Homomorphism Theorem Research Articles

Pseudo Rho-Algebras in Fields with their Applications Using Polynomials with Two Variables

The main purpose of this work is to present the concept of pseudo Rho-algebra as a generalization of the Rho-algebra. We investigate this type behaves nicely in terms of common mappings like homomorphisms and direct products. In fact, we can show identify kernel objects and the fundamental theorem of homomorphisms using Rho-ideals in this theory. Also, some important classes of pseudo Rho-algebra and their applications are investigated and studied. Moreover, the necessary characteristics for or tantamount to membership in major subtypes of pseudo Rho/d-algebras under these cases are identified. The Rho/d-algebras and pseudo Rho/d-algebras are generated using some new classes of polynomials with two variables that are introduced in this work, like commutative identical, associative identical, regular identical, proper conditionally identical, and improper conditionally identical. Finally, the category of pseudo Rho/d-algebras can be classified using a conditionally couple, giving us additional flexibility in dealing with two different fields.

Nimber-preserving reduction: Game secrets and homomorphic Sprague-Grundy theorem

The concept of nimbers—a.k.a. Grundy-values or nim-values—is fundamental to combinatorial game theory. Beyond the winnability, nimbers provide a complete characterization of strategic interactions among impartial games in disjunctive sums. In this paper, we consider nimber-preserving reductions among impartial games, which enhance the winnability-preserving reductions in traditional computational characterizations of combinatorial games. We prove that Generalized Geography is complete for the natural class, IP, of polynomially-short impartial rulesets, under polynomial-time nimber-preserving reductions. We refer to this notion of completeness as Sprague-Grundy-completeness. In contrast, we also show that not every PSPACE-complete ruleset in IP is Sprague-Grundy-complete for IP.By viewing every impartial game as an encoding of its nimber—a succinct game secret richer than its winnability alone—our technical result establishes the following striking cryptography-inspired homomorphic theorem: Despite the PSPACE-completeness of nimber computation for IP, there exists a polynomial-time algorithm to construct, for any pair of games G1,G2 in IP, a Generalized Geography game G satisfying:nimber(G)=nimber(G1)⊕nimber(G2).

Ideals and Homomorphism Theorems of Fuzzy Associative Algebras

Based on the definitions of fuzzy associative algebras and fuzzy ideals, it is proven that the intersections of fuzzy subalgebras are fuzzy subalgebras, and the intersections of fuzzy ideals are fuzzy ideals. Moreover, we prove that the kernels of fuzzy homomorphisms are fuzzy ideals. Using fuzzy ideals, the quotient structures of fuzzy associative algebras are constructed, their corresponding properties are discussed, and their homomorphism theorems are proven.

Teaching Abstract Algebra Concretely via Embodiment

ABSTRACT We describe how an instructor integrated embodiment to teach the Fundamental Homomorphism Theorem (FHT) and preliminary concepts in an undergraduate abstract algebra course. The instructor’s use of embodiment reduced levels of abstraction for formal definitions, theorems, and proofs. The instructor’s simultaneous use of various forms of embodiment primed students for the formalism and symbolism, highlighted and disambiguated students’ referents, amplified students’ contributions to develop conceptual fluency, and linked students’ body form catchments to motivate the FHT. Our results offer practical implications for teaching by illustrating examples of how embodiment can assist in making abstract concepts concrete.

Congruence Relations and Direct Decomposition of Uninorms on Bounded Lattices

In this paper, the concept of [Formula: see text]-compatible congruence and its related properties are studied. We also focus on homomorphism theorems for uninorms. Decomposition of uninorms is investigated via factor congruences, at first. Subsequently, we investigate decomposition of conjunctive and disjunctive uninorms, respectively. After all, we determine necessary and sufficient conditions for decomposition of some uninorms that are neither conjunctive nor disjunctive. In addition to all these, we shed light on the application of our results with the examples.

Fibered universal algebra for first-order logics

We extend Lawvere-Pitts prop-categories (aka. hyperdoctrines) to develop a general framework for providing fibered algebraic semantics for general first-order logics. This framework includes a natural notion of substitution, which allows first-order logics to be considered as structural closure operators just as propositional logics are in abstract algebraic logic. We then establish an extension of the homomorphism theorem from universal algebra for generalized prop-categories and characterize two natural closure operators on the prop-categorical semantics. The first closes a class of structures (which are interpreted as morphisms of prop-categories) under the satisfaction of their common first-order theory and the second closes a class of prop-categories under their associated first-order consequence. It turns out that these closure operators have characterizations that closely mirror Birkhoff's characterization of the closure of a class of algebras under the satisfaction of their common equational theory and Blok and Jónsson's characterization of closure under equational consequence, respectively. These algebraic characterizations of the first-order closure operators are unique to the prop-categorical semantics. They do not have analogues, for example, in the Tarskian semantics for classical first-order logic. The prop-categories we consider are much more general than traditional intuitionistic prop-categories or triposes (i.e., topos representing indexed partially ordered sets). Nonetheless, to the best of our knowledge, our results are new, even when restricted to these special classes of prop-categories.

M-Hazy Module and Its Homomorphism Theorem

Based on a completely distributive lattice M , we propose a new fuzzification approach to a module, which leads to the concept of an M -hazy module. Different from the traditional fuzzification approach that defines a fuzzy algebra as a fuzzy subset of a classical algebra, we introduce an M -hazy module by fuzzifications of algebraic operations. Then, we investigate the fundamental properties of M -hazy modules and M -hazy submodules. In particular, we present the M -hazy module homomorphism theorem.

A Detailed Study of Mathematical Rings in q-Rung Orthopair Fuzzy Framework

Symmetry-related problems can be addressed by means of group theory, and ring theory can be seen as an extension of additive group theory. Ring theory, a significant topic in abstract algebra, is currently active in a diverse range of study domains across the disciplines of mathematics, theoretical physics and coding theory. The study of ideals is vital to the theory of rings in a wide range of ways. The uncertainties present in the information are addressed well by the q-rung orthopair fuzzy set (q-ROFS). Considering the significance of ring theory and the q-ROFS, this article defines q-rung orthopair fuzzy ideals (q-ROFIs) in conventional rings and investigates its various algebraic features. We introduce the notion of q-rung orthopair fuzzy cosets (q-ROFCs) of a q-ROFI and demonstrate that, under certain binary operations, the collection of all q-ROFCs of a q-ROFI forms a ring. In addition, we provide a q-rung orthopair analog of the fundamental theorem of ring homomorphism. Furthermore, we present the notion of q-rung orthopair fuzzy semi-prime ideals (q-ROFSPIs) and provide a comprehensive explanation of their many algebraic properties. Finally, regular rings were characterized using q-ROFIs.

Near-Modul Kuat Faktor

Near-ring is a generalization of ring. In ring theory, let R be a ring over addition and multiplication operation with unity element and G be a commutative group under addition operation. Then G together with scalar multiplication of R and holds several axioms called module over ring. Let N be a near-ring, here we can construct the module over near-ring called near-module. We have three definitions of module over near-ring, such that, near-module, modified near-module, and strong near-module. Then we showed that strong near-module can be constructed into strong near-module factor as well as module factor in G. Furthermore, we generalized the fundamental theorem of homomorphism in module over ring to strong near-module over strong near-ring.

On Pythagorean fuzzy ideals of a classical ring

<abstract><p>The Pythagorean fuzzy set is an extension of the intuitionistic fuzzy set and is an effective approach of handling uncertain situations. Ring theory is a prominent branch of abstract algebra, vibrant in wide areas of current research in mathematics, computer science and mathematical/theoretical physics. In the theory of rings, the study of ideals is significant in many ways. Keeping in mind the importance of ring theory and Pythagorean fuzzy set, in the present article, we characterize the concept of Pythagorean fuzzy ideals in classical rings and study its numerous algebraic properties. We define the concept of Pythagorean fuzzy cosets of a Pythagorean fuzzy ideal and prove that the set of all Pythagorean fuzzy cosets of a Pythagorean fuzzy ideal forms a ring under certain binary operations. Furthermore, we present Pythagorean fuzzy version of the fundamental theorem of ring homomorphism. We also introduce the concept of Pythagorean fuzzy semi-prime ideals and give a detailed exposition of its different algebraic characteristics. In the end, we characterized regular rings by virtue of Pythagorean fuzzy ideals.</p></abstract>

SMSG: Profiling-Free Parallelism Modeling for Distributed Training of DNN

The increasing size of deep neural networks (DNNs) raises a high demand for distributed training. An expert could find good hybrid parallelism strategies, but designing suitable strategies is time and labor-consuming. Therefore, automating parallelism strategy generation is crucial and desirable for DNN designers. Some automatic searching approaches have recently been studied to free the experts from the heavy parallel strategy conception. However, these approaches all rely on a numerical cost model, which requires heavy profiling results that lack portability. These profiling-based approaches cannot lighten the strategy generation work due to the non-reusable profiling value. Our intuition is that there is no need to estimate the actual execution time of the distributed training but to compare the relative cost of different strategies. We propose SMSG (Symbolic Modeling for Strategy Generation), which analyses the cost based on the communication and computation semantics. With SMSG, the parallel cost analyses are decoupled from hardware characteristics. SMSG defines cost functions for each kind of operator to quantitatively evaluate the amount of data for computation and communication, which eliminates the heavy profiling tasks. Besides, SMSG introduces how to apply functional transformation by using the Third Homomorphism theorem to control the high searching complexity. Our experiments show that SMSG can find good hybrid parallelism strategies to generate an efficient training performance similar to the state of the art. Moreover, SMSG covers a wide variety of DNN models with good scalability. SMSG provides good portability when changing training configurations that a profiling-based approach cannot.

An investigation on S-hypersystems over semihypergroups

We recall the definition of an S-hypersystem and the fundamental normal homomorphism theorems on S-hypersystems. We introduce the concept of the category of S-hypersystems and study the product and coproduct of S-hypersystems. We prove that every S-pohypersystem is a subdirect product of subdirectly irreducible S-pohypersystems.

Near Polygroups on Nearness Approximation Spaces

In this paper, we consider the problem of how to establish algebraic structures of near sets on nearness approximation spaces. Essentially, our approach is to define the near polygroup of all weak cosets by considering an hyperoperation on the set of all weak cosets. Afterwards, our aim is to study near homomorphism theorems on near polygroups, and investigate some characterizations of near polygroups.

Some Neutrosophic Triplet Subgroup Properties and Homomorphism Theorems in Singular Weak Commutative Neutrosophic Extended Triplet Group

In 2018, the study of neutrosophic triplet cosets and neutrosophic triplet quotient group of a neutrosophic extended triplet group was initiated with a follow up of the establishment of fundamental homomorphism theorems for neutrosophic extended triplet group. But some lapses in these earlier results were identified and revised through the introduction of special kind of weak commutative neutrosophic extended triplet group (WCNETG) called perfect neutrosophic extended triplet group. Furthermore, neutro-homomorphism basic theorem has been established for commutative neutrosophic extended triplet group. In this current work, the generalization and extention of the above results was done by investigating neutro-homomorphism in singular WCNETG. This was achieved with the introduction and study of some new types of NT-subgroups that are right (left) cancellative, semi-strong, and maximally normal in a singular WCNETG. For any given non-empty subset S and NT-subgroup H of a singular WCNETG X, some of these new NT-subgroups were shown to exist as non-empty neutrosophic triplet normalizer, generated subset and centralizer of S, closure of H, derived subset of X and center of X. With these, the first, second and third neutro-isomorphism and neutro-correspondence theorems were established. This finally led to the proof of the neutro-Zassenhaus Lemma (Neutro-Butterfly Theorem).

Strongest relation, cosets and middle cosets of \(AQFSC(G)\)

The aim of this paper is to introduce strongest relation, cosets and middle cosets of anti \(Q\)-fuzzy subgroups of \(G\) with respect to \(t\)-conorm \(C.\) We investigate equivalent characterizations of them and we construct a new group induced by them, and give the homomorphism theorem between them.

Some Operators on Quantum B-Algebras

The aim of this paper is to investigate several operators on quantum B-algebras. At first, we introduce closure and interior operators on quantum B-algebras and consider their relations on bounded quantum B-algebras. Furthermore, we discuss very true operators on quantum B-algebras by three cases via the unit element, and present some similar conclusions and different results. Finally, by constructing a very true operator on a quotient very true perfect quantum B-algebra, we establish a homomorphism theorem on very true perfect quantum B-algebras.

Application of the Homomorphism Theorem for Rings to Linearization of Systems of Partial Differential Equations

We propose an algorithm for linearizing systems of partial differential equations at constant solutions. The algorithm is based on an isomorphism constructed between the ring of linearized functions and the ring of special matrices, which makes it possible to simplify calculations in the process of linearization. The algorithm is illustrated by applying it to the quasigasdynamic system.

Fuzziness in L-algebras

The aim of this paper is to introduce fuzzy ideals and fuzzy (weak) congruence relations of L-algebras. We give a sufficient and necessary condition for a fuzzy set to be a fuzzy ideal of an L-algebra. After obtaining a necessary condition for a fuzzy set to be a fuzzy ideal of L-algebras, we construct an example to show that the condition is not sufficient. For CKL-algebras, however, we prove that it is both necessary and sufficient. Then we give a process to generate a fuzzy ideal by a fuzzy set of CKL-algebras. Finally, we establish a correspondence between fuzzy ideals and fuzzy weak congruence relations of L-algebras, and get homomorphism theorems induced by fuzzy congruence relations.

Ternary Menger Algebras: A Generalization of Ternary Semigroups

Let n be a fixed natural number. Menger algebras of rank n, which was introduced by Menger, K., can be regarded as the suitable generalization of arbitrary semigroups. Based on this knowledge, an interesting question arises: what a generalization of ternary semigroups is. In this article, we first introduce the notion of ternary Menger algebras of rank n, which is a canonical generalization of arbitrary ternary semigroups, and discuss their related properties. In the second part, we establish the so-called a diagonal ternary semigroup which its operation is induced by the operation on ternary Menger algebras of rank n and then investigate their interesting properties. Moreover, we introduce the concept of homomorphism and congruences on ternary Menger algebras of rank n. These lead us to study the quotient ternary Menger algebras of rank n and to investigate the homomorphism theorem for ternary Menger algebra of rank n with respect to congruences. Furthermore, the characterization of reduction of ternary Menger algebra into Menger algebra is presented.

Intersection soft ideals and their quotients on KU-algebras

<abstract><p>In this paper, we have discussed quotient structures of KU-algebras by using the concept of intersection soft ideals. In general, the soft sets are parameterized families of sets that are used to dealt with uncertainty. In particular, We have given the fundamental homomorphism theorem of quotient KU-algebras. A characterization of commutative quotient KU-algebras, implicative quotient KU-algebras and positive implicative quotient KU-algebras are also presented.</p></abstract>