Finite Nonempty Set Research Articles

Herbrandized modified realizability

AbstractRealizability notions in mathematical logic have a long history, which can be traced back to the work of Stephen Kleene in the 1940s, aimed at exploring the foundations of intuitionistic logic. Kleene’s initial realizability laid the ground for more sophisticated notions such as Kreisel’s modified realizability and various modern approaches. In this context, our work aligns with the lineage of realizability strategies that emphasize the accumulation, rather than the propagation of precise witnesses. In this paper, we introduce a new notion of realizability, namely herbrandized modified realizability. This novel form of (cumulative) realizability, presented within the framework of semi-intuitionistic logic is based on a recently developed star combinatory calculus, which enables the gathering of witnesses into nonempty finite sets. We also show that the previous analysis can be extended from logic to (Heyting) arithmetic.

Variants of distributed reaction systems

AbstractA distributed reaction system consists of a finite set of reaction systems that either interact with a common environment or interact with each other by communicating products or reactions. A reaction system is a well-known qualitative formal model of interactions between biochemical reactions. A reaction is a triplet of nonempty sets representing chemicals, called the set of reactants, the set of inhibitors, and the set of products. A reaction corresponds to a chemical reaction performed on a set of chemicals, and a reaction system is a finite nonempty set of reactions. In this paper, we examine two variants of distributed reaction systems. We introduce the notion of a distributed reaction system with communication by request (a qDRS for short), where sets of products are communicated between the component reaction systems by queries. First, we show that every qDRS can be represented by a reaction system. After that we compare distributed reaction systems with communication by request to extended distributed reaction systems (EDRSs), models that were introduced in a previous paper. We prove that extended distributed reaction systems, where a context automaton provides input for the component reaction systems, simulate distributed reaction systems with communication by request and distributed reaction systems with communication by request simulate special variants of extended distributed reaction systems. Furthermore, we assign languages to these two variants of distributed reaction systems. We prove that the class of agreement languages of extended distributed reaction systems is equal to the class of languages of nondeterministic multihead finite automata and the agreement language of every distributed reaction system with communication by request is an element of a certain subregular language class.

Extended inverse theorems for $h$-fold sumsets in integers

Let $h \geq 2$, $k \geq 5$ be integers and $A$ be a nonempty finite set of $k$ integers. Very recently, Tang and Xing studied extended inverse theorems for $hk-h+1 < \left|hA\right| \leq hk+2h-3$. In this paper, we extend the work of Tang and Xing and study all possible inverse theorems for $hk-h+1<\left|hA\right| \leq hk+2h +1$. Furthermore, we give a range of $|hA|$ for which inverse problems are not possible.

Matroids related to groups and semigroups

Matroid is defined as a pair $(X,\mathcal{I})$, where $X$ is a nonempty finite set, and $\mathcal{I}$ is a nonempty set of subsets of  $X$ that satisfies the Hereditary Axiom and the Augmentation Axiom. The paper investigates for which semigroups (primarily finite) $S$, the pair $(\widehat{S}, \mathcal{I})$ will be a matroid.

The full automorphism groups of general position graphs

Let S be a non-empty finite set. A flag of S is a set f of non-empty proper subsets of S such that X⊆Y or Y⊆X for all X,Y∈f. The set {|X|:X∈f} is called the type of f. Two flags f and f′ are in general position with respect to S if X∩Y=∅ or X∪Y=S for all X∈f and Y∈f′. For a fixed type T, Klaus Metsch defined the general position graph Γ(S,T) whose vertices are the flags of S of type T with two vertices being adjacent when the corresponding flags are in general position. In this paper, we characterize the full automorphism groups of Γ(S,T) in the case that |T|=2. In particular, we solve an open problem proposed by Klaus Metsch.

Hyperplanes in matroids and the axiom of choice

We show that in set theory without the axiom of choice ZF, the statement sH: ``Every proper closed subset of a finitary matroid is the intersection of hyperplanes including it'' implies AC$^{\rm fin}$, the axiom of choice for (nonempty) finite sets. We also provide an equivalent of the statement AC$^{\rm fin}$ in terms of ``graphic'' matroids. Several open questions stay open in ZF, for example: does sH imply the axiom of choice?

On stably finiteness for $$C^*$$-algebras of exponential solvable Lie groups

We study the link between stably finiteness and stably projectionless-ness for $$C^*$$ -algebras of solvable Lie groups. We show that these two properties are equivalent if the dimension of the group is not divisible by 4; otherwise, they are not necessarily equivalent. To provide examples proving the last assertion, we study exponential solvable Lie groups that have nonempty finite open sets in their unitary dual.

On Some Theoretic Aspects of Fuzzy Subsets

Abstract: This article proposes some important theoretic aspects of L -fuzzy power set of a finite set. Here, we take a non-empty finite set E and an ordered subset M of the closed interval 0,1, then the set of mappings from E to M denoted by F is defined as L -fuzzy power set. Considering the disjunctive union ‘ ’ operations between fuzzy subsets of F , the structure F,forms a groupoid, which is defined as special fuzzy groupoid. Later, we try to introduce the product of special fuzzy groupoids and their properties

Rational numbers in ×𝑏-invariant sets

Let b ≥ 2 b \geq 2 be an integer and S S be a finite non-empty set of primes not containing divisors of b b . For any non-dense set A ⊆ [ 0 , 1 ) A \subseteq [0,1) which is invariant under × b \times b operation, we prove the finiteness of rational numbers in A A whose denominators can only be divided by primes in S S . A quantitative result on the largest prime divisors of the denominators of rational numbers in A A is also obtained.

Cocycle superrigidity for profinite actions of irreducible lattices

Let $\Gamma$ be an irreducible lattice in a product of two locally compact groups and assume that $\Gamma$ is densely embedded in a profinite group $K$. We give necessary conditions which imply that the left translation action $\Gamma \curvearrowright K$ is “virtually” cocycle superrigid: any cocycle ${w\colon \Gamma\times K\rightarrow\Delta}$ with values in a countable group $\Delta$ is cohomologous to a cocycle which factors through the map $\Gamma\times K\rightarrow\Gamma\times K\_0$ for some finite quotient group $K\_0$ of $K$. As a corollary, we deduce that any ergodic profinite action of $\Gamma=\mathrm{SL}\_2(\mathbb Z\[S^{-1}])$ is virtually cocycle superrigid and virtually W$^\*$-superrigid for any finite nonempty set of primes $S$.

Families of finite sets in which no set is covered by the union of the others

Let ℱ be a finite nonempty family of finite nonempty sets. We prove the following: (1) ℱ satisfies the condition of the title if and only if for every pair of distinct subfamilies {A1,…,Ar}, {B1,…,Bs} of ℱ, ⋃i=1rAi≠⋃i=1sBi. (2) If ℱ satisfies the condition of the title, then the number of subsets of ⋃A∈ℱA containing at least one set of ℱ is odd. We give two applications of these results, one to number theory and one to commutative algebra.

Dynamic programming and questions of solvability of route bottleneck problem with resource constraints

The article deals with the problem of admissible routing for a system of cycles each of which contains exterior permutation and works connected with megalopolises (non-empty finite sets) visiting. In the initial setting, a resource constraint is given; this constraint should be fulfilled for every cycle under permutation. The solvability conditions in this problem are connected with the extremum of the auxiliary bottleneck routing problem without above-mentioned constraint, in which the apparatus of widely understood dynamic programming (DP) is used. A particular case of the setting is the known bottleneck courier problem which can be used (in particular) for routing a vehicle (airplane or helicopter) aiming to realize the given shipping system with a limited fuel reserve on each flight. An algorithm implemented on a personal computer is constructed.

On Schreier graphs of gyrogroup actions

The notion of a group action can be extended to the case of gyrogroups . In this article, we examine a digraph and graph associated with a gyrogroup action on a finite nonempty set , called a Schreier digraph and graph. We show that algebraic properties of gyrogroups and gyrogroup actions such as being gyrocommutative, being transitive, and being fixed-point-free are reflected in their Schreier digraphs and graphs. We also prove graph-theoretic versions of the three fundamental theorems involving actions: the Cauchy–Frobenius lemma (also known as the Burnside lemma), the orbit-stabilizer theorem, and the orbit decomposition theorem . Finally, we make a connection between gyrogroup actions and actions of symmetric groups by evaluation via Schreier digraphs and graphs.

IWASAWA THEORY FORp-TORSION CLASS GROUP SCHEMES IN CHARACTERISTICp

AbstractWe investigate a novel geometric Iwasawa theory for${\mathbf Z}_p$-extensions of function fields over a perfect fieldkof characteristic$p>0$by replacing the usual study ofp-torsion in class groups with the study ofp-torsion class groupschemes. That is, if$\cdots \to X_2 \to X_1 \to X_0$is the tower of curves overkassociated with a${\mathbf Z}_p$-extension of function fields totally ramified over a finite nonempty set of places, we investigate the growth of thep-torsion group scheme in the Jacobian of$X_n$as$n\rightarrow \infty $. By Dieudonné theory, this amounts to studying the first de Rham cohomology groups of$X_n$equipped with natural actions of Frobenius and of the Cartier operatorV. We formulate and test a number of conjectures which predict striking regularity in the$k[V]$-module structure of the space$M_n:=H^0(X_n, \Omega ^1_{X_n/k})$of global regular differential forms as$n\rightarrow \infty .$For example, for each tower in a basic class of${\mathbf Z}_p$-towers, we conjecture that the dimension of the kernel of$V^r$on$M_n$is given by$a_r p^{2n} + \lambda _r n + c_r(n)$for allnsufficiently large, where$a_r, \lambda _r$are rational constants and$c_r : {\mathbf Z}/m_r {\mathbf Z} \to {\mathbf Q}$is a periodic function, depending onrand the tower. To provide evidence for these conjectures, we collect extensive experimental data based on new and more efficient algorithms for working with differentials on${\mathbf Z}_p$-towers of curves, and we prove our conjectures in the case$p=2$and$r=1$.

A pumping lemma for regular closure of prefix-free languages

Let Σ be an infinite set of distinct symbols. Let PFL⊆Power(Σ⁎) be the set of all prefix-free languages. For N⊆Power(Σ⁎), let ΓREG(N) be the regular closure of N, i.e., let ΓREG(N) be the smallest set such that N⊆ΓREG(N) and L1∪L2, L1L2, L1⁎∈ΓREG(N) for any L1,L2∈ΓREG(N). In this paper, we provide a pumping lemma for ΓREG(PFL). This lemma enables us to prove that some languages do not belong to ΓREG(PFL). It is notable that we immediately obtain a pumping lemma for ΓREG(DCFL) as a corollary, due to the known fact that ΓREG(DCFL)=ΓREG(DCFL∩PFL), where we define DCFL as DCFL:=∪Σ0DCFL(Σ0) with the union taken over all non-empty finite set Σ0⊆Σ, and DCFL(Σ0) is the family of all L⊆Σ0⁎ such that L is accepted by a deterministic pushdown automaton over Σ0 by final state.

Arc-pancyclicity of hypertournaments with irregularity at most two

A k-hypertournament H consists of a pair (V(H),A(H)), where V(H) is a non-empty finite set of vertices and A(H) contains exactly one permutation of S as an arc for all k-subsets S of V(H). The irregularity of hypertournament H=(V(H),A(H)) is defined as maxv∈V(H)⁡{|deg+(v)−deg−(v)|}. A hypertournament with irregularity 0 is called as a ∑-regular or degree regular hypertournament which was introduced by Surmacs and is a generalization of regular tournament. A hypertournament with irregularity 1 is called as an almost ∑-regular or almost degree regular hypertournament which is a generalization of almost regular tournament.It is well known that every regular tournament is arc-pancyclic and every almost regular tournament with at least 8 vertices is arc-4-pancyclic. Surmacs (2017) [7] proved that every ∑-regular k-hypertournament on n-vertices, where 2≤k≤n−2, is arc-pancyclic. In this paper, we prove that every k-hypertournament on n vertices with irregularity at most two, where 3≤k≤n−2, is arc-pancyclic, which is a generalization of above results.

Some applications of optimization routing problems with additional constraints

The paper deals with an extremal routing problem with constraints. In the general formulation, it is assumed that the objects of visiting are any non-empty finite sets — megalopolises. The main applied problem considered in this study is the tool path optimization problem for CNC sheet-cutting machines, known as the Cutting Path Problem. This problem arises at the stage of developing control programs for CNC machines. Other applications are also possible. In particular, the results obtained in the chapter can be used in the problem of minimizing the radiation dose when dismantling a system of radiation-hazardous elements after accidents at nuclear power plants and in transport problems. Among tasks constraints, the precedence constraints are investigated. These constraints can be used to reduce computational complexity. As the main method, the study used broadly understood dynamic programming. The offered realization of the method takes into account the precedence constraints and the dependence of the objective functions on the task list. This dependence belongs to the class of very complex conditions that determine the route admissibility at each routing step, depending on the tasks already completed or, on the contrary, not yet completed. As applied to the Cutting Path Problem, the dependence of the objective function on the task list makes it possible to reduce thermal deformations of the material during cutting. The chapter provides a mathematical formalization of an extremal routing problem with additional constraints, a description of the method, and the exact algorithm obtained with its help. The order of task execution, the specific trajectory of the process, and the starting point are optimized.

Clique and Independence Number of Rough Co-Zero Divisor Graph and Its Applications in Analyzing a Twitter Data

In view of this paper, we have defined Rough Co-zero divisor graph [Formula: see text] of a rough semiring [Formula: see text] for a given approximation space [Formula: see text] where [Formula: see text] is nonempty finite set of objects and [Formula: see text] is an arbitrary equivalence relation on [Formula: see text]. It is proved that the Rough Co-zero divisor graph and the subgraph of Rough Ideal-based rough edge Cayley graph [Formula: see text] are complement to each other. Also the Clique number and Independence number of [Formula: see text] and [Formula: see text] are obtained. The developed concepts are illustrated with suitable examples. A sentiment analysis is also made using [Formula: see text] for a Twitter data.

On the Number of Finite Fuzzy Subsets with Analysis of Integer Sequences

This paper solves the issues of determining the number Fn of fuzzy subsets of a nonempty finite set X. To solve this, this paper incorporates the equivalence relation on the collection of all fuzzy subsets of X. We derive two closed explicit formulas for Fn, which is the sum of a finite series in the product of binomial numbers or the sum of k-level fuzzy subsets Fn,k by introducing a classification technique. Moreover, these explicit formulas enable us to find the number of the maximal chains of crisp subsets of X. Further, this paper presents some elementary properties of Fn,k and Fn.

On the Multiple Conjugacy Problem in Group F/N1 ∩ N2

Let F be a free group generated by a finite alphabet A. Let N1 (N2) be the normal closure of a finite non-empty symmetrized set R1 (respectively, R2) of elements in F. Earlier, one obtained the conditions sufficient for the solvability of the conjugacy problem in the group F/N1 ∩ N2. The present paper is a continuation of this research and is devoted to the solvability of the multiple conjugacy problem in F/N1 ∩ N2. In particular, we get that if R1 ∪ R2 satisfies the small cancellation condition C' (1/6), then the multiple conjugacy problem is solvable in F/N1 ∩ N2.