We propose a semantic foundation for logics for reasoning in settings that possess a distinction between equality of variables, a coarser equivalence of variables, and a notion of conditional independence between variables. We show that such relations can be modelled naturally in atomic sheaf toposes. Equivalence of variables is modelled by an intrinsic relation of atomic equivalence that is possessed by every atomic sheaf. We identify additional structure on the category generating the atomic topos (primarily, the existence of a system of independent pullbacks ) that allows the relation of conditional independence to be interpreted in the topos. We then study the logic of equivalence and conditional independence that is induced by the internal logic of the topos. This atomic sheaf logic is a classical logic that validates a number of fundamental reasoning principles relating equivalence and conditional independence. As a concrete example of this abstract framework, we use the atomic topos over the category of surjections between finite nonempty sets as our main running example. In this category, the interpretations of equivalence and conditional independence coincide with those given by the multiteam semantics of independence logic, in which the role of equivalence is taken by the relation of mutual inclusion. A major difference from independence logic is that, in atomic sheaf logic, the multiteam semantics of the equivalence and conditional independence relations is embedded within a classical surrounding logic. At the end of the paper, we briefly outline two other instances of our framework, to demonstrate its versatility. The first of these is a category of probability sheaves , in which atomic equivalence is equality-in-distribution, and the conditional independence relation is the usual probabilistic one. Our other example is the Schanuel topos (equivalent to nominal sets) where equivalence is orbit equality and conditional independence amounts to a relative form of separatedness.

Highlights A novel non-discrete topology is defined on the unit group using conjugate pairs as basic open sets, where every open set is also closed (clopen). Highlights A novel non-discrete topology is defined on the unit group using conjugate pairs as basic open sets, where every open set is also closed (clopen). The function mapping to is continuous, open, and closed, establishing a strong topological link between the unit group and quadratic residues. The quotient space , under the equivalence , is homeomorphic to the discrete space of quadratic residues . The topology is disconnected and fails the separation axiom, whereas the discrete topology on satisfies stronger separation properties such as . Key topological operators (interior, closure, boundary, limit points) are explicitly characterized for arbitrary subsets of , revealing how structure depends on conjugate-pair symmetryfunction mapping to is continuous, open, and closed, establishing a strong topological link between the unit group and quadratic residues. The quotient space , under the equivalence , is homeomorphic to the discrete space of quadratic residues . The topology is disconnected and fails the separation axiom, whereas the discrete topology on satisfies stronger separation properties such as . Key topological operators (interior, closure, boundary, limit points) are explicitly characterized for arbitrary subsets of , revealing how structure depends on conjugate-pair symmetry This paper introduces a finite topological space on the group of units modulo a prime , defined by its basis of conjugate residue pairs for all units . We investigate the fundamental topological concepts such as point-set topology, separation axioms, and characterise the structure and behaviour of this topology. Additionally, we examine a function from to the topology of quadratic residues , mapping each unit to its square modulo . We analyse the continuity, openness of , and explore its implications for separation properties. Furthermore, we define a quotient topology on based on the equivalence relation if and only if, showing that the resulting quotient space is homeomorphic to .

A graph-theoretical approach to the analysis of motion and rest in many-body systems is developed. Point bodies are represented as vertices of a complete bi-colored graph, termed the motion–rest graph (MRG). Two vertices are connected by a rust-colored edge when the corresponding bodies are at rest relative to each other; that is, when their mutual distance remains constant in time, bodies moving relative to each other are connected by a cyan edge. It is shown that the logical structure of the relation “to be at rest relative to each other” determines the combinatorial structure of the graph. For one-dimensional motion in classical mechanics and special relativity, this relation is reflexive, symmetric, and transitive, and therefore defines an equivalence relation. As a result, rust edges form disjoint complete cliques corresponding to rest-clusters, and the MRG becomes a semi-transitive complete bi-colored graph that is completely determined by the partition of the bodies into equivalence classes. It is proven that any such graph on five vertices necessarily contains a monochromatic triangle. For two- and three-dimensional motion, the transitivity of relative rest generally fails because constant mutual distance does not imply an equality of velocities in the presence of rotational degrees of freedom. In this case, the MRG is non-transitive, and the Ramsey threshold becomes the classical value R(3,3) = 6. The approach is extended to mixed sets containing moving bodies and reference points, including the center of mass of the system. Generalizations to general relativity and quantum mechanics are also discussed. In general relativity, transitivity of relative rest is generically lost because global rigid congruences do not generally exist. In quantum mechanics, exact transitivity survives only at the level of idealized delocalized eigenstates, whereas for physically realizable localized states, the notion of mutual rest becomes only approximate. The results demonstrate that the interplay between kinematics, logical properties of relational motion, and Ramsey-type combinatorial constraints gives rise to unavoidable ordered substructures in many-body systems.

Doubly periodic tangles, or DP tangles, are embeddings of curves in the thickened plane that are periodically repeated in two directions. They are defined as universal covers of their generating cells, the flat motifs, which represent knots and links in the thickened torus, and which can be chosen in infinitely many ways. DP tangles are used in modeling materials and physical systems of entangled filaments. In this paper, we establish the complete mathematical framework of the topological theory of DP tangles. We present an exhaustive analysis of DP tangle isotopies. These are distinguished in local isotopies and global isotopies. Our analysis yields the characterization of DP isotopy as an equivalence relation on the level of their (flat) motifs, called DP tangle equivalence. Along the way, we also discuss motif minimality. We further generalize our results to other diagrammatic categories, namely framed, virtual, welded, singular, pseudo, tied and bonded DP tangles, which could be used in novel applications.

Hoare-style inference rules for program constructs permit the copying of expressions and tests from program text into logical contexts. It is known that this requires care even for sequential programs but much more serious issues arise with concurrent programs because of potential interference to the values of variables. The “rely-guarantee” approach tackles the challenge of recording acceptable interference and offers a way to provide safe inference rules for concurrent constructs. This paper shows how the algebraic presentation of rely-guarantee ideas can clarify and formalise the conditions for safely re-using expressions and tests from program text in logical contexts for reasoning about concurrent programs; crucially this extends to handling expressions that reference more than one shared variable. A non-trivial example related to the Fischer-Galler forest representation of equivalence relations is treated.

Let M be a module over a domain R and M # = { 0 ≠ m ∈ M : Rm ≠ M } be the set of all nonzero nongenerators of M. Consider the following equivalence relation ∼ on M # given by m ∼ n if and only if Rm = Rn for every m , n ∈ M # . Let EC ( M # ) be the set of all equivalence classes of M # with respect to ∼ . In this paper, we construct a topology on EC ( M # ) which is called the divisor topology of M and is denoted by D ( M ) . Actually, D ( M ) is an extension of the divisor topology D ( R ) over domains to modules in the sense of Yiğit and Koç. We investigate separation axioms T i for every 0 ≤ i ≤ 5 , first and second countability, connectivity, compactness, nested property, and Noetherian property on D ( M ) . Also, we characterize some important classes of modules such as uniserial modules, simple modules, vector spaces, and finitely cogenerated modules in terms of D ( M ) . Furthermore, we prove that D ( M ) is a Baire space for factorial modules. Finally, we introduce and study pseudo simple modules which is a new generalization of simple modules, and use them to determine when D ( M ) is a discrete space.

The full-domain partition monoid [Formula: see text] has been discovered independently in two recent studies on connections between diagram monoids and category theory. It is a right restriction Ehresmann monoid, and contains both the full transformation monoid and the join semilattice of equivalence relations. In this paper we give presentations (by generators and relations) for [Formula: see text], its singular ideal, and its planar submonoid. The latter is not an Ehresmann submonoid, but it is a so-called grrac monoid in the terminology of Branco, Gomes and Gould. In particular, its structure is determined in part by a right regular band in one-one correspondence with planar equivalences.

Abstract We consider a large family of theories of equivalence relations, each with finitely many classes, and assuming the existence of an $$\omega $$ ω -Erdős cardinal, we determine which of these theories are Borel complete. We develop machinery, including forbidding nested sequences which implies a tight upper bound on Borel complexity, and admitting cross-cutting absolutely indiscernible sets which in our context implies Borel completeness.

In this article, we study the interactions between so-called fractional relaxations of the integer programs (IPs) which encode homomorphism and isomorphism of relational structures. We give a combinatorial characterization of a certain natural linear programming (LP) relaxation of homomorphism in terms of fractional isomorphism. As a result, we show that the families of constraint satisfaction problems (CSPs) that are solvable by such linear program are precisely those that are closed under an equivalence relation which we call Weisfeiler-Leman invariance . We also generalize this result to the much broader framework of Promise Valued Constraint Satisfaction Problems, which brings together two well-studied extensions of the CSP framework. Finally, we consider the hierarchies of increasingly tighter relaxations of the homomorphism and isomorphism IPs obtained by applying the Sherali-Adams and Weisfeiler-Leman methods, respectively. We extend our combinatorial characterization of the basic LP to higher levels of the Sherali-Adams hierarchy, and we generalize a well-known logical characterization of the Weisfeiler-Leman test from graphs to relational structures.

Thanks to the cryptosystem, confidential information can be transmitted securely. When we perform this encryption using matrices obtained from near soft sets, we increase our security even more. Accordingly, the equivalence relation used in this set will change according to the characteristics of the set that are restricted and the resulting encrypted word will change accordingly.

Abstract We present a novel equivalence between scale-dependent (SD) gravity and scalar-tensor theories (STTs) that have only a single scalar field with a canonical kinetic term in the Einstein frame and a conformal coupling to the metric tensor. In particular, we show that the set of well-behaved SD gravity theories can be fully embedded into STTs in a unique way. Conversely, there are multiple ways to write a STT as a SD theory. This equivalence is established both on the level of the actions and on the level of field equations. We find that, in the context of this equivalence, the scale-setting relation k ( x ) is naturally promoted to a dynamical field, which is made manifest by including a corresponding kinetic term in the SD action. In addition, we demonstrate that the new equivalence fits well into the framework of existing equivalences involving the aforementioned theories and f ( R )-gravity. Finally, we apply the equivalence relations to explicit examples from both SD gravity and STTs.

Hausdorff Morita equivalence is an equivalence relation on singular foliations, which induces a bijection between their leaves. Our main statement is that linearizability along a leaf is invariant under Hausdorff Morita equivalence. The proof relies on a characterization of tubular neighborhood embeddings using Euler-like vector fields.

Fuzzy cellular automata are dynamical systems that are continuous counterparts of the usual cellular automata (CA). Compared with the binary case, defining a fuzzy CA with three or more states is challenging because defining mixed states is difficult. Recently, this difficulty was resolved by representing multiple states as independent vectors in higher dimensions, and the concept of vector-valued fuzzy CA (VFCA) was introduced. In this study, we theoretically analyze and discuss the asymptotic behavior of three-neighbor VFCA. First, we define the weighted-averaging rules of VFCA and show how many rules exist up to the equivalence relations. According to these rules, each state vector in the next step is determined by the weighted average of the vectors in its neighboring cells. Next, we prove that three-state VFCA with weighted-averaging rules converge to a periodic configuration characterized by the symmetric group of order 3. In particular, the non-commutativity of the group action provides an interesting behavior that is not observed in fuzzy CA arising from binary states. Finally, we extend the results to VFCA with more than three states.

While the Yangtze River Economic Belt (YREB) serves as a strategic frontier for ecological civilization, existing scholarship lacks a systemic analysis of the co-evolutionary logic and governance configurations governing the synergistic green development (SGD). Utilizing panel data from 2011 to 2024, this study employs coupling coordination modeling, Social Network Analysis (SNA), and Qualitative Comparative Analysis (QCA) to evaluate its synergistic developmental quality. The findings indicate that SGD levels exhibit a fluctuating upward trajectory, facilitating a paradigmatic shift from "downstream-led breakthroughs" toward "basin-wide inclusive upgrades". However, this progression has simultaneously induced intensified internal "center-periphery" polarization. Such spatial evolution has catalyzed a structural transformation of the association network, migrating from a polarized "dense-East, sparse-West" pattern toward a polycentric, grid-based collaborative ecosystem, as evidenced by the rise of inland hubs like Chongqing and Wuhan. Causally, high-efficiency synergy originates from structural spillovers within the Government-Market-Society (GMS) framework. Through the equivalence relation between capital investment and human capital, modernized social governance effectively compensates for the rigidities of administrative mandates. Consequently, the basin's driving mechanism demonstrates a spatial transition from "government-led compensatory intervention" in the upper reaches to "endogenous synergetic drive" in the lower reaches. This study elucidates a policy pathway for achieving sustainable regional growth through functional compensation and pathway substitution.

Abstract Cryptographic group actions have gained significant attention in recent years for their application on post-quantum Sigma protocols and digital signatures. In NIST’s recent additional call for post-quantum signatures, three relevant proposals are based on group actions: LESS, MEDS, and ALTEQ. This work explores signature optimisations leveraging a group’s factorisation. We show that if the group admits a factorisation as a semidirect product of subgroups, the group action can be restricted on a quotient space under the equivalence relation induced by the factorisation. If the relation is efficiently decidable, we show that it is possible to construct an equivalent Sigma protocol for a relationship that depends only on one of the subgroups. Moreover, if a special class of representative of the quotient space is efficiently computable via a canonical form, the restricted action is effective and does not incur in security loss. Finally, we apply these techniques to the group actions underlying LESS and MEDS, showing how they will affect the length of signatures and public keys.

The relation between Taylor-Dean flow and rotating curved channel flow

We introduce Recognition Geometry (RG), an axiomatic framework in which geometric structure is not assumed a priori but derived. The starting point of the theory is a configuration space together with recognizers that map configurations to observable events. Observational indistinguishability induces an equivalence relation, and the observable space is obtained as a recognition quotient. Locality is introduced through a neighborhood system, without assuming any metric or topological structure. A finite local resolution axiom formalizes the fact that any observer can distinguish only finitely many outcomes within a local region. We prove that the induced observable map R¯:CR→E is injective, establishing that observable states are uniquely determined by measurement outcomes with no hidden structure. The framework connects deeply with existing approaches: C*-algebraic quantum theory, information geometry, categorical physics, causal set theory, noncommutative geometry, and topos-theoretic foundations all share the measurement-first philosophy, yet RG provides a unified axiomatic foundation synthesizing these perspectives. Comparative recognizers allow us to define order-type relations based on operational comparison. Under additional assumptions, quantitative notions of distinguishability can be introduced in the form of recognition distances, defined as pseudometrics. Several examples are provided, including threshold recognizers on Rn, discrete lattice models, quantum spin measurements, and an example motivated by Recognition Science. In the last part, we develop the composition of recognizers, proving that composite recognizers refine quotient structures and increase distinguishing power. We introduce symmetries and gauge equivalence, showing that gauge-equivalent configurations are necessarily observationally indistinguishable, though the converse does not hold in general. A significant part of the axiomatic framework and the main constructions are formalized in the Lean 4 proof assistant, providing an independent verification of logical consistency.

Abstract We introduce a simple equivalence relation on strongly minimal sets in a structure of finite Morley rank, which corresponds, in stability theory, to the non-orthogonality of the associated types. We use it in a group 𝐺 of finite Morley rank to define, for each strongly minimal set 𝑋, two connected normal subgroups M G ⁢ ( X ) M_{G}(X) and W G ⁢ ( X ) W_{G}(X) . When 𝐺 is connected, these subgroups provide a central decomposition of 𝐺 that yields a direct product decomposition of G / Z ⁢ ( G ) G/Z(G) into unidimensional factors, as well as a central decomposition of its derived subgroup into unidimensional subgroups.

Previous papers give accounts of quests for satisfactory formalizations of the classical informal notion of an algorithm and the contemporary informal notion of an interactive algoritm. In this paper, an attempt is made to generalize the results of the former quest to the contemporary informal notion of a concurrent algorithm. The notion of a concurrent proto-algorithm is introduced. The thought is that concurrent algorithms are equivalence classes of concurrent proto-algorithms under an appropriate equivalence relation. As in the previous papers referred to above, three equivalence relations are defined. Two of them are deemed to be bounds for an appropriate equivalence relation and the third is likely an appropriate one. The connection between concurrency and non-determinism in the presented setting is also addressed.

Abstract We introduce the notions of u-amenability and hyper-u-amenability for countable Borel equivalence relations and we show that treeable, hyper-u-amenable countable Borel equivalence relations are hyperfinite. As corollaries of this result, we obtain that if a countable Borel equivalence relation is either: 1. measure-hyperfinite and equal to the orbit equivalence relation of a free continuous action of a virtually free group on a $\sigma $ -compact Polish space, 2. treeable and equal to the orbit equivalence relation of a Borel action of an amenable group on a standard Borel space, 3. treeable, amenable and Borel bounded, then it is hyperfinite.