Search

We say that a signed graph is k $k$‐critical if it is not k $k$‐colorable but every one of its proper subgraphs is k $k$‐colorable. Using the definition of colorability due to Naserasr, Wang, and Zhu that extends the notion of circular colorability, we prove that every 3‐critical signed graph on n $n$ vertices has at least 3n−12 $\frac{3n-1}{2}$ edges, and that this bound is asymptotically tight. It follows that every signed planar or projective‐planar graph of girth at least 6 is (circular) 3‐colorable, and for the projective‐planar case, this girth condition is best possible. To prove our main result, we reformulate it in terms of the existence of a homomorphism to the signed graph C3* ${C}_{3}^{* }$, which is the positive triangle augmented with a negative loop on each vertex. [ABSTRACT FROM AUTHOR]

An essential and quite different class of functions is the triangular norm and its related operators (uninorms, nullnorms, and associative copulas). They are used widely in many disciplines, including fuzzy set theory, probability and statistics, decision sciences, and others. This paper proposes the notion of hybrid Ξ -norm and defines the concepts of Ξ -hybrid ideals, Ξ -hybrid h -ideals in a hemiring . Some equivalent conditions are obtained for a hybrid structure to be a Ξ -hybrid left h -ideal, and it is proved that every imaginable Ξ -hybrid left h -ideal of hemiring is a hybrid left h -ideal, but not conversely by an example. [ABSTRACT FROM AUTHOR]

Fuzzy graphs find numerous applications in real life. One of the extensions of fuzzy graphs is Fermatean fuzzy graphs. Here, we introduce the concepts of Fermatean fuzzy set to the domain of graph theory and obtain a novel type of graph, referred to as Fermatean fuzzy graph (FFG ). The article establishes fundamental terms such as strong FFG , complete FFG , regular FFG , path, degree, total degree, homomorphism, and isomorphism of FFG , as well as the complement of FFG . Alongside the introduction of these concepts, their important properties, theorems, and illustrative examples are defined and discussed. Finally, an application has surfaced suggesting the utilization of FFG s to scrutinize the key factors affecting the productivity of Tata Motors, paving the way for a comprehensive analysis of the company's operational efficiency using score function. [ABSTRACT FROM AUTHOR]

Subgraph densities have been defined, and served as basic tools, both in the case of graphons (limits of dense graph sequences) and graphings (limits of bounded-degree graph sequences). While limit objects have been described for the "middle ranges", the notion of subgraph densities in these limit objects remains elusive. We define subgraph densities in the orthogonality graphs on the unit spheres in dimension d , under an appropriate sparsity condition on the subgraphs. These orthogonality graphs exhibit the main difficulties of defining subgraphs in the "middle" range, and so we expect their study to serve as a key example to defining subgraph densities in more general Markov spaces. Interest in studying homomorphisms of a finite graph G into orthogonality graphs is supported by the fact that such homomorphisms are just the orthonormal representations of the complementary graph. [ABSTRACT FROM AUTHOR]

In this paper, the concept of circular r $r$‐flow in a mono‐directed signed graph (G,σ) $(G,\sigma)$ is introduced. That is a pair (D,f) $(D,f)$, where D $D$ is an orientation on G $G$ and f:E(G)→(−r,r) $f:E(G)\to (-r,r)$ satisfies that ∣f(e)∣∈[1,r−1] $| f(e)| \in [1,r-1]$ for each positive edge e $e$ and ∣f(e)∣∈[0,r2−1]∪[r2+1,r) $| f(e)| \in [0,\frac{r}{2}-1]\cup [\frac{r}{2}+1,r)$ for each negative edge e $e$, and the total in‐flow equals the total out‐flow at each vertex. This is the dual notion of circular colorings of signed graphs and is distinct from the concept of circular flows in bi‐directed graphs associated with signed graphs studied in the literature. We first explore the connection between circular 2kk−1 $\frac{2k}{k-1}$‐flows and modulo k $k$‐orientations in signed graphs. Then we focus on the upper bounds for the circular flow indices of signed graphs in terms of the edge‐connectivity, where the circular flow index of a signed graph is the minimum value r $r$ such that it admits a circular r $r$‐flow. We prove that every 3‐edge‐connected signed graph admits a circular 6‐flow and every 4‐edge‐connected signed graph admits a circular 4‐flow. More generally, for k≥2 $k\ge 2$, we show that every (3k−1) $(3k-1)$‐edge‐connected signed graph admits a circular 2kk−1 $\frac{2k}{k-1}$‐flow, every 3k $3k$‐edge‐connected signed graph has a circular r $r$‐flow with r<2kk−1 $r\lt \frac{2k}{k-1}$, and every (3k+1) $(3k+1)$‐edge‐connected signed graph admits a circular 4k+22k−1 $\frac{4k+2}{2k-1}$‐flow. Moreover, the (6k−2) $(6k-2)$‐edge‐connectivity condition is shown to be sufficient for a signed Eulerian graph to admit a circular 4k2k−1 $\frac{4k}{2k-1}$‐flow, and applying this result to planar graphs, we conclude that every signed bipartite planar graph of negative girth at least 6k−2 $6k-2$ admits a homomorphism to the negative even cycles C−2k ${C}_{-2k}$. [ABSTRACT FROM AUTHOR]

The generic homomorphism problem, which asks whether an input graph \(G\) admits a homomorphism into a fixed target graph \(H\) , has been widely studied in the literature. In this article, we provide a fine-grained complexity classification of the running time of the homomorphism problem with respect to the clique-width of \(G\) (denoted \({\operatorname{cw}}\)) for virtually all choices of \(H\) under the Strong Exponential Time Hypothesis. In particular, we identify a property of \(H\) called the signature number \(s(H)\) and show that for each \(H\) , the homomorphism problem can be solved in time \(\mathcal{O^{*}}(s(H)^{{\operatorname{cw}}})\). Crucially, we then show that this algorithm can be used to obtain essentially tight upper bounds. Specifically, we provide a reduction that yields matching lower bounds for each \(H\) that is either a projective core or a graph admitting a factorization with additional properties—allowing us to cover all possible target graphs under long-standing conjectures. [ABSTRACT FROM AUTHOR]

In this paper, we give another definition of Ruston elements and almost Ruston elements, which is equivalent to the definitions given by Mouton and Raubenheimer in the case that the homomorphism has a closed range and Riesz property. For two homomorphisms, we consider the preserver problems of Fredholm theory and Fredholm spectrum theory. In addition, we study the spectral mapping theorems of Fredholm (Weyl, Browder, Ruston, and almost Ruston) elements relative to a homomorphism. Last but not least, the dependence of Fredholm theory on three homomorphisms is considered, and meanwhile, the transitivity of Fredholm theory relative to three homomorphisms is illustrated. Furthermore, we consider the Fredholm theory relative to more homomorphisms. [ABSTRACT FROM AUTHOR]

In this paper, we discuss an important and nontrivial theorem on evaluation homomorphisms. We state this theorem as a canonical duality between the family of all smooth mappings f ∈ Hom(M,M′) of a smooth real finite-dimensional manifold M into a similar manifold M′ and the family of homomorphisms φ of the algebra C∞ (M′) of smooth scalar-valued functions on M′ into the analogous algebra C∞ (M) on M, φ ∈ Hom (C∞ (M′), C∞ (M)). This formulation possesses the maximum natural generality and, at the same time, allows it to be used in applications in the standard canonical form. [ABSTRACT FROM AUTHOR]

In this research paper, our aim is to introduce the new concept of neutrosophic doubt fuzzy bi-ideal of BSalgebras as an extension of doubt fuzzy bi-ideal of BS-algebras and investigated its algebraic nature. Neutrosophic doubt fuzzy bi-ideal of BS-algebras is also applied in Cartesian product. Finally, we also provide the homomorphic behaviour of Neutrosophic doubt fuzzy bi-ideal of BS-algebras. [ABSTRACT FROM AUTHOR]

We show that for every integer m≥2$$ m\ge 2 $$ and large n$$ n $$, every properly edge‐colored graph on n$$ n $$ vertices with at least n(logn)53$$ n{\left(\log n\right)}^{53} $$ edges contains a rainbow subdivision of Km$$ {K}_m $$. This is sharp up to a polylogarithmic factor. Our proof method exploits the connection between the mixing time of random walks and expansion in graphs. [ABSTRACT FROM AUTHOR]

Copyright of Journal of Jilin University (Science Edition) / Jilin Daxue Xuebao (Lixue Ban) is the property of Zhongguo Xue shu qi Kan (Guang Pan Ban) Dian zi Za zhi She and its content may not be copied or emailed to multiple sites or posted to a listserv without the copyright holder's express written permission. However, users may print, download, or email articles for individual use. This abstract may be abridged. No warranty is given about the accuracy of the copy. Users should refer to the original published version of the material for the full abstract. (Copyright applies to all Abstracts.)

An attribute-based authority system helps in providing file/system access to only authorized users. However, they face specific challenges in securing the dataset from attacks. Hence, a novel hybrid Chimp Optimization -based Elliptical Curve Cryptosystem Model (CbECCM) was designed in this article. The main objective of this model is to overcome these demerits of the existing system and to provide file access to the authorized user. The chimp fitness in the user authorization module improves the system's confidentiality. The presented work was validated with a crop-recommendation dataset, and the outcomes are estimated. Moreover, the homomorphism algorithm increases the data integrity and reduces the time complexity. Finally, the robustness of the developed model was checked by launching the Denial of Service (DoS) attack on the cloud servers. The performance of the designed model is estimated and validated with a comparative analysis. The performance analysis shows that the developed model attained a high confidential rate of 100%. The comparative analysis proves that the developed model achieved better outcomes than others. [ABSTRACT FROM AUTHOR]

The present study investigates the relation between derivations and hyperideals on ordered hyperrings with no zero divisors. Also, we identify some results for the ordered hyperrings induced by the homomorphism of the ordered hyperrings by derivations. The present work explores some aspects of derivations in ordered hyperrings. Also, we establish some results in connection with homomorphisms and hyperideals. Furthermore, we describe prime hyperideals associated to a derivation d on an ordered hyperring T and derive several results about homomorphisms and derivations on ordered hyperrings. [ABSTRACT FROM AUTHOR]

In this paper, we introduce a c-prime fuzzy graph of a nearring with respect to a level ideal of a fuzzy ideal. We find a relation between properties of the fuzzy ideal and properties of the fuzzy graph. We introduce ideal symmetry of the fuzzy graph and obtain conditions under which the graph is ideal symmetric. We find a relation between nearring homomorphisms and graph homomorphisms. We investigate conditions required for the homomorphic image of a c-prime fuzzy ideal to be a c-prime fuzzy ideal. [ABSTRACT FROM AUTHOR]

In this research paper, our aim is to introduce the new concept of neutrosophic doubt fuzzy bi-ideal of BS-algebras as an extension of doubt fuzzy bi-ideal of BS-algebras and investigated its algebraic nature. Neutrosophic doubt fuzzy bi-ideal of BS-algebras is also applied in Cartesian product. Finally, we also provide the homomorphic behaviour of Neutrosophic doubt fuzzy bi-ideal of BS-algebras.

This article has two main parts. In the first part we show that some of the basic theory of generalized polynomials on commutative semi-groups can be extended to all semigroups. In the second part we show that if a sub-semigroup S of a group G generates G in the sense that G = S · S−1, then a generalized polynomial on S with values in an Abelian group H can be extended to a generalized polynomial on G into H. Finally there is a short discussion of the extendability of exponential functions and generalized exponential polynomials.

Neutosophic vague subbisemirings (NSVSBS) are discussed here, as well as their level sets. Subbisemirings are a generalization of bisemirings, and NSVSBSs are a generalization of sub-bisemirings. A number of illustrative examples are provided to illustrate the theory for (ξ, τ )-NSVSBS over bisemiring theory. Following is an outline of the preliminary definitions and results presented in Section 2. The concept of a NSVSBS is introduced in Section 3.

Word-representable graphs, which are the same as semi-transitively orientable graphs, generalize several fundamental classes of graphs. In this paper we propose a novel approach to study word-representability of graphs using a technique of homomorphisms. As a proof of concept, we apply our method to show word-representability of the simplified graph of overlapping permutations that we introduce in this paper. For another application, we obtain results on word-representability of certain subgraphs of simplified de Bruijn graphs that were introduced recently by Petyuk and studied in the context of word-representability. [ABSTRACT FROM AUTHOR]

The aim of this sequence of work is to investigate polynomial equations satisfied by additive functions. As a result of this, new characterization theorems for homomorphisms and derivations can be given. More exactly, in this paper the following type of equation is considered ∑ i = 1 n f i (x p i) g i (x q i) = 0 x ∈ F , where n is a positive integer, F ⊂ C is a field, f i , g i : F → C are additive functions and p i , q i are positive integers for all i = 1 , ... , n . [ABSTRACT FROM AUTHOR]

Isomorphism and homomorphism are fundamental topics in abstract algebra, but how mathematicians characterise the utility of these concepts remains understudied. Based on interviews with nine research-active mathematicians, we confirmed use of previously noted conceptual metaphors and noted new metaphors and metaphor clusters. We expand on prior work by highlighting five thematic contrasts between metaphors for these concepts: context dependent sameness, classification of objects versus computing maps, indistinguishable versus distinct objects, structure-preservation versus structure loss, and information gain versus structure loss. Implications include the centrality of sameness to thinking about isomorphism and homomorphism but the need for care in how that sameness is framed; the value of being able to attend to the object and map; the importance of being able to determine which structures are preserved and which are lost in context; and the existence of a conceptual purpose for homomorphism as a tool for changing perspectives. [ABSTRACT FROM AUTHOR]

In this paper, the group homomorphism graph is introduced and investigated. The group homomorphism graph, denoted by HI(G,G′), is an undirected graph in which the vertex set contains all homomorphisms excluding the monomorphisms and the zero homomorphism from the group G to the group G′, and two distinct vertices are adjacent if and only if the intersection of their kernels is non-trivial. We investigate the interplay between the graph-theoretic properties of this graph with algebraic properties of the groups. In this work, connectedness, diameter, clique number, chromatic number, domination number, independence number, etc. are found. [ABSTRACT FROM AUTHOR]

This paper introduces the idea of a neutrosophic vague subbisemiring (NSVSBS), level sets of NSVSBS, and (p; ƍ)-neutrosophic vague subbisemiring ((p;ƍ)-NSVSBS) of a bisemiring. NSVSBSs are generalizations of neutrosophic subbisemirings and SBS based on bisemirings. Let - be a neutrosophic vague subset in B, we show that V=([Tʌ-, T+ ]ʌ [Iʌʌ ʌ I+ ]ʌ [F&#652ʌ ʌ ;F+ ʌ ]) is a NSVSBS of B if and only if all non empty level set V(t1,t2,s) is a SBS of B for t1, t2, ϵ 2[0,1]. In the case that ʌ is a NSVSBS of a bisemiring B and V is the strongest neutrosophic vague relation of B, we prove that λ is a NSVSBS of B × B. Let Λ be any NSVSBS of B, prove that pseudo neutrosophic vague coset (τΛ)p is a NSVSBS of B, for every τεB. Let τ1;τ2,...,τn be the family of NSV SBSs of B1; B2,...Bn respectively. We show that τ1 × τ2 ×....× τn is a NSVSBS of B1 × B2 &#215,...× Bn. The homomorphic image of every NSVSBS is a NSVSBS. The homomorphic pre-image of every NSVSBS is a NSVSBS. Examples are provided to strengthen our results. [ABSTRACT FROM AUTHOR]

We show that any two Hadamard graphs on the same number of vertices are quantum isomorphic. This follows from a more general recipe for showing quantum isomorphism of graphs arising from certain association schemes. The main result is built from three tools. A remarkable recent result [20] of Mančinska and Roberson shows that graphs G and H are quantum isomorphic if and only if, for any planar graph F , the number of graph homomorphisms from F to G is equal to the number of graph homomorphisms from F to H. A generalization of partition functions called "scaffolds" [23] affords some basic reduction rules such as series-parallel reduction and can be applied to counting homomorphisms. The final tool is the classical theorem of Epifanov showing that any plane graph can be reduced to a single vertex and no edges by extended series-parallel reductions and Delta-Wye transformations. This last sort of transformation is available to us in the case of exactly triply regular association schemes. The paper includes open problems and directions for future research. [ABSTRACT FROM AUTHOR]

Background. The proposed work is devoted to research to clarify the axiomatization of arithmetic in connection with the presence of special assumptions in it. The conceptual problem of analyzing new properties of numbers is formulated under the assumption of the possibility of losing any initial premise. Materials and methods. A solution to the conceptual problem is given, followed by a logical analysis of the meaningful axiomatic theory of natural numbers, which led to the discovery of logical errors. An analysis of logical errors is given based on the meaningful axiomatic theory of natural numbers. It is shown that the properties of an established set of "finite real numbers", or positive real numbers contained in a given segment e, have the property of being discovered depending not only on the "requested numbers" of the actually infinite region, but also on real (positive) numbers infinite half-interval. Results. Conditions have been identified that allow us to establish new properties of numbers. Conclusions. The natural series N begins only with 1, there is no zero in it. It has been proven that modern arithmetic has an incorrect axiomatization and is subject to revision.

Our aim is to explore the idea of neutrosophic N-ideals in near-subtraction semigroups in this article and obtain some outcomes that are equivalent to them. We also illustrate the notion of a neutrosophic κ-intersection. Additionally, in a near-subtraction semigroup, we examine the term homomorphism of a neutrosophic κ-structure and establish some conclusions based on a homomorphic neutrosophic κ-structure preimage of a neutrosophic κ-left (respectively, right) ideal.

This study provides an innovative approach to neutrosophic algebraic structures by introducing a new structure called Neutrosophic Soft Cubic T-ideal (NSCTID), which combines T-ideal (TID) and Neutrosophic Soft Cubic Sets (NSCSs) within the framework of PS-Algebra. Within the already-existing neutrosophic cubic structures, the addition of soft sets with the characteristics of TID makes this structure more desirable. The theoretical development of the proposed structure includes the application of fundamental ideas as union, intersection, the Cartesian product, and homomorphism. We also introduce the notions of NSCTID-translation and NSCTID-multiplication to further enhance the structure of NSCTID. By applying the idea of translation and multiplication, we offer improved algorithm for neutrosophic cubic sets to deal with different parameters that are satisying the TID’s distinctive characteristics. Through this thorough research, we offer an elementary understanding of NSCTID and its capabilities, providing the way to new algebraic structures.

Concepts of generalized commutative intuitionistic fuzzy finite state machines and generalized switching intuitionistic fuzzy finite state machines are proposed in this paper, and properties of these two kinds of intuitionistic fuzzy finite state machines are investigated. By using the conditions associated with generalized commutativity, a kind of congruences of intuitionistic fuzzy finite state machines is defined. Homomorphic properties of generalized commutative intuitionistic fuzzy finite state machine are also discussed. Moreover, products of generalized commutative intuitionistic fuzzy finite state machines and products of generalized switching intuitionistic fuzzy finite state machines are studied, and some related properties are proved. [ABSTRACT FROM AUTHOR]

In this paper we consider generalized monomial functions f , g : F → C (of possibly different degree) that also fulfill f (P (x)) = Q (g (x)) x ∈ F , where P ∈ F [ x ] and Q ∈ C [ x ] are given (classical) polynomials. [ABSTRACT FROM AUTHOR]

Bipolar objects are widespread in nature, and their two attributes describe the opposition and unity of the things. Motivated by characterizing fuzzy topological structures by means of fuzzy graphs, we propose a bipolar fuzzy topological graph to measure the features of bipolar fuzzy systems. The topological characteristics in bipolar neutrosophic and bipolar interval-valued fuzzy settings are discussed as well. Several instances are manifested to clarify new definitions and conclusions. Furthermore, the properties of edge calculating and graph isomorphic are determined. [ABSTRACT FROM AUTHOR]

Our aim is to explore the idea of neutrosophic N−ideals in near-subtraction semigroups in this article and obtain some outcomes that are equivalent to them. We also illustrate the notion of a neutrosophic κ− intersection. Additionally, in a near-subtraction semigroup, we examine the term homomorphism of a neutrosophic κ− structure and establish some conclusions based on a homomorphic neutrosophic κ− structure preimage of a neutrosophic κ− left (respectively, right) ideal. [ABSTRACT FROM AUTHOR]

In this paper, we introduce an equiprime fuzzy graph of a nearring with respect to the level ideal of a fuzzy ideal. We interrelate graph theoretical properties of the graph and ideal theoretical properties of nearring. We show that the properties like vertex cut, connectedness of the graph depend on the properties of the fuzzy ideal. We define ideal symmetry of the graph and find conditions for the graph to be ideal symmetric. If the fuzzy ideal is equiprime then we show that the level set induces a fuzzy clique. We find conditions required for the level set to be the vertex cover of the graph. We find interrelation between equiprime fuzzy graph and fuzzy graph of nearring with respect to level ideal. We study properties of the graph under neaaring homomorphism. We prove that the connectedness of the graph in homomorphic image depends on properties of ideal. We obtain conditions required for homomorphic image of an equiprime fuzzy ideal to be an equiprime fuzzy ideal. [ABSTRACT FROM AUTHOR]

The concept of dual B-Almost distributive fuzzy lattice (dual B-AD FL) interms of its principal ideal fuzzy lattice is introduced. We also prove the equivalency of dual B-ADL and dual B-ADFL and The relations of B-ADFL and Dual B-ADFL are investigated. [ABSTRACT FROM AUTHOR]

Let A and B be unital algebras, that need not be abelian, over fields K and K ′ respectively, let α , a , b , c ∈ K , β ∈ K ′ and λ ∈ C . The present work aims to determine the general solution Φ : A → B of the functional equation Φ (x + y + α x y) = Φ (x) + Φ (y) + β Φ (x) Φ (y) , x , y ∈ A , and to describe the solutions Φ : A → M 2 (C) of the functional equation Φ (a x + b y + c x y) = Φ (x) + Φ (y) + λ Φ (x) Φ (y) , x , y ∈ A. We also show that, when (A , ·) (as a semigroup) is commutative and regular (for instance when dim A = 1 ), the explicit forms of the solutions of the last equation can be given. [ABSTRACT FROM AUTHOR]

A homomorphism of a signed graph (G,σ) $(G,\sigma)$ to (H,π) $(H,\pi)$ is a mapping of vertices and edges of G $G$ to (respectively) vertices and edges of H $H$ such that adjacencies, incidences, and the signs of closed walks are preserved. We observe in this work that, for k≥3 $k\ge 3$, the k $k$‐coloring problem of a given graph G $G$ can be captured by homomorphism to (K2k,σm) $({K}_{2k},{\sigma }_{m})$ from a signed bipartite graph that is built from G $G$. Here σm ${\sigma }_{m}$ assigns a negative sign to the edges of a perfect matching and a positive sign to the rest. Motivated by these reformulations and in connection with results on 3‐colorings of planar graphs, such as Grötzsch's theorem, we prove that any signed graph with the maximum average degree strictly less than 145 $\frac{14}{5}$ admits a homomorphism to (K6,σm) $({K}_{6},{\sigma }_{m})$. For k≥4 $k\ge 4$, we show that the maximum average degree being strictly less than 3 would suffice for a signed graph to admit a homomorphism to (K2k,σm) $({K}_{2k},{\sigma }_{m})$. Both of these bounds are tight. We discuss applications of our work to signed planar graphs and its connection to the study of homomorphisms of 2‐edge‐colored graphs. Among a number of interesting questions that are left open, a notable one is a possible extension of Steinberg's conjecture for the class of signed bipartite planar graphs. [ABSTRACT FROM AUTHOR]

Let Λ and Γ be row finite k-graphs without sources. We show that ∗-algebra homomorphisms ϕ: KPC(Λ) → KPC(Γ) extend to ∗-algebra homomorphisms ¯ϕ: C∗(Λ) → C∗(Γ). We also examine necessary and sufficient conditions for algebra homomorphisms between complex Kumjian-Pask algebras KPC(Λ) and KPC(Γ) which are *-preserving. [ABSTRACT FROM AUTHOR]

This study provides an innovative approach to neutrosophic algebraic structures by introducing a new structure called Neutrosophic Soft Cubic T-ideal (NSCTID), which combines T-ideal (TID) and Neutrosophic Soft Cubic Sets (NSCSs) within the framework of PS-Algebra. Within the already-existing neutrosophic cubic structures, the addition of soft sets with the characteristics of TID makes this structure more desirable. The theoretical development of the proposed structure includes the application of fundamental ideas as union, intersection, the Cartesian product, and homomorphism. We also introduce the notions of NSCTID-translation and NSCTID-multiplication to further enhance the structure of NSCTID. By applying the idea of translation and multiplication, we offer improved algorithm for neutrosophic cubic sets to deal with different parameters that are satisying the TID’s distinctive characteristics. Through this thorough research, we offer an elementary understanding of NSCTID and its capabilities, providing the way to new algebraic structures. [ABSTRACT FROM AUTHOR]

In this paper, we give another definition of Ruston elements and almost Ruston elements, which is equivalent to the definitions given by Mouton and Raubenheimer in the case that the homomorphism has a closed range and Riesz property. For two homomorphisms, we consider the preserver problems of Fredholm theory and Fredholm spectrum theory. In addition, we study the spectral mapping theorems of Fredholm (Weyl, Browder, Ruston, and almost Ruston) elements relative to a homomorphism. Last but not least, the dependence of Fredholm theory on three homomorphisms is considered, and meanwhile, the transitivity of Fredholm theory relative to three homomorphisms is illustrated. Furthermore, we consider the Fredholm theory relative to more homomorphisms.

In this paper, we introduce the notion of bipolar-valued neutrosophic subbisemiring (BVNSBS), level sets of BVNSBS, and bipolar valued neutrosophic normal subbisemiring (BVNNSBS) of a bisemiring. The concept of BVNSBS is a new generalization of subbisemiring over bisemirings. We discussed the theory of (ξ, τ )- BVNSBS and (ξ, τ )-BVNNSBS over bisemirings and presented several illustrative examples to demonstrate the sufficiency and validity of the proposed theorems, lemmas, and propositions.

In this paper, we introduce the notion of intuitionistic fuzzy PMS-subalgebras under homomorphism and Cartesian product and investigate several properties. We study the homomorphic image and inverse image of the intuitionistic fuzzy PMS-subalgebras of a PMS-algebra, which are also intuitionistic fuzzy PMS-subalgebras of a PMS-algebra, and find some other interesting results. Furthermore, we also prove that the Cartesian product of intuitionistic fuzzy PMS-subalgebras is again an intuitionistic fuzzy PMS-subalgebra and characterize it in terms of its level sets. Finally, we consider the strongest intuitionistic fuzzy PMS-relations on an intuitionistic fuzzy set in a PMS-algebra and demonstrate that an intuitionistic fuzzy PMS-relation on an intuitionistic fuzzy set in a PMS-algebra is an intuitionistic fuzzy PMS-subalgebra if and only if the corresponding intuitionistic fuzzy set in a PMS-algebra is an intuitionistic fuzzy PMS-subalgebra of a PMS-algebra.

In this paper, we consider equitable oriented colorings of graphs. Such coloring is a natural combination of two well‐known colorings: oriented coloring and equitable coloring. An oriented k $k$‐coloring ϕ $\phi $ of an oriented graph G→ $\overrightarrow{G}$ is an arc‐preserving homomorphism from G→ $\overrightarrow{G}$ into an oriented graph H→ $\overrightarrow{H}$ on k $k$ vertices, or colors. We say that the coloring is equitable if for every two colors c1,c2∈V(H→) ${c}_{1},{c}_{2}\in V(\overrightarrow{H})$, we have ∣∣ϕ−1(c1)∣−∣ϕ−1(c2)∣∣≤1 $| | {\phi }^{-1}({c}_{1})| -| {\phi }^{-1}({c}_{2})| | \le 1$. This paper presents the following results: we show that there is an undirected graph G $G$ such that every orientation of G $G$ is equitably oriented 5‐colorable, and we show that there is an orientation of G $G$ that cannot be colored with six colors. In other words, there is a gap in the set of numbers of colors that can be used for equitable oriented coloring of G $G$. Additionally, we concentrate on paths and show that there is an orientation of a path that requires four colors to be equitably oriented colorable. On the other hand, we show that every orientation of any path is equitably oriented k $k$‐colorable, for every k≥5 $k\ge 5$. [ABSTRACT FROM AUTHOR]

The purpose of the article is to study about double framed soft fuzzy biideals of gamma near-ring R and their some results. We have proved that the uni-int and int-uni of double framed soft fuzzy bi-ideals of gamma near-ring R are also a double framed soft fuzzy bi-ideal of gamma near-ring R. We investigated some related properties of double framed soft fuzzy bi-ideal of R using homomorphism and anti-homomorphism. We establish the relationship between bi-ideal and double framed soft fuzzy bi-ideal of R. In this paper we also discuss about double framed soft fuzzy interior ideal of gamma near-ring R and their some results. [ABSTRACT FROM AUTHOR]

The complex spherical fuzzy graph (CSFG), which extends the concept of a spherical fuzzy graph (SFG), proves to be a more effective means of depicting relationships among diverse objects when these relationships are subject to uncertainty. In addition, the Dombi operators, featuring an adjustable operational parameter, offer valuable utility by accommodating distinct values. In this research paper, we present the concept of a complex spherical dombi fuzzy graph (CSDFG), an extension of a spherical dombi fuzzy graph (SDFG). Dombi operators are utilized as averaging operators, playing a crucial role in aggregating data into a single value for efficient decision-making. We implement Dombi operators on CSFGs. The complement of a CSDFG is defined, and self-complementry in CSDFGs is discussed. We explore homomorphism, isomorphism, weak isomorphism (W-isomorphism), and co-weak isomorphism (CW-isomorphism) to establish relationships between CSDFGs. We define regular, arc regular, and totally arc regular CSDFGs, explain their key properties, and demonstrate an application of CSDFG in decision-making problems.

We extend two standard theorems on groups to gyrogroups: the direct product theorem and the cancellation theorem for direct products. Firstly, we prove that under a certain condition a gyrogroup G can be decomposed as the direct product of two subgyrogroups. Secondly, we prove that finite gyrogroups can be cancelled in direct products: if A≅B, then A×H≅B×K or H×A≅K×B implies H≅K, where A,B,H, and K are finite gyrogroups.