Given a conditional expectation P P from a C*-algebra B B onto a C*-subalgebra A A , we observe that induction of ideals via P P , together with a map which we call co-induction, forms a Galois connection between the lattices of ideals of A A and B B . Using properties of this Galois connection, we show that, given a discrete group G G and a stabilizer subgroup G x G_x for the action of G G on its Furstenberg boundary, induction gives a bijection between the set of maximal co-induced ideals of C ∗ ( G x ) C^*(G_x) and the set of maximal ideals of C r ∗ ( G ) C^*_r(G) . As an application, we prove that the reduced C*-algebra of Thompson’s group T T has a unique maximal ideal. Furthermore, we show that, if Thompson’s group F F is amenable, then C r ∗ ( T ) C^*_r(T) has infinitely many ideals.

ABSTRACT Modeling knowledge systems by determining relationships among key variables have been and currently is a fundamental and nontrivial challenge in real‐world scenarios. Many approaches have been developed to reach this goal, but many of them are heuristic and require of alternative procedures to provide robust and tractable rules. With this significant aim, attribute implications were introduced in the mathematical framework of Formal Concept Analysis. In this paper, we will introduce a novel procedure to obtain relationships among variables from any dataset in which a Galois connection has been defined. In particular, we will be focused on the general multiadjoint framework, which is one of the most general approached in which a Galois connection have been considered, although the obtained results and methodology can also be used in other well‐known approaches, such as in the residuated or heterogeneous frameworks.

According to a clever but rarely quoted or acknowledged work of E. Vessiot that won the prize of the Académie des Sciences in 1904, “Differential Galois Theory” (DGT) has mainly to do with the study of “Principal Homogeneous Spaces” (PHSs) for finite groups (classical Galois theory), algebraic groups (Picard–Vessiot theory) and algebraic pseudogroups (Drach–Vessiot theory). The corresponding automorphic differential extensions are such that dimK(L)=∣L/K∣<∞, the transcendence degree trd(L/K)<∞ and trd(L/K)=∞ with difftrd(L/K)<∞, respectively. The purpose of this paper is to mix differential algebra, differential geometry and algebraic geometry to revisit DGT, pointing out the deep confusion between prime differential ideals (defined by J.-F. Ritt in 1930) and maximal ideals that has been spoiling the works of Vessiot, Drach, Kolchin and all followers. In particular, we utilize Hopf algebras to investigate the structure of the algebraic Lie pseudogroups involved, specifically those defined by systems of algebraic OD or PD equations. Many explicit examples are presented for the first time to illustrate these results, particularly through the study of the Hamilton–Jacobi equation in analytical mechanics. This paper also pays tribute to Prof. A. Bialynicki-Birula (BB) on the occasion of his recent death in April 2021 at the age of 90 years old. His main idea has been to notice that an algebraic group G acting on itself is the simplest example of a PHS. If G is connected and defined over a field K, we may introduce the algebraic extension L=K(G); then, there is a Galois correspondence between the intermediate fields K⊂K′⊂L and the subgroups e⊂G′⊂G, provided that K′ is stable under a Lie algebra Δ of invariant derivations of L/K. Our purpose is to extend this result from algebraic groups to algebraic pseudogroups without using group parameters in any way. To the best of the author’s knowledge, algebraic Lie pseudogroups have never been introduced by people dealing with DGT in the spirit of Kolchin; that is, they have only been considered with systems of ordinary differential (OD) equations, but never with systems of partial differential (PD) equations.

Exploring the Closure Operator for Minimal a, b-Separators: Insights from Galois Connections

This paper extends Hopf–Galois theory to infinite field extensions and provides a natural definition of subextensions. For separable (possibly infinite) Hopf–Galois extensions, it provides a Galois correspondence. This correspondence also is a refinement of what was known in the case of finite separable Hopf–Galois extensions.

In this paper, the relationship between strong L-fuzzy convex structures and L-fuzzifying interval operators are investigated. It is proved that there is a Galois correspondence between the category of strong L-fuzzy convex spaces and that of L-fuzzifying interval spaces. Also, the concept of arity 2 strong L-fuzzy convex structures is presented, which can be reflectively embedded into the category of L-fuzzifying interval spaces. Finally, the ways of L-fuzzy preorders inducing strong L-fuzzy convex structures and strong L-fuzzy convex structures inducing L-fuzzy preorders are given. It is shown that a strong L-fuzzy convex structure generated by an L-fuzzy preorder is an arity 2 strong L-fuzzy convex structure.

This paper introduces and investigates the fundamental properties of L-primals, a generalization of the primal concept within the framework of L-fuzzy sets and complete lattices. Building upon the established theories of L-topological spaces and L-pre-proximity spaces, this research explores the interrelations among these three generalized topological structures. The study establishes novel categorical links, demonstrating the existence of concrete functors between categories of L-primal spaces and L-pre-proximity spaces, as well as between categories of L-pre-proximity spaces and stratified L-primal spaces. Furthermore, the paper clarifies the existence of a concrete functor between the category of stratified L-primal spaces and the category of L-topological spaces, and vice versa, thereby establishing Galois correspondences between these categories. Theoretical findings are supported by illustrative examples, including applications within the contexts of information systems and medicine, demonstrating the practical aspects of the developed theory.

Galois connections are pairs of functions, defined over ordered sets, that preserve some particular aspects.They are studied in the context of algebraic structures. As a logic topic, we take the four-valued logic PM4N that contemplates, at least, two modal operators for the notions of necessary and possible. In this paper, we develop a particular four-valued implication for PM4N, which constitutes a Galois pair with the two modal operators. Then, we show some properties of the logic just considering the correlates algebraic developments.

This paper introduces a meta-logical framework—based on the theory of institutions (a categorical version of abstract model theory)—to be used as a tool for the formalization of the two main views regarding the structure of scientific theories, namely the syntactic and the semantic views, as they have emerged from the relevant contemporary discussion. The formalization leads to a proof of the equivalence of the two views, which supports the claim that the two approaches are not really in tension. The proof is based on the Galois connection between classes of sentences and classes of models defined over some institution. First, the history of the syntactic–semantic debate is recalled and the theory of institutions formally introduced. Secondly, the notions of syntactic and semantic theories are formalized within the institution and their equivalence proved. Finally, the novelty of the proposed framework is highlighted with respect to existing formalizations.

The purpose of this short note is to fill a gap in the literature: Frobenius reciprocity in the theory of doctrines is closely related to modular connections in projective homological algebra and the notion of a principal element in abstract commutative ideal theory. These concepts are based on particular properties of Galois connections which play an important role also in the abstract study of group-like structures from the perspective of categorical/universal algebra; such role stems from a classical and basic result in group theory: the lattice isomorphism theorem.

Duality and Galois correspondence for vertex superalgebras

Relating macroscopic observables to microscopic interactions is a central challenge in the study of complex systems. While current approaches often focus on pairwise interactions, a complete understanding requires going beyond these to capture the full range of possible interactions. We present a unified mathematical formalism, based on the Möbius inversion theorem, that reveals how different decompositions of a system into parts lead to different, but equally valid, microscopic theories. By providing an exact bridge between microscopic and macroscopic descriptions, this framework demonstrates that many existing notions of interaction, from epistasis in genetics and many-body couplings in physics, to synergy in game theory and artificial intelligence, naturally and uniquely arise from particular choices of system decomposition, or . By revealing the common mathematical structure underlying seemingly disparate phenomena, our paper highlights how the choice of decomposition fundamentally determines the nature of the resulting interactions. We discuss how this unifying perspective can facilitate the transfer of insights across domains, guide the selection of appropriate system decompositions, and enable the search for new notions of interaction. To illustrate the latter in practice, we decompose the Kullback-Leibler divergence, and show that our method correctly identifies which variables are responsible for the divergence. In addition, we use Rota's Galois connection theorem to describe coarse grainings of mereologies, and efficiently derive the renormalized couplings of a 1D Ising model. Our results suggest that the Möbius inversion theorem provides a powerful and practical lens for understanding the emergence of complex behavior from the interplay of microscopic parts, with applications across a wide range of disciplines. Published by the American Physical Society 2025

We prove a Galois correspondence theorem for groupoids acting orthogonally and partially on commutative rings. We also consider partial actions that are not orthogonal, presenting two correspondences in this case: one for strongly Galois partial groupoid actions and one for global groupoid actions (without restriction). Some examples are presented.

Semiconic Idempotent Logic II: Beth Definability and Deductive Interpolation

Abstract Experience in teaching functional programming (FP) on a relational basis has led the author to focus on a graphical style of expression and reasoning in which a geometric construct shines: the (semi) commutative square. In the classroom this is termed the “magic square” (MS), since virtually everything that we do in logic, FP, database modeling, formal semantics and so on fits in some MS geometry. The sides of each magic square are binary relations and the square itself is a comparison of two paths, each involving two sides. MSs compose and have a number of useful properties. Among several examples given in the paper ranging over different application domains, free-theorem MSs are shown to be particularly elegant and productive. Helped by a little bit of Galois connections, a generic, induction-free theory for ${\mathsf{foldr}}$ and $\mathsf{foldl}$ is given, showing in particular that ${\mathsf{foldl} \, {{s}}{}\mathrel{=}\mathsf{foldr}{({flip} \unicode{x005F}{s})}{}}$ holds under conditions milder than usually advocated.

We introduce a continuous domain for function spaces over topological spaces which are not core-compact. Notable examples of such topological spaces include the real line with the upper limit topology, which is used in solution of initial value problems with temporal discretization, and various infinite dimensional Banach spaces which are ubiquitous in functional analysis and solution of partial differential equations. If a topological space $\mathbb{X}$ is not core-compact and $\mathbb{D}$ is a non-singleton bounded-complete domain, the function space $[\mathbb{X} \to \mathbb{D}]$ is not a continuous domain. To construct a continuous domain, we consider a spectral compactification $\mathbb{Y}$ of $\mathbb{X}$ and relate $[\mathbb{X} \to \mathbb{D}]$ with the continuous domain $[\mathbb{Y} \to \mathbb{D}]$ via a Galois connection. This allows us to perform computations in the native structure $[\mathbb{X} \to \mathbb{D}]$ while computable analysis is performed in the continuous domain $[\mathbb{Y} \to \mathbb{D}]$, with the left and right adjoints used for moving between the two function spaces.Comment: 19 pages, 1 figure, Mathematical Foundations of Programming Semantics (MFPS) 2024

Suppose 𝒦 is N-dimensional local field of characteristic p≠0, 𝒢 <p is the maximal quotient of period p and nilpotent class <p of 𝒢=Gal(𝒦 sep /𝒦), and 𝒦 <p ⊂𝒦 sep is such that Gal(𝒦 <p /𝒦)=𝒢 <p . We use nilpotent Artin–Schreier theory to identify 𝒢 <p with the group G(ℒ) obtained from a profinite Lie 𝔽 p -algebra ℒ via the Campbell–Hausdorff composition law. The canonical 𝒫-topology on 𝒦 is used to define a dense Lie subalgebra ℒ 𝒫 in ℒ. The algebra ℒ 𝒫 can be provided with a system of 𝒫-topological generators and we prove that all N-dimensional extensions of 𝒦 in 𝒦 <p are in the bijection with all 𝒫-open subalgebras of ℒ 𝒫 by the Galois correspondence. These results are applied to higher local fields K of characteristic 0 containing a nontrivial p-th root of unity. If Γ=Gal(K alg /K) we introduce similarly the quotient Γ <p and present it in the form G(L), where L is a suitable profinite Lie 𝔽 p -algebra. Then we introduce a dense 𝔽 p -Lie subalgebra L 𝒫 in L, and describe the structure of L 𝒫 in terms of generators and relations. The general result is illustrated by explicit presentation of Γ <p modulo subgroup of third commutators.

Classification of the types for which every Hopf–Galois correspondence is bijective

Inclusions of simple C⁎-algebras arising from compact group actions

The main goal of most static analyses is to prove the absence of bugs : if the analysis reports no alarms, then the program will not exhibit any unwanted behaviours. For this reason, they are designed to over-approximate program behaviours and, consequently, they can report some false alarms. O’Hearn’s recent work on incorrectness has renewed the interest in the use of under-approximations for bug finding , because they only report true alarms. In principle, Abstract Interpretation techniques can handle under-approximations as well as over-approximations, but, in practice, few attempts were developed for the former, notwithstanding the much wider literature on the latter. In this article, we investigate the possibility of exploiting under-approximation abstract domains for bug-finding analyses. First we restrict to consider concrete powerset domains and highlight some intuitive asymmetries between over- and under-approximations. Then, we prove that the effectiveness of abstract domains defined by under-approximation Galois connection is limited because the analysis is likely to return trivial results whenever common transfer functions are encoded in the program. To this aim, we introduce the original concepts of non-emptying functions and highly surjective function family , and we prove the nonexistence of abstract domains able to under-approximate such functions in a non-trivial way. We show many examples of finite and infinite numerical domains, as well as other generic domains. In all such cases, we prove the impossibility of performing non-trivial analyses via under-approximation Galois connections.