Kleene Star Research Articles

On a particular subspace of homogeneous preserving star operations

On a particular subspace of homogeneous preserving star operations

On star Rothberger spaces modulo an ideal

In this article, we introduce the ideal star-Rothberger property by coupling the notion of a star operator to that of an ideal Rothberger space, after which some of its topological characteristics are analysed. By creating relationships between a numbers of topological features with structures similar to the ideal star-Rothberger space, we reinforce the concept. In order to illustrate the differences between a number of related topological properties, we also provide several counter examples. Certain preservation-related properties under subspaces and functions are investigated. Lastly we find a way to express ideal star-Rothberger space by means of families of closed sets by bringing some modifications to the SSI^I property.

Star γ Covers and Their Applications to Selection Principles

Abstract By imposing the star operator on γ covers, we establish the notion of st-γ covers. The interrelationships among many recent variants of γ covers have been identified, and various counterexamples have been shown to distinguish them. An investigation has been conducted on the basic topological attributes of these covers. We have also examined the implications between various selection principles and topological games in the context of the class of st-γ coverings.

On generalized Star-GCD extensions

– Let ⋆ be a star operation on a ring extension R ⊆ S . The ring extension is said to be a generalized ⋆ -greatest common divisor (G- ⋆ GCD) extension if the intersection of two ⋆ -invertible ideals of R is ⋆ -invertible. We investigate properties of G- ⋆ GCD extensions. We give special attention to polynomial rings and Nagata rings.

On quasinormal Fitting classes

Let Zp be a group of order p and G ≀ Z p the regular wreath product of the group G with Zp . A Fitting class F is said to be X -quasinormal if there exists a natural number m such that G m ≀ Z p ∈ F whenever F ⊆ X , p is a prime, G ∈ F and G ≀ Z p ∈ X . In this paper, we generalize the well known theorem of Blessenohl and Gaschütz and prove that the intersection of any set of nonidentity X -quasinormal Fitting classes is a nonidentity X -quasinormal Fitting class. For a Fitting class F , we also discuss the relation between its X -normality and its X -quasinormality, and give the condition of X * = X ⊛ , where “ * ” is Lockett star operation and X ⊛ is the smallest nonidentity X -quasinormal Fitting class.

Hyperarithmetical complexity of infinitary action logic with multiplexing

Abstract In 2023, Kuznetsov and Speranski introduced infinitary action logic with multiplexing $!^{m}\nabla \textrm{ACT}_{\omega }$ and proved that the derivability problem for it lies between the $\omega $ level and the $\omega ^{\omega }$ level of the hyperarithmetical hierarchy. We prove that this problem is $\varDelta ^{0}_{\omega ^{\omega }}$-complete under Turing reductions. Namely, we show that it is recursively isomorphic to the satisfaction predicate for computable infinitary formulas of rank less than $\omega ^{\omega }$ in the language of arithmetic. As a consequence we prove that the closure ordinal for $!^{m}\nabla \textrm{ACT}_{\omega }$ equals $\omega ^{\omega }$. We also prove that the fragment of $!^{m}\nabla \textrm{ACT}_{\omega }$ where Kleene star is not allowed to be in the scope of the subexponential is $\varDelta ^{0}_{\omega ^{\omega }}$-complete. Finally, we present a family of logics, which are fragments of $!^{m}\nabla \textrm{ACT}_{\omega }$, such that the complexity of the $k$-th logic lies between $\varDelta ^{0}_{\omega ^{k}}$ and $\varDelta ^{0}_{\omega ^{k+1}}$.

A NEW SOFT SET OPERATION: COMPLEMENTARY EXTENDED STAR OPERATION

Soft set theory has established itself as a valuable mathematical framework for tackling issues marked by uncertainty, demonstrating its applicability across a range of theoretical and practical fields since its inception. Central of this theory is the operations of soft sets. To enhance the theory and to make a theoretical contribution to the theory, a new type of soft set operation, called “complementary extended star operation” for soft set, is proposed. An exhaustive examination of the properties of this operation has been undertaken, including its distributions over other soft set operations, with the goal of clarifying the relationship between the complementary extended star operation and other soft set operations. This paper also attempts to make a contribution to the literature of soft sets in the sense that studying the algebraic structure of soft sets from the standpoint of soft set operations offers a comprehensive understanding of their application as well as an appreciation of how soft set can be applied to classical and nonclassical logic.

The Skolem property in rings of integer-valued rational functions

Let D be a domain and let Int(D) and IntR(D) be the ring of integer-valued polynomials and the ring of integer-valued rational functions, respectively. Skolem proved that if I is a finitely-generated ideal of Int(Z) with all the value ideals of I not being proper, then I=Int(Z). This is known as the Skolem property, which does not hold in Z[x]. One obstruction to Int(D) having the Skolem property is the existence of unit-valued polynomials. This is no longer an obstruction when we consider the Skolem property on IntR(D). We determine that the Skolem property on IntR(D) is equivalent to the maximal spectrum being contained in the ultrafilter closure of the set of maximal pointed ideals. We generalize the Skolem property using star operations and determine an analogous equivalence under this generalized notion.

The intermediate neutron capture process

Context. The intermediate neutron capture process (i-process) can develop during proton ingestion events (PIE), potentially during the early stages of low-mass low-metallicity asymptotic giant branch (AGB) stars. Aims. We examine the impact of overshoot mixing on the triggering and development of i-process nucleosynthesis in AGB stars of various initial masses and metallicities. Methods. We computed AGB stellar models, with initial masses of 1, 2, 3, and 4 M⊙ and metallicities in the −2.5 ≤ [Fe/H] ≤ 0 range, using the stellar evolution code STAREVOL with a network of 1160 nuclei coupled to the transport equations. We considered different overshooting profiles below and above the thermal pulses, and below the convective envelope. Results. The occurrence of PIEs is found to be primarily governed by the amount of overshooting at the top of pulse (ftop) and to increase with rising ftop. For ftop = 0, 0.02, 0.04, and 0.1, we find that 0%, 6%, 24%, and 86% of our 21 AGB models with −2 < [Fe/H] < 0 experience a PIE, respectively. Variations of the overshooting parameters during a PIE leads to a scatter on abundances of 0.5 − 1 dex on elements, with 36 < Z < 56; however, this barely impacts the production of elements with 56 < Z < 80, which therefore appear to be a reliable prediction of our models. Actinides are only produced if the overshooting at the top of pulse is small enough. We also find that PIEs leave a 13C-pocket at the bottom of the pulse that can give rise to an additional radiative s-process nucleosynthesis. In the case of the 2 M⊙ models with [Fe/H] = −1 and −0.5, it produces a noticeable mixed i + s chemical signature at the surface. Finally, the chemical abundance patterns of 22 observed r/s-stars candidates (18 dwarfs or giants and 4 post-AGB) with −2 < [Fe/H] < −1 are found to be in reasonable agreement with our AGB model predictions. The binary status of the dwarfs/giants being unclear, we suggest that these stars have acquired their chemical pattern either from the mass transfer of a now-extinct AGB companion or from an early generation AGB star that polluted the natal cloud. Conclusions. The occurrence of PIEs and the development of i-process nucleosynthesis in AGB stars remains sensitive to the overshooting parametrization. A high (yet realistic) ftop value triggers PIEs at (almost) all metallicities. The existence of r/s-stars at [Fe/H] ≃ −1 is in favour of an i-process operating in AGB stars up to this metallicity. Stricter constraints from multi-dimensional hydrodynamical models on overshoot coefficients could deliver new insights into the contribution of AGB stars to heavy elements in the Universe.

Star operations related to polynomial closure

We study polynomial closure of ideals from the point of view of star operations. We show that, if D is an integrally closed domain or a domain of residue characteristic 0, the polynomial closure of an ideal coincides with the divisorial closure. We also analyze what happens in characteristic p.

Relational Models for the Lambek Calculus with Intersection and Constants

We consider relational semantics (R-models) for the Lambek calculus extended with intersection and explicit constants for zero and unit. For its variant without constants and a restriction which disallows empty antecedents, Andreka and Mikulas (1994) prove strong completeness. We show that it fails without this restriction, but, on the other hand, prove weak completeness for non-standard interpretation of constants. For the standard interpretation, even weak completeness fails. The weak completeness result extends to an infinitary setting, for so-called iterative divisions (Kleene star under division). We also prove strong completeness results for product-free fragments.

A New Soft Set Operation: Complementary Soft Binary Piecewise Star (*) Operation

Soft set theory, introduced by Molodtsov, is as an important mathematical tool to deal with uncertainty and it has been conveyed to many fields both as theoretical and application aspect. Since its inception, different kinds of soft set operations are defined and used in various types. In this paper, we define a new kind of soft set operation called, complementary soft binary piecewise star operation and we investigate its basic algebraic properties. Moreover, it is aimed to contribute to the soft set literature by obtaining the relationships between this new soft set operation and some other types of soft set operations by examing the distribution of complementary soft binary piecewise star operation over extended soft set operations, complementary extended soft set operations, soft binary piecewise operations, complementary soft binary piecewise operations and restricted soft set operations

Equivalence between an [formula omitted]-valued GOA with ε-moves and its ε-reduction

The present study aims to develop the notion of L-valued general orthomodular automata (L-valued GOA) to a new one which is known as “L-valued GOA with ε-moves”. Subsequently, a clear relationship is attained between the language accepted by an L-valued GOA with ε-moves and the language accepted by ε-reduction. Moreover, it is demonstrated that certain commutativity which exists between basic actions of the automaton implies the equivalence between an automaton with ε-moves and its ε-reduction. Also, it is shown that the predicate commutative regularity is preserved by the Kleene closure and that consequently both the predicate regularity and commutative regularity are preserved by the pre-image of a homomorphism between the two languages. Finally, the quantum subset construction of L-valued GOA is presented as L-valued deterministic GOA (L-valued DGOA) and then the equivalence between L-valued GOA and L-valued DGOA is demonstrated. We further establish pumping lemma in the frame of quantum logic. To clarify the notions and the results obtained in this study, some examples are also submitted.

A variation of the class of statisticalγcovers

AbstractIn this article, we introduce s-s-γ\gammacover using the notion of star operator, which is an extension of the previous results on s-γ\gammacovers. We also define s-s-dense set to address the issue that a statistical dense subset of an s-s-γ\gammacover is not an s-s-γ\gammacover. Furthermore, we study some results on s-γ\gammacovers in subspace topology.

High order geometric methods with splines: Fast solution with explicit time-stepping for Maxwell equations

We introduce a high-order spline geometric approach for the initial boundary value problem for Maxwell's equations. The method is geometric in the sense that it discretizes in structure preserving fashion the two de Rham sequences of differential forms involved in the formulation of the continuous system. Both the Ampère–Maxwell and the Faraday equations are required to hold strongly, while to make the system solvable two discrete Hodge star operators are used. By exploiting the properties of the chosen spline spaces and concepts from exterior calculus, a non-standard explicit in time formulation is introduced, based on the solution of linear systems with matrices presenting Kronecker product structure, rather than mass matrices as in the standard literature. These matrices arise from the application of the exterior (wedge) product in the discrete setting, and they present Kronecker product structure independently of the geometry of the domain or the material parameters. The resulting scheme preserves the desirable energy conservation properties of the known approaches. The computational advantages of the newly proposed scheme are studied both through a complexity analysis and through numerical tests in three dimensions.

Novel Free Differential Algebras for Supergravity

We develop the theory of Free Integro-Differential Algebras (FIDA) extending the powerful technique of Free Differential Algebras constructed by D. Sullivan. We extend the analysis beyond the superforms to integral- and pseudo-forms used in supergeometry. It is shown that there are novel structures that might open the road to a deeper understanding of the geometry of supergravity. We apply the technique to some models as an illustration and we provide a complete analysis for D = 11 supergravity. There, it is shown how the Hodge star operator for supermanifolds can be used to analyze the set of cocycles and to build the corresponding FIDA. A new integral form emerges which plays the role of the truly dual to 4-form F(4) and we propose a new variational principle on supermanifolds.

Nondeterministic operational complexity in subregular languages

We study the nondeterministic state complexity of basic regular operations on subregular language families. In particular, we focus on the classes of combinational, singleton, finite, finitely generated left ideal, symmetric definite, group, star, comet, two-sided comet, ordered, and power-separating languages, and consider the operations of intersection, union, concatenation, power, Kleene star, reversal, and complementation. We get the exact complexity in all cases, except for complementation of singleton languages where we only have a lower bound n and an upper bound n. The complexity of all operations on combinational languages is given by a constant function, except for the k-th power where it is k+1. For all considered operations, the known upper bounds for left ideals are met by finitely generated left ideal languages, and the known lower bound 2n−1 for complementation on left ideals is tight also for symmetric definite languages. The complexity of intersection on singleton languages is min⁡{m,n}, and for all other operations, except for complementation, the known upper bounds for finite languages are met by singleton languages. The nondeterministic state complexity of the k-th power, star, and reversal on star languages is n, and the nondeterministic state complexity of complementation on group languages is max⁡{n,(n−1⌊n/2⌋)}. In all the remaining cases, the nondeterministic state complexity of all considered operations is the same as in the regular case, although sometimes we need to use a larger alphabet to describe the corresponding witnesses.

Kazhdan–Lusztig cells of a-value 2 in a(2)-finite Coxeter systems

A Coxeter group is said to be a(2)-finite if it has finitely many elements of a-value (in the sense of Lusztig) equal to 2. In this paper, we give explicit combinatorial descriptions of the left, right, and two-sided Kazhdan–Lusztig cells of a-value 2 in an irreducible a(2)-finite Coxeter group. In particular, we introduce elements we call stubs to parameterize the one-sided cells and we characterize the one-sided cells via both star operations and weak Bruhat orders. We also compute the cardinalities of all the one-sided and two-sided cells of a-value 2 in irreducible a(2)-finite Coxeter groups.

INEv: In-Network Evaluation for Event Stream Processing

Complex event processing (CEP) detects situations of interest by evaluating queries over event streams. Once CEP is used in networked applications, the distribution of query evaluation among the event sources enables performance optimization. Instead of collecting all events at one location for query evaluation, sub-queries are placed at network nodes to reduce the data transmission overhead. Yet, existing techniques either place such sub-queries at exactly one node in the network, which neglects the benefits of truly distributed evaluation, or are agnostic to the network structure, which ignores transmission costs due to the absence of direct network links. To overcome the above limitations, we propose INEV graphs for in-network evaluation of CEP queries with rich semantics, including Kleene closure and negation. Our idea is to introduce fine-granular routing of partial results of sub-queries as an additional degree of freedom in query evaluation: We exploit events already disseminated in the network as part of one sub-query, when evaluating another one. We show how to instantiate INEv graphs by splitting a query workload into sub-queries, placing them at network nodes, and forwarding of their results to other nodes. Also, we characterize INEv graphs that guarantee correct and complete query evaluation, and discuss their construction based on a cost model that unifies transmission and processing latency. Our experimental results indicate that INEv graphs can reduce transmission costs for distributed CEP by up to eight orders of magnitude compared to baseline strategies.

A Unified Interpretation of the Semantics of Relevance Logic

Abstract I introduce a novel and quite intuitive interpretation of the ternary relation that figures in the relational semantics of many relevance logics. Conceptually, my interpretation makes use only of incompatibility and parthood relations, defined over a set of states. In this way, the proposed interpretation—of the ternary relation and the conditional—extends Dunn’s and Restall’s works on negation and the Routley star operator. Therefore, the interpretation is unified, and hence not only intuitive but also parsimonious. Additionally, the interpretation provides us with a cogent argument to the effect that a ternary relation is indispensable, and we cannot merely do with a binary relation.