Canonical Extension Research Articles

Linking invariants for valuations and orderings on fields

The mod-2 arithmetic Milnor invariants, introduced by Morishita, provide a decomposition law for primes in canonical Galois extensions of Q\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mathbb {Q}$$\\end{document} with unitriangular Galois groups and contain the Legendre and Rédei symbols as special cases. Morishita further proposed a notion of mod-q arithmetic Milnor invariants, where q is a prime power, for number fields containing the qth roots of unity and satisfying certain class field theory assumptions. We extend this theory from the number field context to general fields, by introducing a notion of a linking invariant for discrete valuations and orderings. We further express it as a Magnus homomorphism coefficient and relate it to Massey product elements in Galois cohomology.

Algebraic Aspects and Functoriality of the Set of Affiliated Operators

Abstract In this article, we aim to provide a satisfactory algebraic description of the set of affiliated operators for von Neumann algebras. Let $\mathscr{M}$ be a von Neumann algebra acting on a Hilbert space $\mathcal{H}$, and let $\mathscr{M}_{\text{aff}}$ denote the set of unbounded operators of the form $T = AB^{\dagger }$ for $A, B \in \mathscr{M}$ with $\text{null} (B) \subseteq \text{null} (A)$, where $(\cdot )^{\dagger }$ denotes the Kaufman inverse. We show that $\mathscr{M}_{\text{aff}}$ is closed under sum, product, Kaufman-inverse, and adjoint, and has the structure of a (right) near-semiring. Moreover, the above quotient representation of an operator in $\mathscr{M}_{\text{aff}}$ is essentially unique. Thus, we may view $\mathscr{M}_{\text{aff}}$ as the multiplicative monoid of unbounded operators on $\mathcal{H}$ generated by $\mathscr{M}$ and $\mathscr{M}^{\dagger }$. We further show that our definition of affiliation, as reflected in $\mathscr{M}_{\text{aff}}$, subsumes the traditional one. Let $\Phi $ be a unital normal *-homomorphism between represented von Neumann algebras $(\mathscr{M}; \mathcal{H})$ and $(\mathscr{N}; \mathcal{K})$. Using the quotient representation, we obtain a canonical extension of $\Phi $ to a mapping $\Phi _{\text{aff}}: \mathscr{M}_{\text{aff}} \to \mathscr{N}_{\text{aff}}$ which is a near-semiring homomorphism that respects Kaufman-inverse and adjoint; in addition, $\Phi _{\text{aff}}$ respects Murray–von Neumann affiliation of operators and also respects strong sum and strong product. Thus, $\mathscr{M}_{\text{aff}}$ is intrinsically associated with $\mathscr{M}$ and transforms functorially as we change representations of $\mathscr{M}$. Furthermore, $\Phi _{\text{aff}}$ preserves operator properties such as being symmetric, or positive, or accretive, or sectorial, or self-adjoint, or normal, and also preserves the Friedrichs and Krein–von Neumann extensions of densely defined closed positive operators. As a proof of concept, we transfer some well-known results about closed unbounded operators to the setting of closed affiliated operators for properly infinite von Neumann algebras, via “abstract nonsense”.

Multi-omic integration of microbiome data for identifying disease-associated modules

Multi-omic studies of the human gut microbiome are crucial for understanding its role in disease across multiple functional layers. Nevertheless, integrating and analyzing such complex datasets poses significant challenges. Most notably, current analysis methods often yield extensive lists of disease-associated features (e.g., species, pathways, or metabolites), without capturing the multi-layered structure of the data. Here, we address this challenge by introducing “MintTea”, an intermediate integration-based approach combining canonical correlation analysis extensions, consensus analysis, and an evaluation protocol. MintTea identifies “disease-associated multi-omic modules”, comprising features from multiple omics that shift in concord and that collectively associate with the disease. Applied to diverse cohorts, MintTea captures modules with high predictive power, significant cross-omic correlations, and alignment with known microbiome-disease associations. For example, analyzing samples from a metabolic syndrome study, MintTea identifies a module with serum glutamate- and TCA cycle-related metabolites, along with bacterial species linked to insulin resistance. In another dataset, MintTea identifies a module associated with late-stage colorectal cancer, including Peptostreptococcus and Gemella species and fecal amino acids, in line with these species’ metabolic activity and their coordinated gradual increase with cancer development. This work demonstrates the potential of advanced integration methods in generating systems-level, multifaceted hypotheses underlying microbiome-disease interactions.

Fano Threefolds With Affine Canonical Extensions

Fano Threefolds With Affine Canonical Extensions

S-Embedding of Lie Superalgebras and Its Implications for Fuzzy Lie Algebras

This paper performed an investigation into the s-embedding of the Lie superalgebra (→S1∣1), a representation of smooth vector fields on a (1,1)-dimensional super-circle. Our primary objective was to establish a precise definition of the s-embedding, effectively dissecting the Lie superalgebra into the superalgebra of super-pseudodifferential operators ( SψD⊙) residing on the super-circle S1|1. We also introduce and rigorously define the central charge within the framework of (→S1∣1), leveraging the canonical central extension of SψD⊙. Moreover, we expanded the scope of our inquiry to encompass the domain of fuzzy Lie algebras, seeking to elucidate potential connections and parallels between these ostensibly distinct mathematical constructs. Our exploration spanned various facets, including non-commutative structures, representation theory, central extensions, and central charges, as we aimed to bridge the gap between Lie superalgebras and fuzzy Lie algebras. To summarize, this paper is a pioneering work with two pivotal contributions. Initially, a meticulous definition of the s-embedding of the Lie superalgebra (→S1|1) is provided, emphasizing the representationof smooth vector fields on the (1,1)-dimensional super-circle, thereby enriching a fundamental comprehension of the topic. Moreover, an investigation of the realm of fuzzy Lie algebras was undertaken, probing associations with conventional Lie superalgebras. Capitalizing on these discoveries, we expound upon the nexus between central extensions and provide a novel deformed representation of the central charge.

Lax Extensions of Conical I-Semifilter Monads

For a quantale I, the unit interval endowed with a continuous triangular norm, we introduce the canonical, op-canonical and Kleisli extensions of the conical I-semifilter monad to I-Rel. It is proved that the op-canonical extension coincides with the Kleisli extension.

New perspectives on semi-primal varieties

We study varieties generated by semi-primal lattice-expansions by means of category theory. We provide a new proof of the Keimel-Werner topological duality for such varieties and, using similar methods, establish its discrete version. We describe multiple adjunctions between the variety of Boolean algebras and the variety generated by a semi-primal lattice-expansion, both on the topological side and explicitly algebraic. In particular, we show that the Boolean skeleton functor has two adjoints, both defined by taking certain Boolean powers, and we identify properties of these adjunctions which fully characterize semi-primality of an algebra. Lastly, we give a new characterization of canonical extensions of algebras in semi-primal varieties in terms of their Boolean skeletons.

Coherency for monoids and purity for their acts

This article examines the three-way relationship between right coherency of a monoid S, solutions of equations over S-acts, and injectivity properties of S-acts. A monoid S is right coherent if every finitely generated subact of every finitely presented (right) S-act itself has a finite presentation. Purity properties of an S-act A may either be expressed in terms of solutions in A of certain consistent sets of equations over A, or in terms of injectivity properties. For example, an S-act A is absolutely pure (almost pure) if every finite consistent set of equations over A (in one variable) has a solution in A. Equivalently, A is absolutely pure (almost pure) if it is injective with respect to inclusions of finitely generated subacts into finitely presented (monogenic finitely presented) S-acts.Our first main result shows that for a right coherent monoid S the classes of almost pure and absolutely pure S-acts coincide. Our second main result is that a monoid S is right coherent if and only if the classes of mfp-pure and absolutely pure S-acts coincide: an S-act is mfp-pure if it is injective with respect to inclusions of finitely presented subacts into monogenic finitely presented S-acts. We give specific examples of monoids S that are not right coherent yet are such that the classes of almost pure and absolutely pure S-acts coincide. Finally we give a condition on a monoid S for all almost pure S-acts to be absolutely pure in terms of finitely presented S-acts, their finitely generated subacts, and certain canonical extensions.

Affine Subspace Concentration Conditions

We define a new notion of affine subspace concentration conditions for lattice polytopes, and prove that they hold for smooth and reflexive polytopes with barycenter at the origin. Our proof involves considering the slope stability of the canonical extension of the tangent bundle by the trivial line bundle and with the extension class $c_1(\mathcal{T}_X)$ on Fano toric varieties.

On conditional probabilities and their canonical extensions to Boolean algebras of compound conditionals

In this paper we investigate canonical extensions of conditional probabilities to Boolean algebras of conditionals. Before entering into the probabilistic setting, we first prove that the lattice order relation of every Boolean algebra of conditionals can be characterized in terms of the well-known order relation given by Goodman and Nguyen. Then, as an interesting methodological tool, we show that canonical extensions behave well with respect to conditional subalgebras. As a consequence, we prove that a canonical extension and its original conditional probability agree on basic conditionals. Moreover, we verify that the probability of conjunctions and disjunctions of conditionals in a recently introduced framework of Boolean algebras of conditionals are in full agreement with the corresponding operations of conditionals as defined in the approach developed by two of the authors to conditionals as three-valued objects, with betting-based semantics, and specified as suitable random quantities. Finally we discuss relations of our approach with nonmonotonic reasoning based on an entailment relation among conditionals.

Global pluripotential theory on hybrid spaces

Let (X,L) be a polarized scheme over a Banach ring A. We define and study a class PSH(X,L) of plurisubharmonic metrics on the Berkovich analytification X an . We focus mainly on the case where A is a hybrid ring of power series, so that X an is the hybrid space associated to a degeneration of complex manifolds X. We then prove that any plurisubharmonic metric on (X,L) with logarithmic growth at zero admits a canonical plurisubharmonic extension to the hybrid space X hyb . We also discuss the continuity of the family of Monge-Ampère measures associated to a continuous plurisubharmonic hybrid metric.

Canonical Witt formal scheme extensions and p-torsion groups

We study the nth arithmetic jet space of the p-torsion subgroup attached to a smooth commutative formal group scheme. We show that the nth jet space above fits in the middle of a canonical short exact sequence between a power of the formal scheme of Witt vectors of length n and the p-torsion subgroup we started with. This result generalizes a result of Buium on roots of unity.

Invariants of formal pseudodifferential operator algebras and algebraic modular forms

We study from an algebraic point of view the question of extending an action of a group Γ on a commutative domain R to a formal pseudodifferential operator ring B = R((x ; d)) with coefficients in R, as well as to some canonical quadratic extension C = R((x 1/2 ; 1 2 d))2 of B. We give a necessary and sufficient condition of compatibility between the action and the derivation d of R for such an extension to exist, and we determine all possible extensions of the action to B and C. We describe under suitable assumptions the invariant subalgebras B Γ and C Γ as Laurent series rings with coefficients in R Γ. The main results of this general study are applied in a numbertheoretical context to the case where Γ is a subgroup of SL(2, C) acting by homographies on an algebra R of functions in one complex variable. Denoting by Mj the vector space of algebraic modular forms in R of weight j (even or odd), we build for any nonnegative integer k a linear isomorphism between the subspace C Γ k of invariant operators of order ≥ k in C Γ and the product space M k = j≥k Mj, which can be identified with a space of algebraic Jacobi forms of weight k. It results in particular a structure of noncommutative algebra on M0 and an algebra isomorphism Ψ : M0 → C Γ 0 , whose restriction to the particular case of even weights was previously known in the litterature. We study properties of this correspondence combining arithmetical arguments and the use of the algebraic results of the first part of the article.

A Point-Free Approach to Canonical Extensions of Boolean Algebras and Bounded Archimedean $$\ell$$-Algebras

Recently W. Holliday gave a choice-free construction of a canonical extension of a boolean algebra B as the boolean algebra of regular open subsets of the Alexandroff topology on the poset of proper filters of B. We make this construction point-free by replacing the Alexandroff space of proper filters of B with the free frame $$\mathcal {L}_B$$ generated by the bounded meet-semilattice of all filters of B (ordered by reverse inclusion) and prove that the booleanization of $$\mathcal {L}_B$$ is a canonical extension of B. Our main result generalizes this approach to the category $$\varvec{ ba \ell }$$ of bounded archimedean $$\ell$$ -algebras, thus yielding a point-free construction of canonical extensions in $$\varvec{ ba \ell }$$ . We conclude by showing that the algebra of normal functions on the Alexandroff space of proper archimedean $$\ell$$ -ideals of A is a canonical extension of $$A\in \varvec{ ba \ell }$$ , thus providing a generalization of the result of Holliday to $$\varvec{ ba \ell }$$ .

A Topological Duality for Monotone Expansions of Semilattices

In this paper we provide a Stone style duality for monotone semilattices by using the topological duality developed in S. Celani, L.J. González (Appl Categ Struct 28:853–875, 2020) for semilattices together with a topological description of their canonical extension. As an application of this duality we obtain a characterization of the congruences of monotone semilattices by means of monotone lower-Vietoris-type topologies.

Constellations with range and IS-categories

Constellations are asymmetric generalisations of categories. Although they are not required to possess a notion of range, many natural examples do. These include commonly occurring constellations related to concrete categories (since they model surjective morphisms), and also others arising from quite different sources, including from well-studied classes of semigroups . We show how constellations with a well-behaved range operation are nothing but ordered categories with restrictions. We characterise abstractly those categories that are canonical extensions of constellations with range, as so-called IS-categories. Such categories contain distinguished subcategories of insertions (which are monomorphisms) and surjections (in general different to the epimorphisms) such that each morphism admits a unique factorisation into a surjection followed by an insertion. Most familiar concrete categories are IS-categories, and we show how some of the well-known properties of these categories arise from the fact that they are IS-categories. For appropriate choices of morphisms in each, the category of IS-categories is shown to be equivalent to the category of constellations with range.

Solutions of a Class of Multiplicatively Advanced Differential Equations II: Fourier Transforms

For a wide class of solutions to multiplicatively advanced differential equations (MADEs), a comprehensive set of relations is established between their Fourier transforms and Jacobi theta functions. In demonstrating this set of relations, the current study forges a systematic connection between the theory of MADEs and that of special functions. In a large subset of the general case, we introduce a new family of Schwartz wavelet MADE solutions W μ , λ t for μ and λ rational with λ > 0 . These W μ , λ t have all moments vanishing and have a Fourier transform related to theta functions. For low parameter values derived from λ , the connection of the W μ , λ t to the theory of wavelet frames is begun. For a second set of low parameter values derived from λ , the notion of a canonical extension is introduced. A number of examples are discussed. The study of convergence of the MADE solution to the solution of its analogous ODE is begun via an in depth analysis of a normalized example W − 4 / 3 , 1 / 3 t / W − 4 / 3 , 1 / 3 0 . A useful set of generalized q -Wallis formulas are developed that play a key role in this study of convergence.

Logarithmic Riemann–Hilbert correspondences for rigid varieties

On any smooth algebraic variety over a p p -adic local field, we construct a tensor functor from the category of de Rham p p -adic étale local systems to the category of filtered algebraic vector bundles with integrable connections satisfying the Griffiths transversality, which we view as a p p -adic analogue of Deligne’s classical Riemann–Hilbert correspondence. A crucial step is to construct canonical extensions of the desired connections to suitable compactifications of the algebraic variety with logarithmic poles along the boundary, in a precise sense characterized by the eigenvalues of residues; hence the title of the paper. As an application, we show that this p p -adic Riemann–Hilbert functor is compatible with the classical one over all Shimura varieties, for local systems attached to representations of the associated reductive algebraic groups.

A Wavelet Plancherel Theory with Application to Multipliers and Sparse Approximations

We introduce an extension of continuous wavelet theory that enables an efficient implementation of multiplicative operators in the coefficient space. In the new theory, the signal space is embedded in a larger abstract signal space – the so called window–signal space. There is a canonical extension of the wavelet transform to an isometric isomorphism between the window–signal space and the coefficient space. Hence, the new framework is called a wavelet-Plancherel theory, and the extended wavelet transform is called the wavelet-Plancherel transform. Since the wavelet-Plancherel transform is an isometric isomorphism, any operation in the coefficient space can be pulled-back to an operation in the window–signal space. It is then possible to improve the computational complexity of methods that involve a multiplicative operator in the coefficient space, by performing all computations directly in the window–signal space. As one example application, we show how continuous wavelet multipliers (also called Calderón–Toeplitz operators), with polynomial symbols, can be implemented with linear complexity in the resolution of the 1D signal. As another example, we develop a framework for efficiently computing greedy sparse approximations to signals based on elements of continuous wavelet systems.

End extending models of set theory via power admissible covers

Motivated by problems involving end extensions of models of set theory, we develop the rudiments of the power admissible cover construction (over ill-founded models of set theory), an extension of the machinery of admissible covers invented by Barwise as a versatile tool for generalising model-theoretic results about countable well-founded models of set theory to countable ill-founded ones. Our development of the power admissible machinery allows us to obtain new results concerning powerset-preserving end extensions and rank extensions of countable models of subsystems of ZFC. The canonical extension KPP of Kripke-Platek set theory KP plays a key role in our work; one of our results refines a theorem of Rathjen by showing that Σ1P-Foundation is provable in KPP (without invoking the axiom of choice).