Group Homomorphism Research Articles

T-norms over Fuzzy Multigroups

In this paper, we propose the notion of fuzzy multigroups under t-norms. Some properties of them are explored and some related results are obtained. Also inverse, product, intersection and sum of them will be defined and investigated properties of them. Finally under group homomorphisms, image and pre image of them will be introduced and investigated.

New types of soft rough sets in groups based on normal soft groups

Hybridization of soft sets and rough sets is an important way to deal with uncertainties. This paper aims to study the concept of roughness in soft sets over groups. In this regard, a pair of two soft sets, viz. soft lower and soft upper approximation spaces, are introduced by applying the normal soft groups corresponding to each parameter. Some important results related to these soft approximation spaces over groups are studied with examples. Furthermore, this paper presents a relationship between the soft approximation spaces based on the soft image and soft pre-image of a normal soft group via group homomorphisms. This work can be applicable in the field of information technology to connect two information systems.

Class group of the ring of invariants of an exponential map on an affine normal domain

Let k be a field and let B be an affine normal domain over k. Let $$\phi $$ be a non-trivial exponential map on B and let $$A = B^{\phi }$$ be the ring of $$\phi $$ -invariants. Since A is factorially closed in B, $$A = K \cap B$$ where K denotes the field of fractions of A. Hence A is a Krull domain. We investigate here a relation between the class group $$\mathrm{Cl}(A)$$ of A and the class group $$\mathrm{Cl}(B)$$ of B. In this direction, we give a sufficient condition for an injective group homomorphism from $$\mathrm{Cl}(A)$$ to $$\mathrm{Cl}(B)$$ . We also give an example to show that $$\mathrm{Cl}(A)$$ may not be realized as a subgroup of $$\mathrm{Cl}(B)$$ .

Homological properties of quotient divisible Abelian groups and compact groups dual to them

Homological properties of quotient divisible Abelian groups are studied. These groups form an important class of groups, which has been extensively studied in recent years. The first part of the paper is devoted to conditions for the triviality of extension groups in which one of the arguments is a quotient divisible group. Under certain additional assumptions, groups of homomorphisms from quotient divisible groups to reduced Abelian groups are described. Universality properties of quotient divisible Abelian groups are investigated. The second part of the paper considers homological properties of compact Abelian groups dual to quotient divisible groups in the sense of L.S. Pontryagin. Such groups are said to be “quotient toroidal.” Conditions for the triviality of group extensions in which one of the arguments is a quotient toroidal group are studied. Certain groups of continuous homomorphisms in which the second argument is a quotient toroidal group are described. The last part of the paper is devoted to conditions for the triviality of the groups of extensions of quotient divisible groups by compact quotient toroidal ones. The fundamental group of the topological space of a quotient toroidal group is characterized.

A Certain Class of t-Intuitionistic Fuzzy Subgroups

In this study, the t-intuitionistic fuzzy normalizer and centralizer of t intuitionistic fuzzy subgroup are proposed. The t-intuitionistic fuzzy centralizer is normal subgroup of t-intuitionistic fuzzy normalizer and investigate various algebraic properties of this phenomena. We also introduce the concept of t-intuitionistic fuzzy Abelian and cyclic subgroups and prove that every t-intuitionistic fuzzy subgroup of Abelian (cyclic) group is t-intuitionistic fuzzy Abelian (cyclic) subgroup. We show that the image and pre-image of t-intuitionistic fuzzy Abelian (cyclic) subgroup are t-intuitionistic fuzzy Abelian (cyclic) subgroup under group homomorphism.

A Novel Applications of Complex Intuitionistic Fuzzy Sets in Group Theory

In this paper, we initiate the novel concept of complex intuitionistic fuzzy subgroups and prove that every complex intuitionistic fuzzy subgroup generates two intuitionistic fuzzy subgroups. We extend this ideology to define the concept of level subsets of complex intuitionistic fuzzy set and discuss its various fundamental algebraic characteristics. We also show that the level subset of the complex intuitionistic fuzzy subgroups is a subgroup. Furthermore, we investigate the homomorphic image and pre-image of complex intuitionistic fuzzy subgroup under group homomorphism. Moreover, we prove that the product of two complex intuitionistic fuzzy subgroups is also a complex intuitionistic fuzzy subgroup and develop some new results about direct product of complex intuitionistic fuzzy subgroups.

Introduction to AntiGroups

The notion of AntiGroups is formally presented in this paper. A particular class of AntiGroups of type-AG[4] is studied with several examples and basic properties presented. In AntiGroups of type-AG[4], the existence of an inverse is taking to be totally false for all the elements while the closure law, the existence of identity element, the axioms of associativity and commutativity are taking to be either partially true, partially indeterminate or partially false for some elements. It is shown that some algebraic properties of the classical groups do not hold in the class of AntiGroups of type-AG[4]. Specifically, it is shown that intersection of two AntiSubgroups is not necessarily an AntiSubgroup and the union of two AntiSubgroups may be an AntiSubgroup. Also, it is shown that distinct left(right) cosets of AntiSubgroups of AntiGroups of type-AG[4] do not partition the AntiGroups; and that Lagranges’ theorem and fundamental theorem of homomorphisms of the classical groups do not hold in the class of AntiGroups of type-AG[4].

Relative geometric assembly and mapping cones Part II: Chern characters and the Novikov property

We study Chern characters and the assembly mapping for free actions using the framework of geometric K-homology. The focus is on the relative groups associated with a group homomorphism phi: Gamma(1) -> Gamma(2) along with applications to Novikov type properties. In particular, we prove a relative strong Novikov property for homomorphisms of hyperbolic groups and a relative strong l(1)-Novikov property for polynomially bounded homomorphisms of groups with polynomially bounded cohomology in C. As a corollary, relative higher signatures on a manifold with boundary W, with pi(1)(partial derivative W) -> pi(1) (W) belonging to the class above, are homotopy invariant.

Naturality and definability II

We regard an algebraic construction as a set-theoretically defined map taking structures A to structures B which have A as a distinguished part, in such a way that any isomorphism from A to A' lifts to an isomorphism from B to B'. In general the construction defines B up to isomorphism over A. A construction is uniformisable if the set-theoretic definition can be given in a form such that for each A the corresponding B is determined uniquely. A construction is natural if restriction from B to its part A always determines a map from the automorphism group of B to that of A which is a split surjective group homomorphism. We prove that there is no transitive model of ZFC (Zermelo-Fraenkel set theory with Choice) in which the uniformisable constructions are exactly the natural ones. We construct a transitive model of ZFC in which every uniformisable construction (with a restriction on the parameters in the formulas defining the construction) is ‘weakly’ natural. Corollaries are that the construction of algebraic closures of fields and the construction of divisible hulls of abelian groups have no uniformisations definable in ZFC without parameters.

The inversion characterizations of $C(\mathcal {L})$ for a locale $\mathcal {L}$

The category $\mathbf {W}$ is comprised of archimedean $\ell $-groups $G$ with distinguished weak unit and unit-preserving $\ell $-group homomorphisms. For $G \in \mathbf {W}$ there is the canonical Yosida representation $G \leq D(\mathcal {Y}G)$ as extended-real valued functions, with the unit $e_G$ represented as the constant $1$ function. We define $a \in G$ to be kernel-maximal if the $\mathbf {W}$-kernel generated by $a$ in $G$ is all of $G$, and we define $a$ to be Yosida invertible if there is some $b \in G$ with $ab = e_G$ in the partial multiplication inherited from $D(\mathcal {Y}G)$. The main theorem is that $G$ is isomorphic to $C(\mathcal {L})$ for some (identifiable) locale $\mathcal {L}$ iff $G$ is divisible, uniformly complete, and every kernel-maximal element of $G$ is Yosida invertible.

Applications of roughness in soft-intersection groups

In this research article, the notion of roughness in soft sets is investigated. Its applications are found on soft-intersection groups defined by Cagman et al. (Neural Comput Appl 21(1):S151–S158, 2012). The group homomorphism is manipulated to establish the connection between the approximation spaces of soft sets of groups.

Lattice structure on bounded homomorphisms between topological lattice rings

Suppose X is a topological ring. It is known that there are three classes of bounded group homomorphisms on X whose topological structures make them again topological rings. First, we show that if X is a Hausdorff topological ring, then so are these classes of bounded group homomorphisms on X. Now, assume that X is a locally solid lattice ring. In this paper, our aim is to consider lattice structure on these classes of bounded group homomorphisms more precisely, we show that, under some mild assumptions, they are locally solid lattice rings. In fact, we consider bounded order bounded homomorphisms on X. Then we show that under the assumed topology, they form locally solid lattice rings. For this reason, we need a version of the remarkable RieszKantorovich formulae for order bounded operators in Riesz spaces in terms of order bounded homomorphisms on topological lattice groups.

Artin’s braids, braids for three space, and groups Γn4 and Gnk

We construct a group [Formula: see text] corresponding to the motion of points in [Formula: see text] from the point of view of Delaunay triangulations. We study homomorphisms from pure braids on [Formula: see text] strands to the product of copies of [Formula: see text]. We will also study the group of pure braids in [Formula: see text], which is described by a fundamental group of the restricted configuration space of [Formula: see text], and define the group homomorphism from the group of pure braids in [Formula: see text] to [Formula: see text]. At the end of this paper, we give some comments about relations between the restricted configuration space of [Formula: see text] and triangulations of the 3-dimensional ball and Pachner moves.

On the behavior of π-submaximal subgroups under homomorphisms

We construct examples that show the difference in behavior of π-maximal and π-submaximal subgroups under group homomorphisms.

Stability of Lie group homomorphisms and Lie subgroups

We discuss a Moser type argument to show when a deformation of a Lie group homomorphism and of a Lie subgroup is trivial. For compact groups we obtain stability results

Imprints of the underlying structure of physical theories

In the context of scientific realism, this paper intends to provide a formal and accurate description of the structural-based ontology posited by classical mechanics, quantum mechanics and special relativity, which is preserved across the empirical domains of these theories and explain their successful predictions. Along the lines of ontic structural realism, such a description is undertaken by a particular ontological commitment: the belief in the existence of a freestanding actual structure, approximately represented by a subgroup of the Inhomogeneous Symplectic Group (up to group homomorphisms), and their corresponding state-space representations. Accordingly, the hierarchy and the complexity of this group-theoretical structure is represented by appropriate philosophical tools, namely, by the language of partial structures. Upon this approach, the lack of knowledge of some relations that hold at the boundary between mathematics and physics, and the presence of surplus structure within the structural edifice are explored and represented. The conclusive issue appeals to an interesting example of a surplus but fruitful structure, where superposition of states with different mass are suggested to be actual relativistic remnants within non-relativistic quantum mechanics, as opposed to the standard interpretation in which they are empirically meaningless.

AUTOMATIC CONTINUITY OF ABSTRACT HOMOMORPHISMS BETWEEN LOCALLY COMPACT AND POLISH GROUPS

We prove results about automatic continuity and openness of abstract surjective group homomorphisms \( K\overset{\varphi }{\to }G, \) where G and K belong to a certain class К of topological groups, and where the kernel of φ satisies a certain topological countability condition. Our results apply in particular to the case where G is a semisimple Lie group or a semisimple compact group, and where К is either the class of all locally compact groups or the class of all Polish groups.

On Fully Idempotent Homomorphisms of Abelian Groups

We provide some examples of irregular fully idempotent homomorphisms and study the pairs of abelian groups A and B for which the homomorphism group Hom(A, B) is fully idempotent. We show that if B is a torsion group or a mixed split group and if at least one of the groups A or B is divisible then the full idempotence of the homomorphism group implies its regularity. If at least one of the groups A or B is a reduced torsion-free group and their homomorphism groups are nonzero then the group is not fully idempotent. The study of fully idempotent groups Hom(A, A) comes down to reduced mixed groups A with dense elementary torsion part.

On fully idempotent homomorphisms of Abelian groups

On fully idempotent homomorphisms of Abelian groups

Representations of the Kauffman bracket skein algebra III: closed surfaces and naturality

This is the third article in the series begun with [8, 10], devoted to finite-dimensional representations of the Kauffman bracket skein algebra of an oriented surface S. In [8] we associated a classical shadow to an irreducible representation \rho of the skein algebra, which is a character represented by a group homomorphism \pi_1(S) \to \mathrm {SL}_2(\mathbb C) . The main result of the current article is that, when the surface S is closed, every character r\in \mathcal R(S) occurs as the classical shadow of an irreducible representation of the Kauffman bracket skein algebra. We also prove that the construction used in our proof is natural, and associates to each group homomorphism r\colon \pi_1(S) \to \mathrm {SL}(\mathbb C) a representation of the skein algebra \mathcal S^A(S) that is uniquely determined up to isomorphism.