Catalog

Repositories

Latest version of each Zenodo concept by Logica Artificialis. A row shows only the files on that record, and a GitHub address only when the manuscript names one.

34 repositories

• 10.5281/zenodo.238380122026

  Order extension in Chehata's group

  Martino Lupini · Orderability, Groups

  In the real case, Chehata's group consists of the increasing piecewise-linear homeomorphisms of the real line with finitely many pieces and bounded support. We prove that every bi-invariant partial order on this group extends to a bi-invariant total order, answering the real-field form of the question recorded in the 2009 survey of Bludov, Glass, Kopytov, and Medvedev. More precisely, a set of nonidentity elements generates a normal semigroup containing the identity if and only if the logarithms of its initial slopes and those of its final slopes each admit a nonempty positive integer relation. Every partial order therefore extends to an endpoint-slope order induced by an additive total order on the abstract group of real numbers. These orders form two disjoint families and exhaust all bi-orders; their number is 2^{2^{aleph_0}}. The proof creates an arbitrary endpoint germ after cancellation of a positive germ relation and then applies an equal-germ positive factorization argument. Over an Archimedean ordered field F, the analogous criterion and classification use additive orders on the logarithmic slope group and give 2^{|F|} bi-orders.

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• 10.5281/zenodo.238353112026

  Groups with prescribed finite numbers of lattice orders

  Martino Lupini · Orderability, Groups

  We prove that the possible finite numbers of lattice group orders on an ℓ-orderable group are exactly 1 and the positive even integers. We also construct, for every n ≥ 1, a countable nonabelian simple group with exactly 2n bi-invariant total orders. These results answer the finite-spectrum questions in the 2009 problem list of Bludov, Glass, Kopytov, and Medvedev. The lattice-order construction rests on a classification of all lattice orders on certain groups of homeomorphisms of the line. For the complementary examples, we show that every lattice order on the affine family used here, due to Kopytov, is total.

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• 10.5281/zenodo.238280862026

  Normal-semigroup generators for the bi-order cones of Thompson's group F′

  Martino Lupini · Thompson groups, Orderability, Groups

  The commutator subgroup F′ of Thompson's group has four bi-invariant total orders, determined by the slope at the first or last point of support. We prove that the positive cone of each order is generated by two elements as a normal semigroup: every positive element is a nonempty product of conjugates of those two elements. An explicit pair f, g works for all four cones, using the four choices of signs (f, g), (f⁻¹, g⁻¹), (f⁻¹, g), and (f, g⁻¹). This answers the finite-generation question posed by Navas and Rivas in 2008, including their stronger requirement that the same pair yield both prescribed partitions of F′\{1}. Two generators are necessary for each cone.

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• 10.5281/zenodo.238276522026

  Failure of rational injectivity for minimal principal étale groupoids

  Martino Lupini · Orderability, Groupoids, Groups

  Applying OpenAI's torsion-free group counterexample, we construct a second-countable locally compact Hausdorff minimal principal ample groupoid with Cantor unit space whose ordinary reduced Baum--Connes assembly map has an infinite-order kernel class in degree zero. Its rationalization is therefore not injective, with the natural coefficient algebra of continuous functions on the unit space. The groupoid is a transformation groupoid of a finitely generated torsion-free group and is compactly generated. We use the invariant-measure trace class of Antonini, Azzali and Skandalis to preserve an ordinary assembly kernel class under extension of coefficients to continuous functions. Elek's free minimal measure-preserving Cantor action supplies the required dynamics. The standard compact-action comparison identifies group assembly with coefficients and transformation-groupoid assembly. Thus minimality and principality do not restore the natural-coefficient reduced Baum--Connes assertion.

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• 10.5281/zenodo.238040062026

  Order-universal free subgroups and bi-order extension

  Martino Lupini · Orderability, Groups

  We construct fixed free subgroups along which every prescribed bi-order extends to the ambient group. Such subgroups exist inside every nontrivial normal subgroup of a free group of rank two. A finite-support localization theorem transfers this property to other bi-orderable groups, using one visible element and an invariant order on the actual intermediate quotient. We apply it to every bi-orderable rank-two free-by-cyclic group, to higher-rank monodromies with positive real homology spectrum not entirely equal to one, and to an explicit IA mapping torus. The last example and all its height cyclic covers admit no compatible cyclic-cover free quotient in any integral character direction, but still have the required universal free subgroups. Consequently total Bi-order Extension is undecidable in each of these fixed groups, already for a fixed finite background and one varying word. The argument also distinguishes the order-transfer problem from the existence of free quotients in finite covers.

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• 10.5281/zenodo.236826802026

  Jordan spectra of nilpotent derivations from scalar geometry

  Martino Lupini · Orderability, Groups

  We determine the complete Jordan spectrum of nilpotent derivations of a nonabelian class-two Lie algebra I over a characteristic-zero field. We assume that its scalar bracket space has a decomposable basis over the ground field and that its reduced geometric decomposable locus admits no nontrivial additive-group action. A plane-preserver algebra and a hyperplane arrangement determine the horizontal rank set R; an integer criterion then decides every full Jordan type. Every admitted type occurs over the ground field, and the spectrum remains unchanged under field extension. For u = dim(I/Z(I)), d = dim I′, f = dim(Z(I)/I′), and r = max R, the greatest partition is (2^r, 1^(u-2r)) + (f) + (1^d), with componentwise addition. The full spectrum recovers R and is a dominance order ideal exactly when R has no gaps. For fixed free central dimension f, finite scalar loci with identical support-sum rank functions can have different spectra; support dimension six is minimal. A family has Jordan strata and commuting varieties whose component dimensions differ without bound. Geometric consequences include a smooth rational greatest-type stratum, a crepant resolution, and a classification of nilpotent linear spaces of largest dimension. Hall polynomials give uniform Jordan-stratum point counts at almost all places for number-field models.

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• 10.5281/zenodo.236405402026

  Positive normal identities and bi-order extension

  Martino Lupini · Orderability, Groups

  We prove that it is undecidable whether the identity in a free group of any fixed finite rank at least two is a nonempty product of conjugates of given elements, with only positive occurrences of those elements allowed. It is also undecidable whether a finite subset of that group can be made positive in a total order invariant under multiplication on both sides. Both conclusions hold for a fixed finite background together with one varying word, all lying in the commutator subgroup. These give negative answers to the normal identity and bi-order questions stated online in 2025 by Bodart, Ciobanu and Metcalfe, and to the bi-order extension question formulated by Colacito and Metcalfe (2017). The second result also settles their 2019 question about the word problem for free representable lattice-ordered groups: it is undecidable already on two generators.

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• 10.5281/zenodo.235230912026

  Simple left-orderable groups with prescribed finiteness length

  Martino Lupini · Orderability, Groups

  For every integer n ≥ 2 we construct an infinite simple left-orderable group of type F_n and not of type FP_{n+1}(ℤ), and hence not of type F_{n+1}, where type F_n means that a classifying space has finite n-skeleton. This answers affirmatively the left-orderable interpretation of a question of Xiaolei Wu in the problem list associated with the 2024 CIRM meeting. The construction inserts arithmetic groups of one-sided germs into a group of homeomorphisms of the line. A forest complex controls the finiteness of the germ groups, and an actual interval-lift argument controls the singularity stabilizers. An (n−1)-connected band in the germ complex then gives type F_n for the simple derived subgroup. A Lipschitz retraction onto a kernel of endpoint characters retains the homological finiteness obstruction.

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• 10.5281/zenodo.235206832026

  Semiprojectivity and norm-controlled lifting in Banach lattices

  Martino Lupini · Orderability

  We characterize semiprojectivity of a real Banach lattice by almost contractive lifting through ideal quotients of separable Banach lattices, with no separability assumption on the domain. Semiprojectivity is equivalent to ℓ-openness, implies ℓ-closedness, and is unchanged if the finite ideal stage is allowed to depend on the norm tolerance. For separable domains it coincides with 1⁺-projectivity. We prove that arbitrary ℓ₁-sums preserve semiprojectivity without an order-unit hypothesis on the summands. In contrast, ℓ₁(ℕ) ⊕∞ ℝ is not semiprojective, although both summands are 1⁺-projective. Thus semiprojectivity need not pass to finite ℓ∞-sums or split extensions. The example is ∞-projective with infimum of uniform projectivity constants equal to 2, admits an equivalent 1⁺-projective lattice norm, and is not ℓ-closed.

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• 10.5281/zenodo.234699552026

  Order, norming faces, and symmetries of Tursi's Banach lattice

  Martino Lupini · Orderability, Groups

  Let L be Tursi’s separable universal homogeneous real Banach lattice. The set of decreasing positive sequences with infimum zero is coanalytic complete, even for sequences of components of one fixed quasi-interior vector, providing a fixed-lattice coanalytic-complete example for the decreasing-order-complexity question raised in Tursi’s 2024 survey. This gives a negative answer to the 2024 question of Avilés, Rosendal, Taylor, and Tradacete whether every separable Banach lattice belongs to their countable Fatou hierarchy. Unrestricted sequential order convergence has the exact complete complexity coA(Pi^1_1), the dual of one Souslin operation on coanalytic sets. We classify quasi-interior positive vectors by their norming Choquet simplices and answer Tursi’s documented 2021 automorphism-group universality question affirmatively. We prove universality of embedding orbits for every nonzero separable domain and of automorphism conjugacy, and establish concentration, lifting, and representation results for pointwise face kernels. Further results describe generic embedding orbits, ideals, and finite-dimensional approximate disposition.

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• 10.5281/zenodo.234696252026

  Forests as Heyting spectra: representation criteria, constructions, and obstructions

  Martino Lupini · Orderability

  Our main result shows that an Esakia space whose order is a downward forest has a locally finite clopen-upset algebra whenever its maximal boundary is scattered. More precisely, failure of local finiteness forces a nonempty perfect principal maximal boundary. We also construct locally finite representations of immediately splitting trees without assuming well-foundedness, and classify the cardinal multiplicities of a uniform family of binary canopies. The countable-forest inverse-sequence characterization and failure of infinitary order axiomatizability are consequences of these results, using established inverse-limit and model-theoretic methods. We give finite-partition certificates as explicit forms of the existing duality and compactification machinery. The unrestricted intrinsic structural classification of forest spectra remains unresolved here.

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• 10.5281/zenodo.234611402026

  Endomorphism counts from primitive orbits and synchronized block actions

  Martino Lupini · Orderability, Posets, Groups

  For a finite poset P, let R(P) = |End(P)|/|Aut(P)|, where endomorphisms preserve the reflexive order. We prove R(P) ≥ 2^{|P|/(64 log₂ |P|)} for |P| ≥ 4 when every chain has at most two elements and the action on each nonsingleton automorphism orbit is primitive. The same bound holds for a class of imprimitive actions: on each interaction component the action must be the full automorphism group of a rooted tree, acting identically on all the identified orbits. The proof combines simultaneous collapses of indices with maps into principal ideals and filters. We give counting lemmas that retain the whole kernel of a block action, and a quantitative collapse, degree, or orbit-size alternative for normal products of alternating groups with natural orbits. We also construct connected twin-free posets with group S₃ wr Sₜ, constant minimum of the two maximum principal-set sizes, and imprimitive action on every orbit, although every orbit-preserving endomorphism is an automorphism. A restricted three-level extension has lower bound 2^{|P|/(128 (log₂ |P|)²)}. These results give partial answers to the finite-poset automorphism problem of Rival and Rutkowski; the unrestricted question remains beyond the arguments here.

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• 10.5281/zenodo.234584502026

  A countable scattered poset without a strongly maximal chain

  Martino Lupini · Orderability, Posets

  A chain C in a partially ordered set is strongly maximal if |D \ C| ≤ |C \ D| for every chain D. We construct a countably infinite poset with no infinite antichain and no strongly maximal chain. The poset is scattered: none of its chains contains a copy of the rational order. This gives a negative answer, already in the countable scattered case, to the strong-chain question posed by Hollom in 2024 and restated in 2026. The construction replaces each point (x, y, n) of Hollom’s layered poset by a finite chain of length 2^x + 2^y. A finite path exchange restricts the possible chains, after which a further finite exchange always improves them. We also analyze a width-two example of Hollom showing that the chain-extension inclusion used in an earlier claimed positive proof need not hold.

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• 10.5281/zenodo.234405222026

  A countable norm construction for free Banach lattices under arbitrary convexity implications

  Martino Lupini · Orderability

  We give an explicit countable construction of the norm of the free Banach lattice of an arbitrary prescribed convexity type over a pre-ordered Banach space. Starting with the ordinary free Banach lattice, each step takes the gauge of a closed solid absolutely convex hull determined by the original convexity implications. If $p_k$ is the seminorm obtained after $k$ steps, then the free norm is $p_\omega=\inf_{k<\omega}p_k$. No continuity is required of the increasing numerical functions in the implications, and every antecedent is retained in full. The proof uses disjoint cutoffs that repair the antecedents while preserving the value of one positive functional. The same estimate gives quotient closure and an exact restriction of each successor gauge to finite-dimensional vector sublattices. Applying classical compactness to the resulting class yields a local norm formula over finite-dimensional linear input spaces and finite sets of complete implications. Consequently every strict decrease of the free norm is witnessed at a finite input dimension, a finite set of implications, and a finite stage. For each fixed separable input, a countable subfamily of complete conditions already determines the same free norm. For one presentation of the AM-space type, no finite stage computes the norm on all lattice expressions. The formula also gives an intrinsic isometry criterion, without providing the convenient geometric test asked for by de Jeu and Jiang.

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• 10.5281/zenodo.234404832026

  Peripheral eigenspace dimensions for positive operators and semigroups

  Martino Lupini · Orderability, Operator algebras

  Let $T$ be a positive contraction on a monotonically complete complex Banach lattice with the Fatou property. We prove $$\dim\ker(\lambda I-T)\leq\dim\ker(\lambda^k I-T) \qquad (|\lambda|=1,\ k\in\mathbb Z),$$ where all infinite dimensions are identified with infinity. This answers an open problem posed by Jochen Glück in 2021 and extends the previously established root-of-unity case. The bounded bilateral trajectories of $T$ form an intrinsic Banach lattice by the fixed-space theorem. On this lattice, the shift is a lattice automorphism, and evaluation at zero identifies its unimodular eigenspaces with those of $T$. We also prove the corresponding multiplicity inequality for imaginary generator eigenspaces of positive contractive $C_0$-semigroups, and for positive power-bounded operators that are order continuous.

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• 10.5281/zenodo.234388512026

  Universal minimal flows from pointwise completions and atomic decomposition

  Martino Lupini · Orderability, Operator algebras, Descriptive set theory

  We organize explicit universal minimal flows of mapping and transformation groups around two standard reductions: passage to a compact group completion and removal of a closed extremely amenable normal subgroup. Finite interpolation identifies the pointwise completions of path, loop, gauge, and section groups; for disconnected compact Lie targets the completion also records the clopen information in the source. For measurable maps into a compact metrizable group K, the established atomless fixed-point theorem gives M(L0(Y, μ; K)) = KAμ, where Aμ is the set of atoms. Combining these reductions with known structure and compact-extension theorems yields atomic formulas for measure-space transformations, Lp-isometries, and automorphism groups of operator algebras. The general Banach-valued statement uses complex fibres; real Hilbert fibres are treated separately. We also give the formula M(FΩ ⋊ Sym(Ω)) = M(F)Ω × LO(Ω) for a Polish group F with metrizable universal minimal flow and a countable set Ω, with the natural skew action. Its ordered relational special cases have earlier Ramsey-theoretic treatments. The results are presented as explicit consequences and extensions of the cited completion, extreme-amenability, and structural results.

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• 10.5281/zenodo.234376442026

  Spectral criteria for extending partial left orders

  Martino Lupini · Orderability, Groups

  We answer affirmatively two extension questions posed by Bodart, Ciobanu, and Metcalfe in 2024: every partial left order extends to a total left order on the group of positive rational affine transformations, and on every group ℤ² ⋊T ℤ with T ∈ SL₂(ℤ) and tr(T) > 2. Both follow from a metabelian criterion: it suffices that the group be torsion-free, its derived subgroup have finite rational rank, and all conjugation eigenvalues on that subgroup be positive real. We derive more general extension results from structural theorems of Dong and Bodart–Dong. For groups with a finite-rank nilpotent normal subgroup and abelian quotient, extension of arbitrary partial left orders is equivalent to extension of total left orders from subgroups. This connects the cyclic case with Pierce’s classical theorem. A unified spectral criterion allows a locally nilpotent derived subgroup: under local finiteness and positive spectrum modulo its center, full extension is equivalent to the requirement that a central eigenvalue whose positive power is positive real is itself positive real. The proofs explain how local affine certificates yield orders on the prescribed ambient group.

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• 10.5281/zenodo.234257012026

  Pairwise mixing layouts and self-joinings of weakly mixing transformations

  Martino Lupini · Orderability

  Let T be an invertible weakly mixing transformation of a standard probability space (X, μ). We prove that pairwise mixing along any prescribed finite family of integer sequences implies joint mixing: if every two-coordinate marginal of (T^{d₁(b)}x, …, T^{dₖ(b)}x), for x distributed according to μ, converges to μ², then the full distribution converges to μᵏ. No entropy assumption is needed. This answers Ryzhikov's question on mixing sequences. The proof depends essentially on structural results from OpenAI's multiple-mixing preprint. For weakly mixing rank-one transformations, a tower argument shows that every pairwise-independent finite self-joining is product, answering the corresponding questions in Ryzhikov's problem list. Finally, for invertible ergodic transformations of standard nonatomic spaces, two-fold minimal self-joinings imply minimal self-joinings of all finite orders. This follows from OpenAI's theorem and Ryzhikov's equivalence with the restricted multiple-mixing problem.

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• 10.5281/zenodo.234200872026

  Bi-orderable soluble groups with only Conradian left orders

  Martino Lupini · Orderability, Groups

  We prove that a soluble bi-orderable group all of whose left orders are Conradian is locally nilpotent, without any finite-generation or countability assumption. Consequently, every subgroup of such a group also has only Conradian left orders. This answers the soluble, bi-orderable case of the subgroup question posed by A. H. Rhemtulla and recorded in 2009.

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• 10.5281/zenodo.234199962026

  Thompson's group F′ is an O*-group: Positive conjugacy semigroups in Thompson's group F′

  Martino Lupini · Thompson groups, Orderability, Groups

  We give an explicit countable simple example for the Mal’cev–Fuchs question whether a simple O*-group exists, posed in 1963 and 1965. Every bi-invariant partial order on the commutator subgroup F′ of Thompson’s group F extends to a bi-invariant total order. Every partial positive cone lies in one of four cones determined by slopes at the leftmost or rightmost support endpoint. The central ingredient is a factorization theorem for right-moving one-bump maps with the same support: equality of their two endpoint slopes is equivalent to a sufficiently large power of one being a product of the corresponding number of conjugates of the other, supported on that interval. The proof combines bounded drift in orbit coordinates with a uniform correction bound from the commutator width of F′.

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• 10.5281/zenodo.234196352026

  Saturation and equational hulls of ordered semigroup varieties

  Martino Lupini · Orderability

  We prove that an inequality-defined variety of partially ordered semigroups is saturated if and only if the ordinary semigroup variety generated by its reducts is saturated. Here saturation excludes proper epimorphic extensions in the ambient category of all semigroups or all partially ordered semigroups. This transfers ordinary equational saturation criteria to compatible partial orders without finiteness, permutation, or convexity assumptions. It recovers a positive answer to Ahanger, Shah, and Khan’s question about ordered normal bands and gives a general equational framework for relaxing convexity in related saturation results. The proof uses the completely regular core E(S)SE(S), a bounded-exponent variety symmetry theorem, and a countable construction for arbitrary inequational theories.

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• 10.5281/zenodo.234182732026

  Isomorphism and bi-embeddability of countable Archimedean ordered groups

  Martino Lupini · Orderability, Descriptive set theory, Groups

  Let E and F denote order-preserving isomorphism and mutual order-preserving embeddability of countable Archimedean ordered groups. We prove that E

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• 10.5281/zenodo.234114932026

  Polarities, voltages, and capacitors: a categorical approach to hulls, envelopes, and completions

  Ivan Di Liberti, Martino Lupini · Orderability, C*-algebras

  This article provides a general framework in the context of category theory where one can recognize as particular instances of the same abstract construction several notions of completion, envelope, and hull, such as the completion of a Boolean algebra, the Dedekind–MacNeille completion of an ordered set, the multiplier ring of a ring, and the multiplier algebra of a C*-algebra.

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• 10.5281/zenodo.233423112026

  Dynamical length of Cartan pairs: comparison, rigidity, and invariance

  Martino Lupini · Orderability, C*-algebras, Groupoids

  A Cartan subalgebra permits a separable unital C*-algebra to be studied through its twisted étale Weyl groupoid. We combine five cohomological, derivation, and automorphism lengths into the dynamical length ℓ(A,D) of a Cartan pair. These lengths measure the accumulation of coboundaries and inner symmetries through the canonical filtration of phantom subgroups.

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• 10.5281/zenodo.233421812026

  Bounded geometry in Polish groups: locally compact models, amenability, and simple examples

  Martino Lupini · Orderability, Descriptive set theory, Amenability

  We develop locally compact models and applications for Polish groups of bounded geometry. Building on Hrushovski's quasi-model theorem and the ultrafilter framework of Krupiński–Pillay, we give an explicit coarse-density calculation for a model with compact conjugation-invariant multiplication defect and exact control of the intrinsic coarsely bounded bornology. Applications yield the Haagerup property under topological amenability and two-sided entropy estimates under metric amenability. Packing and model obstructions treat the stated classes of groups. For McCleary's classical simple periodic coloured-order group, we compute the pointwise Polish topology and intrinsic coarse geometry, obtaining a non-locally compact quasi-line.

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• 10.5281/zenodo.233411642026

  Solution to Dixmier's Problem about spectra of C*-algebras

  Martino Lupini · Orderability, C*-algebras, Descriptive set theory

  Dixmier asked in 1967 whether the spectra of simple separable non-type-I C*-algebras all have the same Borel structure. We solve the formulation using ordinary isomorphisms of quotient Borel σ-algebras by proving that its universal assertion is independent of ZFC, relative to the consistency of ZFC. Under the continuum hypothesis, all separable non-type-I spectra are measurably isomorphic, without simplicity or unitality assumptions. In the model obtained by iterating Sacks forcing for ω₂ stages with countable supports over a GCH ground model, a fixed simple unital universal spectrum is not measurably isomorphic to the CAR spectrum. Neither consistency direction requires a large cardinal.

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• 10.5281/zenodo.232847082026

  A three-generated abelian lattice-ordered group without an existentially closed prime extension

  Martino Lupini · Orderability, Groups

  No abstract on this deposit.

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• 10.5281/zenodo.232775982026

  Distributivity of lattice-ordered pregroups

  Martino Lupini · Orderability

  No abstract on this deposit.

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• 10.5281/zenodo.231768272026

  Convex normal subgroups of ℓ-simple lattice-ordered groups

  Martino Lupini · Orderability, Groups

  We construct a countable nonabelian normal-valued lattice-ordered group G which is ℓ-simple and has a nontrivial proper ordinary convex normal subgroup N. Both N and its abelian lattice hull H are free abelian of countably infinite rank. The subgroup N is an antichain normally generated by one element, while H is proper, normal, cofinal and order dense. This answers Problem 1.1 of Bludov, Glass, Kopytov and Medvedev negatively. Finite triangular coefficient tables on lexicographically ordered blocks preserve the antichain. A positive element forces an ordinary convex normal subgroup to contain the entire next finite-table group; iteration therefore gives simplicity. A conjugating extension and order density show that every positive nonidentity element of G has a cofinal conjugacy class. For an ℓ-simple group, a strong order unit guarantees an affirmative answer. We derive strong order units from finite lattice generation, a finite-index commensurator of a nontrivial finitely generated subgroup, or a nontrivial normal subgroup whose elements have powers in a fixed finitely generated subgroup.

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• 10.5281/zenodo.231257522026

  The Artin group of type $E_8$ and the left-orderability of spherical Artin groups

  Martino Lupini · Orderability, Groups

  No abstract on this deposit.

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• 10.5281/zenodo.230466982026

  A countable lattice-ordered group not latticeisomorphic to any abelian one

  Martino Lupini · Orderability, Groups

  No abstract on this deposit.

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• 10.5281/zenodo.229325952026

  Lattice-Ordered Groups as Full Automorphism Groups of Posets

  Martino Lupini · Orderability, Posets, Groups

  No abstract on this deposit.

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• 10.5281/zenodo.228833532026

  Isomorphism of finitely generated bi-orderable elementary amenable groups is universal

  Martino Lupini · Orderability, Amenability, Groups

  No abstract on this deposit.

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• 10.5281/zenodo.228526312026

  Ordered Compactifications and Circular Ellis Semigroups

  Martino Lupini · Orderability

  No abstract on this deposit.

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