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In this paper, we establish the idea of fuzzy congruences on Abel-Grassmann's group (AG-group). We investigate different outcomes of fuzzy-congruences on AG-groups in detail and give some examples to illustrate the newly established results. We develop the relation between fuzzy congruence and fuzzy normal subgroup. In the end, we also provide some results of fuzzy homomorphism on AG-groups.

For any positive integer $ n $, let $ \mathbb Z_n: = \mathbb Z/n\mathbb Z = \{0, \ldots, n-1\} $ be the ring of residue classes module $ n $, and let $ \mathbb{Z}_n^{\times}: = \{x\in \mathbb Z_n|\gcd(x, n) = 1\} $. In 1926, for any fixed $ c\in\mathbb Z_n $, A. Brauer studied the linear congruence $ x_1+\cdots+x_m\equiv c\pmod n $ with $ x_1, \ldots, x_m\in\mathbb{Z}_n^{\times} $ and gave a formula of its number of incongruent solutions. Recently, Taki Eldin extended A. Brauer's result to the quadratic case. In this paper, for any positive integer $ n $, we give an explicit formula for the number of incongruent solutions of the following cubic congruence \begin{document}$ x_1^3+\cdots +x_m^3\equiv 0\pmod n\ \ \ {\rm with} \ x_1, \ldots, x_m \in \mathbb{Z}_n^{\times}. $\end{document}

A planar (upper) semimodular lattice $L$ is slim if the five-element nondistributive modular lattice $M_3$ does not occur among its sublattices. (Planar lattices are finite by definition.) Slim rectangular lattices as particular slim planar semimodular lattices were defined by G. Gr\"atzer and E. Knapp in 2007. In 2009, they also proved that the congruence lattices of slim planar semimodular lattices with at least three elements are the same as those of slim rectangular lattices. In order to provide an effective tool for studying these congruence lattices, we introduce the concept of lamps of slim rectangular lattices and prove several of their properties. Lamps and several tools based on them allow us to prove in a new and easy way that the congruence lattices of slim planar semimodular lattices satisfy the two previously known properties. Also, we use lamps to prove that these congruence lattices satisfy four new properties including the two-pendant four-crown property and the forbidden marriage property., Comment: 26 pages, 11 figures

We construct a free abelian trioid and describe the least abelian congruence on a free trioid.

Pressupondo que as organizações e os guardas prisionais detêm um conjunto de valores sobre como reduzir o crime, correspondente a uma abordagem reabilitativa e/ou punitiva, que podem, ou não, coincidir, pretende-se explorar com base no Modelo Person-Environment fit, a relação entre a congruência de valores e o stress ocupacional vivenciado por guardas prisionais portugueses. Os dados foram recolhidos através da análise documental e entrevista semiestruturada individual realizada a oito guardas prisionais. Conclui-se que os valores punir e ressocializar são independentes e interagem entre si e que a congruência de valores, que se manifesta através de três conceitos independentes, fit, misfit e conflito de valores, é um elemento crítico da vida nas prisões. Constata-se de facto, a existência de dois tipos distintos de conflito e que a ausência de fit não é indicador de conflito. Estes resultados são interpretados considerando-se as implicações teóricas e práticas para as organizações e guardas prisionais.

In this paper, we consider a generalization of Horadam sequence $\left\{w_{n}\right\} $ which is defined by the recurrence $w_{n}=aw_{n-1}+cw_{n-2},$ if $n$ is even, $w_{n}=bw_{n-1}+cw_{n-2},$ if $n$ is odd with arbitrary initial conditions $w_{0},w_{1}$ and nonzero real numbers $a,b,$ and $c.$ We investigate some congruence properties of the generalized Horadam sequence $\{ w_{n}\}$.

The notion of a congruence pair for principal MS-algebras, simpler than the one given by Beazer for K 2-algebras [6], is introduced. It is proved that the congruences of the principal MS-algebras L correspond to the MS-congruence pairs on simpler substructures L °° and D(L) of L that were associated to L in [4]. An analogy of a well-known Grätzer’s problem [11: Problem 57] formulated for distributive p-algebras, which asks for a characterization of the congruence lattices in terms of the congruence pairs, is presented here for the principal MS-algebras (Problem 1). Unlike a recent solution to such a problem for the principal p-algebras in [2], it is demonstrated here on the class of principal MS-algebras, that a possible solution to the problem, though not very descriptive, can be simple and elegant. As a step to a more descriptive solution of Problem 1, a special case is then considered when a principal MS-algebra L is a perfect extension of its greatest Stone subalgebra L S . It is shown that this is exactly when de Morgan subalgebra L °° of L is a perfect extension of the Boolean algebra B(L). Two examples illustrating when this special case happens and when it does not are presented.

Although collecting data from multiple informants is highly recommended, methods to model the congruence and incongruence between informants are limited. Bauer and colleagues suggested the trifactor model that decomposes the variances into common factor, informant perspective factors, and item-specific factors. This study extends their work to the trifactor mixture model that combines the trifactor model and the mixture model. This combined approach allows researchers to investigate the common and unique perspectives of multiple informants on targets using latent factors and simultaneously take into account potential heterogeneity of targets using latent classes. We demonstrate this model using student self-rated and teacher-rated academic behaviors ( N = 24,094). Model specification and testing procedures are explicated in detail. Methodological and practical issues in conducting the trifactor mixture analysis are discussed.

It is well-known that the congruence ∑ i = 1 n i n ≡ 1 ( mod n ) has exactly five solutions: { 1 , 2 , 6 , 42 , 1806 } . In this work, we characterize the solutions to the congruence 1 n + 2 n + ⋯ + n n ≡ p ( mod n ) for every prime p . This characterization leads to an algorithm for computing all such solutions, when there is a finite number of them. More generally, our algorithm enables computing all the solutions below a much higher bound as compared to what can be achieved by a naive exhaustive search.

We show that a dihedral congruence prime for a normalised Hecke eigenform f in $$S_k(\Gamma _0(D),\chi _D)$$ , where $$\chi _D$$ is a real quadratic character, appears in the denominator of the Bloch–Kato conjectural formula for the value at 1 of the twisted adjoint L-function of f. We then use a formula of Zagier to prove that it appears in the denominator of a suitably normalised $$L(1,{\mathrm {ad}}^0(g)\otimes \chi _D)$$ for some $$g\in S_k(\Gamma _0(D),\chi _D)$$ .

We show that if c is a positive integer satisfying $$2c-1=3p^l\ \hbox{or}\ 2c-1=5p^l$$ with p prime and l positive integer, then the equation $${x^2 + (2c-1)^m}=c^n$$ has only the positive integer solution $$(x,m,n)=(c-1,1,2)$$ without any congruence condition on a prime p.

Let p be an odd prime and $$H_k^{(n)}=\sum _{j=1}^k1/j^n$$ denote the generalized harmonic number. In this paper, the authors establish a kind of congruences involving $$\sum _{k=1}^{p-1}k^mH_{k}^{(n)}\pmod {p^2}$$ , where m, n are positive integers. Furthermore, the authors prove a congruence for $$\sum _{k=1}^{p-1}k^{p-2}(H_{k}^{(2)})^2\pmod {p^2}$$ .

A finite computational process using arithmetic operations only is called a rational algorithm. Presently, no rational algorithm is available that is able to check the congruence of arbitrary complex matrices A and B. In this paper, we propose a rational algorithm of verification that works under less stringent conditions than all the conditions known by now.

A finite algorithm that uses arithmetic operations only is said to be rational. There exist rational methods for checking the congruence of a pair of Hermitian matrices or a pair of unitary ones. We propose a rational algorithm for checking the congruence of general normal matrices.

Established media preference theories implicitly presume that audiences know a media product’s content or effects. Brands can fill this logical gap, with brands that media creators craft around the...

A basic sanity property of a process semantics is that it constitutes a congruence with respect to standard process operators. This issue has been traditionally addressed by developing, for a specific process semantics, a syntactic format for operational semantics specifications. We suggest a novel, orthogonal approach, which focuses on a specific process operator and determines a class of congruence relations for this operator. To this end, we impose syntactic restrictions on Hennessy–Milner logic, so that a process semantics whose modal characterization satisfies those criteria is guaranteed to be a congruence with respect to the operator in question. We investigate alternative composition, action prefix, projection, encapsulation, renaming, and parallel composition with communication, in the context of both concrete and weak process semantics.

Background Values are of high importance to the nursing profession. Value congruence is the extent to which an individual’s values align with the values of their organisation. Value congruence has important implications for job satisfaction. Aim This study explored nurse values, value congruence and potential implications for individual nurses and organisations in terms of wellbeing and patient care and safety. Method Fifteen nurses who worked in acute hospital settings within the UK participated in semi-structured telephone interviews. Thematic analysis was utilised to analyse the data. Results Four themes were identified: organisational values incongruent with the work environment; personal and professional value alignment; nurse and supervisor values in conflict; nurses’ values at odds with the work environment. Perceived value incongruence was related to poorer wellbeing, increased burnout and poorer perceived patient care and safety. The barriers identified for nurses being able to work in line with their values are described. Conclusions Value congruence is important for nurse wellbeing and patient care and safety. Improving the alignment between the values that organisations state they hold, and the values implied by the work environment may help improve patient care and safety and support nurses in practice.

The negative emotion of Foreign Language Classroom Anxiety (FLCA) and the positive emotion of Foreign Language Enjoyment (FLE) have both been linked to foreign language (FL) learners’ Self-Perceived Proficiency (SPP). In linear correlational studies, SPP has been found to be negatively affected by FLCA and positively affected by FLE. However, linear correlational research methods are limited in the extent to which they can depict these interrelations. In this study, we examined how different levels of FLCA and FLE influenced SPP in concert in a sample of n = 1039 FL learners by utilising polynomial regression with response surface analysis. The response model generated visually depicted a complex SPP pattern as a result of differing levels of FLCA and FLE. This three-dimensional model provides considerable insight into the interaction of positive and negative emotions above and beyond what could be derived from ‘classical’ methods of correlation and linear regression.

This paper is a survey on sheaves of arithmetic differential operators with congruence level on formal schemes. We present first results about these sheaves and discuss some examples.

In 2012, Chan posed six conjectures on congruences for Appell–Lerch sums and five of them were proved by mathematicians. In this paper, we confirm the remaining conjectural congruence of Chan by utilizing an identity due to Hickerson and Mortenson and the theory of modular forms.

It has been well established that congruences between automorphic forms have far-reaching applications in arithmetic. In this paper, we construct congruences for Siegel–Hilbert modular forms defined over a totally real field of class number 1. As an application of this general congruence, we produce congruences between paramodular Saito–Kurokawa lifts and non-lifted Siegel modular forms. These congruences are used to produce evidence for the Bloch–Kato conjecture for elliptic newforms of square-free level and odd functional equation.

For several classes of special p-groups G, of exponent p, p > 2, we show that the near-ring, C 0 ( G ) , of congruence preserving functions on G is a ring if and only if G is a 1-affine complete group.

The accessible writing is occupied with affirmation proposing the necessity for picking and utilizing proper individuals for the organization's, likewise as reporting the significance of congruence among people at workplaces with the organization and making this relationship among the duo stronger for the general accomplishment of the great number of objectives of the organization and all the stakeholders associated directly or indirectly with the organizations. Likewise, researchers, experts and practitioners in the field of organizational behaviour and its allied fields across the globe have demonstrated and proved a lot of enthusiasm in examining the domain of person-organizational congruence in relation to several individual level organizational level outcomes like, organizational commitment, work satisfaction, organizational citizenship behaviour, work performance, turnover intention and intention to stay. Research’s that have been carried out as of now on person-organizational congruence and its relationship with several individual level and organizational level outcomes were limited in context to the Indian settings and growing number of studies advised a need to further investigate person-organizational congruence and its relationship with individual and organizational level outcomes. In light of this developing essentialness associated to the phenomenon of person-organizational congruence domain the present examination is as such a modest undertaking toward this way. KEY WORDS: person-organizational congruence, organizational outcome variables, individual outcome variables

In congruence with the remarkable trend of consumer’s adoption of online hotel booking services, this study applies the Stimulus-Organism-Response (S-O-R) framework to analyse the direct triadic ef...

Loday and Ronco introduced the notion of a trioid. We construct a trioid which is isomorphic to the free $$n$$ -nilpotent trioid, and characterize the least dimonoid congruences and the least semigroup congruence on it. We also study separately singly generated free $$n$$ -nilpotent trioids.

Hyper logical algebras were first studied in 2000 by Borzooei et al. They applied the concept of hyperstructures to one of the logical algebraic structures known as the BCK-algebra, and introduced two generalizations of them called the hyper BCK-algebra and hyper K-algebra. Then many researchers in this field continued their research and used hyperstructures on other logical algebras and introduced the concepts of hyper residuated lattices, hyper BL-algebras, hyper MV-algebras, hyper EQ-algebras, hyper BE-algebras, hyper equality algebras, hyper hoops and etc. Moreover, they defined some new notions such as different kinds of hyper ideals, hyper filters and hyper congruence relations on these structures and studied some properties, the relation among them and the quotient structure. Now, in this paper, we review the definitions of all those hyper logical algebras and investigate relations among them.

This paper is a review of the recently published article by Martin O’Loughlin and Hovhannes Demirchian [1] on covariant description of gravitational memory effect. There, a new approach was introduced to study geodesic congruences and their evolution after traversing impulsive signals, containing a mixture of material sources and gravitational waves. It was shown that the effect of the wave on the congruence is a discontinuity in the $$\mathcal{B}$$ -tensor of the congruence which provides a detector independent and covariant characterization of gravitational memory. It was also shown how different components of the impulsive signal effect the evolution of the congruence in the future.

Hilbert algebras are important tools for certain investigations in intuitionistic logic and other non-classical logic and as a generalization of Hilbert algebra a new algebraic structure, called a GE-algebra (generalized exchange algebra), is introduced and studied its properties. We consider filters, upper sets and congruence kernels in a GE-algebra. We also characterize congruence kernels of transitive GE-algebras.

This article deals with the process of adaptation that NATO has gone through over the last decade. It contends that while seeking congruence with its environment, NATO is facing a risk of maladapta...

The set $A$ is an asymptotic nonbasis of order $h$ for an additive abelian group $X$ if there are infinitely many elements of $X$ not in the $h$-fold sumset $hA$. For all $h \geq 2$, this paper constructs new classes of asymptotic nonbases of order $h$ for $\mathbf{Z}$ and for $\mathbf{N}_0$ that are not subsets of maximal asymptotic nonbases., Comment: 10 pages; corrected a few minor typos

Complex n×n matrices A and B are said to be T-congruent if B = STAS and ∗-congruent if B = S∗AS, where S is an arbitrary nonsingular matrix. For several facts related to normal matrices and ∗-congruences, their counterparts in the theory of T-congruences concerning conjugate-normal matrices are found. Bibliography: 6 titles.

The paper discusses the possibility of reducing a square complex matrix A to a direct sum of smaller matrices by using ∗-congruence transformations. It turns out that this possibility is related to appropriate partitions of the spectrum of the cosquare of A. This makes it possible to associate the direct summands of the sum with subsets of the latter spectrum.

The detection of deformation is one of the major tasks in surveying engineering. It is meaningful only if the statistical significance of the distortions is correctly investigated, which often underlies a parametric modelization of the object under consideration. So-called regression B-spline approximation can be performed for point clouds of terrestrial laser scanners, allowing the setting of a specific congruence test based on the B-spline surfaces. Such tests are known to be strongly influenced by the underlying stochastic model chosen for the observation errors. The latter has to be correctly specified, which includes accounting for heteroscedasticity and correlations. In this contribution, we justify and make use of a parametric correlation model called the Matérn model to approximate the variance covariance matrix (VCM) of the residuals by performing their empirical mode decomposition. The VCM obtained is integrated into the computation of the congruence test statistics for a more trustworthy test decision. Using a real case study, we estimate the distribution of the test statistics with a bootstrap approach, where no parametric assumptions are made about the underlying population that generated the random sample. This procedure allows us to assess the impact of neglecting correlations on the critical value of the congruence test, highlighting their importance.

It is widely conjectured that every sufficiently large integer satisfying certain necessary congruence conditions is the sum of 4 cubes of prime numbers. As an approximation to this conjecture, we shall establish two results in this paper. Firstly, we show that every large odd integer is the sum of a prime, 4 cubes of primes and 15 powers of 2. Secondly, we show that the conjecture is true for at least $$8.25\%$$ of the positive integers.

We extend the computations in [2] , [3] , [4] to find the cohomology in degree five of a congruence subgroup of SL 4 ( Z ) with coefficients in a field twisted by a nebentype character, along with the action of the Hecke algebra on the cohomology. This is the top cuspidal degree. For each Hecke eigenclass we find, we produce the unique Galois representation that appears to be attached to it. The computations require serious modifications to our previous algorithms. Nontrivial coefficients add a layer of complication to our data structures. New possibilities must be taken into account in the Galois Finder, the code that finds the Galois representations. We have improved the Galois Finder to report when the attached Galois representation is uniquely determined by our data.

Dyson’s famous conjectures (proved by Atkin and Swinnerton-Dyer) gave a combinatorial interpretation of Ramanujan’s congruences for the partition function. The proofs of these results center on one of the universal mock theta functions that generate partitions according to Dyson’s rank. George Beck has generalized the study of partition function congruences related to rank by considering the total number of parts in the partitions of [Formula: see text]. The related generating functions are no longer part of the world of mock theta functions. However, George Beck has conjectured that certain linear combinations of the related enumeration functions do satisfy congruences modulo 5 and 7. The conjectures are proved here.

The purpose of this paper is to consider the computation of the Fourier coefficients of real-analytic Eisenstein series associated to general cusps of Hecke congruence subgroups of arbitrary level. Explicit formulas for some coefficients in the Fourier expansion of the Eisenstein series of weight zero for general cusps are given.

In this paper, we study the hyper versions of Green’s relations in ordered semihypergroups in detail. The Green’s relations $${{\mathcal {R}}}, {{\mathcal {L}}}, {{\mathcal {J}}}$$ and $$ {\mathcal H}$$ in ordered semihypergroups are first introduced, and the relations between them are given. Furthermore, we investigate the properties of Green’s relations in ordered semihypergroups. Particularly, we illustrate the Green’s relation $${{\mathcal {R}}}$$ (resp. $${{\mathcal {L}}}$$ ) in an ordered semihypergroup S is not necessarily a left (resp. right) congruence on S by counterexamples. Meanwhile, we also provide a sufficient condition that makes the above conclusion true. Finally, we introduce the concept of a-maximal hyperideals of an ordered semihypergroup, and discuss its related properties by terms of the Green’s relation $${{\mathcal {J}}}.$$

Consider the system x 2 − ay 2 = b, P (x, y) = z 2, where P is a given integer polynomial. Historically, the integer solutions of such systems have been investigated by many authors using the congruence arguments and the quadratic reciprocity. In this paper, we use Kedlaya’s procedure and the techniques of using congruence arguments with the quadratic reciprocity to investigate the solutions of the Diophantine equation 7X 2 + Y 7 = Z 2 if (X, Y) = (L n , F n ) (or (X, Y) = (F n , L n )) where {F n } and {L n } represent the sequences of Fibonacci numbers and Lucas numbers respectively.