 3.3 is correct, then the proof implies the opposite of its conclusion; if Lemma 3.3 is misquoted, then the descent of a Hamel subspace has not been shown to satisfy the required P-invariance and lateral completeness. Either way, the central argument is unsupported at its key step. The deciding check is external to this paper (the cited monograph), so I do not upgrade the reader's conditional verdict to acceptance; I also would not reject outright, since the authors may be able to correct the lemma or supply the missing descent verification. Keeping the conditional verdict is therefore the honest outcome.","tokens_in":12570,"tokens_out":25425,"duration_ms":270172,"concrete_test":"Check [2, Theorem 2.5.1] and verify whether Lemma 3.3's implication (4) ⇒ (1), with order completeness included, is accurate. If it is, then Theorem 3.5(4) is contradicted by the authors' own proof. If it is not, identify the precise conditions under which the descent of an internal R∧-vector sublattice is order complete, and test the specific Xk of Lemma 3.4 against those conditions. A useful test case is Y=(R∧)↓ for a non-σ-distributive complete Boolean algebra B: R∧ is a proper internal subfield and is not order complete internally, so Y should not be order complete externally. Determine which clauses of Lemma 3.3(3) fail for Y, and repeat for the Xk of Lemma 3.4; this settles whether the 'By Lemma 3.3' step is valid or whether Theorem 3.5(4) directly contradicts it.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The proof of Theorem 3.5 descends the internal R∧-linear subspaces Xk of Lemma 3.4 to X_k := Xk↓. Lemma 3.3(4) says that a sublattice of X=R↓ is of the form X0↓ for some vector sublattice X0 of R over R∧ iff it is order complete, laterally complete, and fragment closed. The Xk are exactly such internal vector sublattices, so Lemma 3.3(4) ⇒ (1) makes X1 and X2 order complete. This directly contradicts Theorem 3.5(4), which asserts that none of X1, X2 is order complete. The paper never explains how order completeness behaves under descent; internally, the Xk of Lemma 3.4 are explicitly not order complete as ordered vector spaces over R∧, and descent reflection would make X_k not order complete externally. Thus as written, the proof proves the opposite of its own conclusion (4). If Lemma 3.3 is meant without the order-completeness clause, then the proof lacks the required verification of lateral completeness and P(X)-invariance for the descended Hamel subspaces. In either reading, the central construction is not supported. All four conclusions of Theorem 3.5 depend on this descent step.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper develops two applications of Boolean valued analysis to vector lattice theory. The first main result, Theorem 3.5, claims that any universally complete vector lattice without locally one-dimensional bands can be decomposed as a direct sum of two fragment-closed, laterally complete vector sublattices, each of which admits a band-preserving linear bijection to the original lattice but is not order complete. The second main result, Theorem 6.4, establishes a Boolean valued analogue of Ando's theorem: a B-cyclic Banach lattice of Boolean dimension at least 3 has contracting positive projections onto every B-complete closed sublattice if and only if it decomposes into pieces isometrically isomorphic to Lp(Φ) and C#(Qγ, c0(γ)) over a partition of unity. The paper builds on the Gordon representation of the reals in V(B), Gutman's theorem on locally one-dimensional lattices, and the Boolean valued transfer principle for injective Banach lattices.","tokens_in":12596,"tokens_out":18363,"duration_ms":184817,"significance":"If correct, Theorem 3.5 would give a striking strengthening of the Abramovich-Kitover counterexample, showing that the failure of lattice isomorphism under band-preserving linear isomorphism is total in the non-locally-one-dimensional case. Theorem 6.4 is a natural and potentially useful extension of Ando's classical characterization to B-cyclic Banach lattices, and the paper correctly identifies the relevant transfer machinery (Gordon's theorem, Maharam operators, injective Banach lattices, and the descent of c0(Γ)). The paper is clearly organized around a coherent Boolean valued methodology and invokes appropriate classical inputs. However, the central descent argument supporting Theorem 3.5 is internally inconsistent as written, and the second theorem has an unproved transfer step, so the current manuscript does not establish its advertised results.","major_comments":[{"comment":"The proof of Theorem 3.5 applies Lemma 3.3 to the descents Xk↓ of the internal R∧-linear subspaces Xk. Lemma 3.3(4) implies that any such descent is order complete, laterally complete, and fragment closed, since condition (4) is equivalent to condition (1). The proof of Theorem 3.5 invokes Lemma 3.3 precisely to conclude that X1 and X2 are fragment closed and laterally complete, thereby committing to their order completeness. But Theorem 3.5(4) explicitly asserts that X1 and X2 are not order complete. The paper nowhere explains how the internal non-order-completeness of the subspaces Xk from Lemma 3.4 is compatible with the external order completeness of their descents Xk↓. If Lemma 3.3 is meant to apply only to internally order-complete sublattices X0, then the proof of Theorem 3.5 fails to verify that hypothesis for the Xk, which are not order complete by Lemma 3.4. If Lemma 3.3 is meant as stated, then Theorem 3.5(4) is contradicted by the very lemma used to prove the other conclusions. In either reading, the load-bearing descent correspondence is not established and the central construction is unsupported.","section":"Section 3, Lemma 3.3 and Theorem 3.5"},{"comment":"The transfer step between norm-closed sublattices in the internal Banach lattice X and B-complete norm-closed sublattices in its bounded descent X⇓ is asserted rather than proved. In direction (1)=>(2), the paper says \"It is easy to check\" that a closed sublattice X0 in X corresponds to a B-complete norm-closed sublattice X0⇓ in X, and in direction (2)=>(1) it refers to \"As in Lemma 3.3\". Lemma 3.3, however, concerns sublattices of R↓ and their descents; it does not address bounded descents of arbitrary Banach lattices or norm-closed sublattices. This correspondence is load-bearing for the equivalence in Theorem 6.4, and no proof or appropriate citation is supplied for it.","section":"Section 6, proof of Theorem 6.4"}],"minor_comments":[{"comment":"The last sentence contains a typo: \"if and only if p1 and p1 are order bounded\" should read \"if and only if p1 and p2 are order bounded\".","section":"Section 3, proof of Theorem 3.5"},{"comment":"The sentence \"The real R is a finite extension of no proper subfield P ⊂ R\" is awkward; it should read \"R is not a finite extension of any proper subfield P ⊂ R\".","section":"Section 3, Lemma 3.4"},{"comment":"In the proof, after introducing the partition of unity (πξ), the formula \"bθx↓(γ) = mixξ∈Ξ πξt∧θ,ξ\" mixes two partitions (bθ and πξ) without explanation, and the notation appears inconsistent; please clarify.","section":"Section 5, Lemma 5.1"},{"comment":"In the first paragraph of the proof, \"Proposition 6.4(1)\" should read \"Theorem 6.4(1)\".","section":"Section 6, proof of Theorem 6.4"}],"recommendation":"reject","confidential_remarks":"The paper relies heavily on the authors' own monograph [2] for key technical lemmas, especially Lemma 3.3, but the exact statement of [2, Theorem 2.5.1] is not reproduced, and the lemma as stated is inconsistent with its use in Theorem 3.5. This is not a presentation issue but a load-bearing gap in the first main result. The second main result may be repairable, but as submitted the proof transfers the core closed-sublattice correspondence without adequate justification. I recommend rejection, while noting that a substantially revised version with a corrected Lemma 3.3 and full proofs of the transfer steps could be reconsidered."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"The two headline results are well-motivated and the Boolean valued framing is genuinely illuminating. The Section 3 insight—that the d-basis machinery from Abramovich-Kitover is, inside the model, just a Hamel basis over a proper subfield—is the kind of conceptual simplification that makes the method feel right. Theorem 6.4's Ando transfer to B-cyclic Banach lattices is also new and the Lp(Phi)/C#(Q,c0(gamma)) dichotomy is a plausible and interesting target. The external inputs (Gordon, Gutman, Ando) are invoked correctly and the transfer skeletons look structurally sound. Credit where due: these are not recycled results and the paper is clearly written for those who know the machinery.\n\nThe soft spot is in the proof of Theorem 3.5, and it is load-bearing. Lemma 3.3(4) says that a sublattice X0 of X=R↓ is the descent of some internal vector sublattice of R over R∧ iff it is order complete, laterally complete, and fragment closed. The proof of Theorem 3.5 uses exactly that equivalence to get fragment closed and laterally complete for the descended Hamel subspaces Xk↓. But the same equivalence forces Xk↓ to be order complete, which directly contradicts conclusion (4) of Theorem 3.5. The proof never reconciles these. If Lemma 3.3 is correct, the proof proves the opposite of its own assertion; if the lemma is misstated or missing a hypothesis, then the verification of lateral completeness and P(X)-invariance is missing. There is also a quieter issue: the internal R∧-linear subspaces from Lemma 3.4 are only shown to be linear subspaces, not sublattices, so it is not even clear that the descent is a vector sublattice as required by Lemma 3.3. I don't think the contradiction is unrecoverable—the authors' monograph may contain a more careful statement—but as written the central construction of Theorem 3.5 is unsupported. The gap in Theorem 6.4 is lesser: the correspondence between closed sublattices and B-complete sublattices is asserted with \"it is easy to check\" and deserves a few lines, but that transfer is standard and I would expect it to hold up.\n\nWho benefits? Someone working in Boolean valued analysis or the theory of universally complete lattices gets a novel transfer template and two interesting conjectures. But the first theorem needs a serious fix before it can be used. I would send this to a knowledgeable referee rather than desk reject it; the results are significant enough that the gap is worth addressing, and the authors are likely to supply the missing argument.\n\nRecommendation: engage with it, but ask for a rewrite of the Lemma 3.3-to-Theorem 3.5 step.","headline":"Two genuinely new Boolean transfer results, but the first main theorem's proof has a load-bearing contradiction between Lemma 3.3 and Theorem 3.5(4) that needs to be resolved before the paper is publishable.","tokens_in":13383,"tokens_out":9281,"would_cite":false,"duration_ms":97709,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["46B42","46A40","03C90"],"pacs":[],"model":"deepseek-v4-flash","headline":"Every universally complete vector lattice without locally one-dimensional bands splits as a direct sum of two sublattices, each band-preservingly isomorphic to the original and neither order complete.","keywords":["Boolean valued analysis","universally complete vector lattice","band preserving operator","disjointness preserving operator","Ando theorem","injective Banach lattice","cyclic Banach lattice","Maharam operator"],"falsifier":"Take $X = \\mathbb{R}^{\\downarrow}$ where $B$ is the regular-open algebra of $[0,1]$ (not $\\sigma$-distributive), split $\\mathbb{R}$ as a vector space over its standard-name subfield by a Hamel basis $E = E_1 \\sqcup E_2$ with $|E_1| = |E_2|$, and check directly whether the two descended sublattices $X_1^{\\downarrow}$ and $X_2^{\\downarrow}$ are laterally complete and non-order-complete; any violation would contradict the first main theorem.","tokens_in":12136,"feed_emoji":"🧩","tokens_out":18533,"duration_ms":154062,"temperature":0.7,"pith_summary":"This paper develops two applications of Boolean valued models of set theory to the structure of vector and Banach lattices. The first result shows that if a universally complete vector lattice contains no locally one-dimensional bands, then it can be decomposed as a direct sum of two vector sublattices that are laterally complete, invariant under all band projections, and band-preservingly linearly isomorphic to the original lattice; nevertheless neither sublattice is order complete, so neither is lattice-isomorphic to the whole space. This settles a strong form of the problem of whether an invertible operator whose inverse preserves disjointness forces lattice isomorphism. The second result transfers the classical Ando dichotomy to cyclic Banach lattices: a $B$-cyclic Banach lattice with Boolean dimension at least three has a contracting positive projection onto every $B$-complete closed sublattice exactly when, over a partition of unity, it is a $B$-cyclic $L_p(\\Phi)$-space or a space $C^\\#(Q_\\gamma, c_0(\\gamma))$. The interest is that the proofs convert an elementary algebraic fact about Hamel bases over proper subfields of the reals into genuine lattice-theoretic structure, and that they extend a central classification theorem of Banach lattice theory to a Boolean valued setting.","feed_headline":"Vector lattice splits into two copies, neither order complete","feed_subtitle":"Boolean valued analysis answers the disjointness-preserving operator question and transfers the Ando theorem","key_machinery":"The engine is the descent functor between a Boolean valued model $V^{(B)}$ and the ordinary universe, combined with two recognition theorems. The Gordon theorem identifies the reals $\\mathbb{R}$ inside $V^{(B)}$ with a universally complete vector lattice $\\mathbb{R}^{\\downarrow}$ whose band projections correspond to the elements of $B$. The Gutman theorem says $\\mathbb{R}$ equals its standard-name $\\mathbb{R}^{\\wedge}$ if and only if $B$ is $\\sigma$-distributive, equivalently $\\mathbb{R}^{\\downarrow}$ is locally one-dimensional; the absence of locally one-dimensional bands therefore forces a proper field extension inside the model. The load-bearing bridge is Lemma 3.3: a sublattice of $\\mathbb{R}^{\\downarrow}$ is order complete, laterally complete, and fragment closed exactly when it is the descent of some vector sublattice of $\\mathbb{R}$ over the subfield $\\mathbb{R}^{\\wedge}$. Applying this to the Hamel-basis subspaces produced inside the model yields the decomposition. For the Ando part, the machinery is the Boolean valued transfer principle for injective Banach lattices, the construction of $L_p(\\Phi)$ via a strictly positive Maharam operator with the Levy property, and the identification of $c_0(\\Gamma^{\\wedge})^{\\downarrow}$ with $C^\\#(Q, c_0(\\Gamma))$.","core_discovery":"The central discovery is that the negative phenomenon behind the first result is, after Boolean valued transfer, nothing but a Hamel basis over a proper subfield. When the Boolean algebra $B$ is not $\\sigma$-distributive, the reals $\\mathbb{R}$ inside the Boolean valued model differ from their standard-name copy $\\mathbb{R}^{\\wedge}$; the real field is then infinite-dimensional over the subfield $\\mathbb{R}^{\\wedge}$, so it splits as $\\mathbb{R} = X_1 \\oplus X_2$ into two $\\mathbb{R}^{\\wedge}$-linear subspaces that are isomorphic to $\\mathbb{R}$ as vector spaces over $\\mathbb{R}^{\\wedge}$ but not as ordered vector spaces over $\\mathbb{R}^{\\wedge}$. Descent turns this algebraic splitting into a decomposition $X = X_1 \\oplus X_2$ of the universally complete vector lattice $X = \\mathbb{R}^{\\downarrow}$, with $X_1$ and $X_2$ fragment closed, laterally complete, and related to $X$ by band-preserving linear bijections. The second discovery, for the Ando transfer, is that every injective Banach lattice is the bounded descent of an AL-space, that the spaces $L_p(\\Phi)$ built from a Maharam operator are exactly the bounded descents of Boolean valued $AL_p$-spaces, and that $c_0(\\Gamma^{\\wedge})^{\\downarrow}$ coincides with $C^\\#(Q, c_0(\\Gamma))$; applying the classical Ando theorem inside the Boolean valued model then yields the dichotomy over a partition of unity.","pith_inferences":["The same Boolean valued Hamel-basis trick should produce decompositions into non-order-complete sublattices for any universally complete lattice whose Boolean algebra is not $\\sigma$-distributive, with the recognition of descended sublattices as the only obstruction; this suggests a general splitting-by-cardinal-absorption principle.","The Ando dichotomy in the second theorem suggests a splicing paradigm: a $B$-cyclic Banach lattice whose $B$-complete sublattices are all projection-complemented is a mix of a continuous spectral type ($L_p$-type) and a discrete spectral type ($c_0$-type) along a partition of unity; the same Boolean valued transfer could be applied to other classical characterizations of Banach lattices.","Because the classical Ando theorem fails in dimensions one and two, the Boolean valued analogue should likewise fail for Boolean dimension at most two; transferring the classical counterexamples would yield explicit low-dimensional $B$-cyclic counterexamples.","The cardinal-shift nonuniqueness in the $c_0$-piece points to a deeper interplay between set-theoretic cardinal arithmetic and descended Banach lattice structure: distinct cardinals inside the model can collapse to isomorphic descended spaces, so dimension is not a Boolean valued invariant."],"forward_implications":["The motivating disjointness-preserving operator question receives a strongly negative answer: whenever $X$ is universally complete with no locally one-dimensional bands, there are two non-order-complete sublattices each band-preservingly linearly isomorphic to $X$ and together spanning $X$.","The decomposition can be refined to countably or infinitely many summands, since the relevant cardinal identity $\\kappa = \\sum_{\\alpha \\in A}\\kappa$ holds for infinite $\\kappa$ and $|A| \\le \\kappa$.","In the $B$-cyclic setting, the Ando dichotomy holds in exact form: a $B$-cyclic Banach lattice with Boolean dimension at least three has contractive positive projections onto every $B$-complete closed sublattice if and only if it is, over a partition of unity, a $B$-cyclic $L_p(\\Phi)$-space or a space $C^\\#(Q_\\gamma, c_0(\\gamma))$.","The $B$-atomic part admits an explicit $\\ell^\\infty$-sum representation $\\rho X \\simeq (\\oplus\\sum_{\\gamma \\in \\Delta} C^\\#(P_\\gamma, \\ell^{p(\\gamma)}(\\gamma)))_{\\ell^\\infty}$, while the $c_0$-piece is generally nonunique because of the cardinal-shift phenomenon.","The classical spaces $c_0(\\Gamma)$, $\\ell^1$, and $\\ell^\\infty$ acquire natural $B$-cyclic analogues $C^\\#(Q,c_0)$, $C^\\#(Q,\\ell^1)$, and $C^\\#(Q,\\ell^\\infty)$ that play the same embedding role as their classical counterparts."],"supporting_citations":[{"why":"Supplies the Gordon theorem that the reals inside $V^{(B)}$ descend to a universally complete vector lattice, giving the identification $X = \\mathbb{R}^{\\downarrow}$.","marker":"[7]"},{"why":"Supplies the Gutman theorem characterizing locally one-dimensional K-spaces via $\\sigma$-distributivity; this forces $\\mathbb{R} \\neq \\mathbb{R}^{\\wedge}$ under the theorem's hypothesis.","marker":"[9]"},{"why":"The monograph cited for Lemma 3.3, the characterization of order-complete, laterally complete, fragment-closed sublattices as descents of vector sublattices over $\\mathbb{R}^{\\wedge}$, and for descent rules on band-preserving operators.","marker":"[2]"},{"why":"Poses the problem of whether an invertible disjointness-preserving operator forces lattice isomorphism and gives the first counterexample that the paper's first theorem strengthens.","marker":"[5]"},{"why":"The classical Ando theorem on Banach lattices whose closed sublattices are images of contracting positive projections; the second theorem transfers this statement inside $V^{(B)}$.","marker":"[19]"},{"why":"Provides the Boolean valued transfer principle for injective Banach lattices, identifying them as bounded descents of AL-spaces.","marker":"[11]"},{"why":"Supplies the structure theorem for injective Banach lattices used for the Maharam-operator representation and the continuous-bundle picture in Section 4.","marker":"[16]"}],"fun_headline_variants":["Boolean valued transfer splits lattices into disjoint twins","Ando dichotomy for cyclic Banach lattices via Boolean valued models","Hamel basis trick yields lattice decomposition","Two applications: lattice splitting and Ando transfer","Lattice splits into two band-invariant copies"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The argument's load-bearing premise is that, in the Boolean valued model, the reals form a proper extension of their standard-name subfield precisely when the lattice has no locally one-dimensional bands, and that descended vector subspaces of the reals automatically produce sublattices of $\\mathbb{R}^{\\downarrow}$ with the required lateral completeness, fragment closure, and lack of order completeness; failure of either half would break the decomposition theorem.","fun_headline_variants_meta":{"raw":{"variants":["Boolean valued transfer splits lattices into disjoint twins","Ando dichotomy for cyclic Banach lattices via Boolean valued models","Hamel basis trick yields lattice decomposition","Two applications: lattice splitting and Ando transfer","Lattice splits into two band-invariant copies"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000665,"raw_usage":{"total_tokens":3037,"prompt_tokens":951,"completion_tokens":2086,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":567,"completion_tokens_details":{"reasoning_tokens":2013}},"tokens_in":567,"tokens_out":2086,"duration_ms":16911,"temperature":1.0,"reasoning_tokens":2013,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-14T14:46:32.355083+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Take $X = \\mathbb{R}^{\\downarrow}$ where $B$ is the regular-open algebra of $[0,1]$ (not $\\sigma$-distributive), split $\\mathbb{R}$ as a vector space over its standard-name subfield by a Hamel basis $E = E_1 \\sqcup E_2$ with $|E_1| = |E_2|$, and check directly whether the two descended sublattices $X_1^{\\downarrow}$ and $X_2^{\\downarrow}$ are laterally complete and non-order-complete; any violation would contradict the first main theorem.","supporting_citations":[{"cited_title":"I., Real numbers in Boolean-valued models of set theory and K-spaces, Dokl","cited_arxiv_id":null,"evidence_quote":"Supplies the Gordon theorem that the reals inside $V^{(B)}$ descend to a universally complete vector lattice, giving the identification $X = \\mathbb{R}^{\\downarrow}$."},{"cited_title":"E., Locally one-dimensional K-spaces and σ -distributive Boolean algebras , Sib","cited_arxiv_id":null,"evidence_quote":"Supplies the Gutman theorem characterizing locally one-dimensional K-spaces via $\\sigma$-distributivity; this forces $\\mathbb{R} \\neq \\mathbb{R}^{\\wedge}$ under the theorem's hypothesis."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"The monograph cited for Lemma 3.3, the characterization of order-complete, laterally complete, fragment-closed sublattices as descents of vector sublattices over $\\mathbb{R}^{\\wedge}$, and for descent rules on band-preserving operators."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Poses the problem of whether an invertible disjointness-preserving operator forces lattice isomorphism and gives the first counterexample that the paper's first theorem strengthens."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"The classical Ando theorem on Banach lattices whose closed sublattices are images of contracting positive projections; the second theorem transfers this statement inside $V^{(B)}$."},{"cited_title":"G., The Boolean transfer principle for injective Banach lattic es, Sib","cited_arxiv_id":null,"evidence_quote":"Provides the Boolean valued transfer principle for injective Banach lattices, identifying them as bounded descents of AL-spaces."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Supplies the structure theorem for injective Banach lattices used for the Maharam-operator representation and the continuous-bundle picture in Section 4."}],"review_version":1}