S. Anis

COMSATS University Islamabad Abbottabad Campus · Mathematics

In this paper, we introduce cubic soft matrices and define some new operations on these matrices. Moreover, we develop a cubic soft decision-making model based on the cubic soft matrices and apply it to a decision-making problem.

A complex fuzzy distance measure (CFDMs) plays a significant role in applications involving complex or high-dimensional data where traditional distance measures may not adequately capture the nuances of the data relationships. The significance of CFDMs lies in their ability to handle uncertainty, imprecision, and complexity in various domains. Nume...

In this paper, we introduce some set-theoretic operations and laws of the IV-CFSSs, such as interval-valued complex fuzzy soft complement, union, intersection, t-norm, s-norm, simple product, Cartesian product, probabilistic sum, simple difference, and the convex linear sum of min and max operators. We define the distance measure of two IV-CFSSs. T...

A fuzzy soft matrix is a type of mathematical matrix that combines the principles of fuzzy set theory and soft set theory. It is used to handle uncertainty and vagueness in decision-making problems. Fuzzy soft matrix theory cannot handle negative information. To overcome this difficulty, we define the notion of bipolar fuzzy soft (BFS) matrices and...

The concept of complex fuzzy set (CFS) and complex intuitionistic fuzzy set (CIFS) is two recent developments in the field of fuzzy set theory. The significance of these concepts lies in the fact that these concepts assigned membership grades from the unit circle in the plane, i.e., in the form of a complex number instead of [0,1] interval. CFS can...

In this paper, we discuss the further development of the theory of complex fuzzy sets (CFSs). The motivation for this extension is the utility of complex-valued function in membership grade which can express the two-dimensional ambiguous information that is prevalent in time-periodic phenomena. We introduce partial order relation on complex fuzzy s...

In this paper, we established some new operations and formulas of set theory for complex fuzzy sets (CFSs). We introduced the basic results of CFSs with their examples using union, intersection, complement, dot product, complex fuzzy probalistic sum, complex fuzzy bold sum, complex fuzzy bold sum over associative law of union, etc. Moreover, we int...

We characterize the intra-regular AG-groupoids in terms of their (∈, ∈ ∨q k)-fuzzy ideals. Mathematics subject class Classification: (2000) 20M10 and 20N99

In this paper, we have established some new operations and results of set theory for complex fuzzy sets (CFSs). Moreover, we introduced the basic results of complex fuzzy sets with their examples such as complex fuzzy union, complex fuzzy intersection, complex fuzzy complement, complex fuzzy Cartesian product, complex fuzzy bounded sum, complex fuz...

In this paper, we introduce the concept of complex neutrosophic soft matrices. We define some basic operations including complement, union, and intersection on these matrices. We extend the concept of complex neutrosophic soft sets to complex neutrosophic soft matrices and prove related properties. Moreover, we develop an algorithm using complex ne...

In a wide spectrum of mathematical issues, the presence of a fixed point (FP) is equal to the presence of a appropriate map solution. Thus in several fields of math and science, the presence of a fixed point is important. Furthermore, an interesting field of mathematics has been the study of the existence and uniqueness of common fixed point (CFP)...

In this paper, we studied the action of Picard modular group on the biquadratic field. We found patern of algebraic integers formed by this action.

In this paper, we introduce complex fuzzy soft matrices and define some new operations on these matrices. Moreover we develop an algorithm using complex fuzzy soft matrices and apply it to a decision making problem in signal processing.

In this paper, we studied the action of Picard modular group \(PSL(2, \mathbb {Z} [i])\) denoted by \(\Gamma \) on the biquadratic field \( \mathbb {Q} \left( i,\sqrt{3}\right) \). We found patern of algebraic integers formed by this action. To prove results we used coset diagrams for the action of \(\Gamma \) on \( \mathbb {Q} \left( i,\sqrt{3}\ri...

We introduced a new nonassociative algebra, namely, left almost algebra, and discussed some of its genetic properties. We discussed the relation of this algebra with flexible algebra, Jordan algebra, and generalized Jordan algebra.

The notions of hybrid subsemigroups and hybrid left (resp., right) ideals in semigroups are introduced, and several properties are investigated. Using these notions, characterizations of subsemigroups and left (resp., right) ideals are discussed. The concept of hybrid product is also introduced, and characterizations of hybrid subsemigroups and hyb...

It is common knowledge that various models with their limited boundaries of truth and falsehood are not sufficient to detect the reality so there is a need to discover other systems which are able to address the daily life problems. In every branch of science problems arise which abound with uncertainties and imprecisions. Some of these problems ar...

This book consists of seven chapters. In chapter one we introduced neutrosophic ideals (bi, quasi, interior, (m,n) ideals) and discussed the properties of these ideals. Moreover, we characterized regular and intra-regular AG-groupoids using these ideals. In chapter two we introduced neutrosophic minimal ideals in AG-groupoids and discussed several...

In this paper coset diagrams, propounded by Higman, are used to investigate the behavior of elements as words in orbits of the action of the Picard group Γ=PSL(2,ℤ[i]) on . Graphical interpretation of amalgamation of the components of Γ is also given. Some elements of and their conjugates over ℚ(i) have different signs in the orbits of the biquadra...

This paper aims to apply fuzzy sets and soft sets in combination to investigate algebraic properties of regular AG-groupoids. We initiate (∈γ, ∈γ Vqδ)-fuzzy soft left ideals (right ideals, biideals and quasiideals) over AG-groupoids and explore some related properties. Moreover, we give a number of characterizations for regular AG-groupoids by virt...

p>An AG-groupoid is an algebraic structure that lies in between a groupoid and a commutative semigroup. It has many characteristics similar to that of a commutative semigroup. If we consider x2y2= y2x2, which holds for all x, y in a commutative semigroup, on the other hand one can easily see that it holds in an AG-groupoid with left identity e and...

In this paper, we have study the concept of (∈,∈ ∨qk)-fuzzy ideals in an AG-groupoids. We characterize the intra-regular AG-groupoids in terms of their (∈,∈ ∨qk)-fuzzy ideals.

In this paper, we introduce the concepts of generalized cubic soft sets, generalized cubic soft AG-subgroupoids and generalized cubic soft left (resp., right) ideals to study the algebraic structures and properties of AG-groupoids. We also give some examples of generalized cubic soft AG-subgroupoids and generalized cubic soft left (resp., right) id...

In this paper, we have study the concept of (∈, ∈ ∨q k)-fuzzy ideals in an AG-groupoids. We characterize the intra-regular AG-groupoids in terms of their (∈, ∈ ∨q k)-fuzzy ideals.

In this paper, we have study the concept of (∈, ∈ ∨q k)-fuzzy ideals in an AG-groupoids. We characterize the intra-regular AG-groupoids in terms of their (∈, ∈ ∨q k)-fuzzy ideals.

In this paper, we have study the concept of (∈, ∈ ∨q k)-fuzzy ideals in an AG-groupoids. We characterize the intra-regular AG-groupoids in terms of their (∈, ∈ ∨q k)-fuzzy ideals.

Let S be an inverse AG-groupoid (Abel-Grassmann groupoid) and define a relation gamma on S by a gamma b if and only if there exist some positive integers n and m such that b(m) is an element of (Sa) S and a(n) is an element of (Sb)S. We prove that S/gamma is a maximal semilattice homomorphic image of S. Thus, every inverse AG-groupoid S is uniquely...

In this study we introduce a new class of a non-associative algebraic structure namely intra-regular Abel Grassmann's groupoid (AG-groupoid in short). We apply generalized fuzzy ideal theory to this class and discuss its related properties. We introduce,(∈,∈ ∨ qk)-fuzzy semiprime ideals in AG-groupoids and characterize it. Specifically we have char...

In this paper, we investigate some characterizations of regular and intraregular Abel-Grassmann's groupoids in terms of (qk)-fuzzy ideals and (qk)-fuzzy quasi-ideals.

In this study, we have introduced the notion of γ -fuzzification in γ -AG-groupoids which is in fact the generalization of fuzzy AG-groupoids. We have studied several properties of an intra-regular γ -Aγ *-groupoids in terms of fuzzy γ -left (right, two-sided, quasi, interior, generalized bi-, bi-) ideals. We have proved that all fuzzy γ -ideals co...

We first give some useful characterizations of regular semigroups and right weakly regular semigroups by the properties of their bi-ideals, interior ideals, left ideals and right ideals. Based on these characterizations, we also characterize regular semigroups and right weakly regular semigroups by the properties of their (is an element of, is an e...

We first give some useful characterizations of regular semigroups and right weakly regular semigroups by the properties of their bi-ideals, interior ideals, left ideals and right ideals. Based on these characterizations, we also characterize regular semigroups and right weakly regular semigroups by the properties of their (εε Vqk)-fuzzy (generalize...

In this paper, we have decomposed an AG-groupoid. Let S be an AG-groupoid with left identity and a relation γ be defined on S as: aγb if and only if there exist positive integers m and n such that b m ∈ (Sa)S and a n ∈ (Sb)S for all a and b in S. We have proved that S/γ is a maximal separative semilattice homomorphic image of S. Every AG-groupoid S...

We have introduced a new nonassociative class of Abel-Grassmann's groupoid, namely, intraregular and characterized it in terms of its $(\in ,\in {\vee }_{q})$-fuzzy quasi-ideals.

The action of a subgroup G1 = 〈A, C, D: A 3 = C 3 = D 2 = (AC) 2 = (AD) 2 = 1〉 of the Picard group on biquadratic field ℚ(i,√2) gives some interesting and useful results. In this paper, we have find the exact number of ambiguous number in a closed path in the orbit αG 1, by using coset diagram, where α is ambiguous. We can find the closed path of a...

The Picard group {\Gamma} is PSL(2,Z[i]). We have defined coset diagram for the Picard group. It has been observed that some elements of Q(i,/surd3) of the form ((a+b/surd3)/c) and their conjugates ((a-b/surd3)/c) over \mathbb{Q}(i) have different signs in the coset diagram for the action of {\Gamma} on the biquadratic field Q(i,/surd3), these are...

In this paper, we introduce the concept of (∈, ∈ ∨q)-fuzzy ideals in an ordered AG-groupoid. We will also characterize a left regular ordered AG-groupoid by using its (∈, ∈ ∨q)-fuzzy ideals.

In this paper, we introduce the concept of (∈, ∈ ∨q)-fuzzy ideals in an ordered AG-groupoid. We will also characterize a left regular ordered AG-groupoid by using its (∈, ∈ ∨q)-fuzzy ideals.

In this paper, we introduce the concept of (∈, ∈ ∨q)-fuzzy ideals in an ordered AG-groupoid. We will also characterize a left regular ordered AG-groupoid by using its (∈, ∈ ∨q)-fuzzy ideals.

In this paper, we introduce the concept of (∈, ∈ ∨q)-fuzzy ideals in an ordered AG-groupoid. We will also characterize a left regular ordered AG-groupoid by using its (∈, ∈ ∨q)-fuzzy ideals.

In this paper, we introduce the concept of (∈, ∈ ∨q)-fuzzy ideals in an ordered AG-groupoid. We will also characterize a left regular ordered AG-groupoid by using its (∈, ∈ ∨q)-fuzzy ideals.

In this paper, we have studied action of a subgroup G1 =<A,C,D: A3 = C3 = D2 = (AC)2 = (AD)2 = 1> of the Picard group on biquadratic field ℚ(i,√2) by using coset diagrams. We have studied ambiguous numbers in the coset diagram and investigated the path it forms and also found how many paths are present in one orbit. In this way we can find the numb...

In this paper,The concept of a (∈, ∈ ∨q)-fuzzy ideal in an ordered Abel-Grassmann groupoid is introduced. In partcicular, left regular ordered Abel-Grassmann groupoids are characterized by using the (∈ , ∈ ∨q)-fuzzy ideals.

In this paper,The concept of a (∈, ∈ ∨q)-fuzzy ideal in an ordered Abel-Grassmann groupoid is introduced. In partcicular, left regular ordered Abel-Grassmann groupoids are characterized by using the (∈ , ∈ ∨q)-fuzzy ideals.

In this paper,The concept of a (∈, ∈ ∨q)-fuzzy ideal in an ordered Abel-Grassmann groupoid is introduced. In partcicular, left regular ordered Abel-Grassmann groupoids are characterized by using the (∈ , ∈ ∨q)-fuzzy ideals.

In this paper,The concept of a (∈, ∈ ∨q)-fuzzy ideal in an ordered Abel-Grassmann groupoid is introduced. In partcicular, left regular ordered Abel-Grassmann groupoids are characterized by using the (∈ , ∈ ∨q)-fuzzy ideals.

In this paper,The concept of a (∈, ∈ ∨q)-fuzzy ideal in an ordered Abel-Grassmann groupoid is introduced. In partcicular, left regular ordered Abel-Grassmann groupoids are characterized by using the (∈ , ∈ ∨q)-fuzzy ideals.

In this paper,The concept of a (∈, ∈ ∨q)-fuzzy ideal in an ordered Abel-Grassmann groupoid is introduced. In partcicular, left regular ordered Abel-Grassmann groupoids are characterized by using the (∈ , ∈ ∨q)-fuzzy ideals.

In this paper,The concept of a (∈, ∈ ∨q)-fuzzy ideal in an ordered Abel-Grassmann groupoid is introduced. In partcicular, left regular ordered Abel-Grassmann groupoids are characterized by using the (∈ , ∈ ∨q)-fuzzy ideals.

In this paper,The concept of a (∈, ∈ ∨q)-fuzzy ideal in an ordered Abel-Grassmann groupoid is introduced. In partcicular, left regular ordered Abel-Grassmann groupoids are characterized by using the (∈ , ∈ ∨q)-fuzzy ideals.

In this paper,The concept of a (∈, ∈ ∨q)-fuzzy ideal in an ordered Abel-Grassmann groupoid is introduced. In partcicular, left regular ordered Abel-Grassmann groupoids are characterized by using the (∈ , ∈ ∨q)-fuzzy ideals.

In this paper,The concept of a (∈, ∈ ∨q)-fuzzy ideal in an ordered Abel-Grassmann groupoid is introduced. In partcicular, left regular ordered Abel-Grassmann groupoids are characterized by using the (∈ , ∈ ∨q)-fuzzy ideals.

In this paper,The concept of a (∈, ∈ ∨q)-fuzzy ideal in an ordered Abel-Grassmann groupoid is introduced. In partcicular, left regular ordered Abel-Grassmann groupoids are characterized by using the (∈ , ∈ ∨q)-fuzzy ideals.

Coset diagrams defined for the transitive actions of PSL(2, Z) on projective line over a Galois field Fp, PL(Fp), where p is prime, are used to obtain a formula for finding the number of subgroups of index p + 1 of the modular group PSL(2, Z). Some intransitive actions of PSL(2, Z) on PL(Fp) for some special values of θ, when θ ∈ Fp, are also studi...