Rough semigroups and rough fuzzy semigroups based on fuzzy ideals
-
Qiumei Wang
Abstract
In this paper, we firstly introduce a special congruence relation U(μ, t) induced by a fuzzy ideal μ in a semigroup S. Then we define the lower and upper approximations based on a fuzzy ideal in semigroups. We can establish rough semigroups, rough ideals, rough prime ideals, rough fuzzy semigroups, rough fuzzy ideals and rough fuzzy prime ideals according to the definitions of rough sets and rough fuzzy sets. Furthermore, we shall consider the relationships among semigroups and rough semigroups, fuzzy semigroups and rough fuzzy semigroups, and some relative properties are also discussed.
1 Introduction
In 1965, the notion of fuzzy sets was introduced by Zadeh [1] and he gave the way to study uncertainty problems. Thereafter, the theory has developed quickly and has been applied in many areas of our real world. Up to now, fuzzy mathematics has remained an important branch of mathematics. Meanwhile, in the early 20th century, some researchers began to research semigroups formally. Since 1950, the investigations of finite semigroups have become rather important in theoretical computer science because there are natural relations between finite semigroups and finite automatons. With the fuzzy mathematics being deeply studied, fuzzy algebras were also investigated. The study of fuzzy algebras is also one of the most important directions in fuzzy mathematics. In particular, a great number of studies on fuzzy groups and fuzzy semigroups were performed. It should be pointed out that the studies on fuzzy semigroups are deeply influencing the development of fuzzy theory. In recent years, a large number of fuzzy algebras have been developed by some scholars, for instances see [2–4]. Especially, Davvaz [5], Dudek [6] and Zhan [7] applied fuzzy theory to n-ary semigroups. Recently, Zhan [8, 9] studied some types of fuzzy k-ideals and fuzzy h-ideals of hemirings, respectively.
In 1982, Pawlak [10] firstly developed a rough set theory. It is rather important to deal with inexact, uncertain, or vague situations. More and more researchers have been attracted to study rough sets and their applications. While mentioning rough sets, we can not neglect the lower and upper approximations. In Pawlak rough sets, the equivalent relations are essential to construct the lower and upper approximations. To deal with imperfect data, rough set theory is a powerful theory [11]. In recent years, Bonikowaski [12], Iwinski [13], and Pomykala [14] have studied algebraic properties of rough sets. The notion of rough subgroups was introduced by Biswas and Nanda [15]. Then, Kuroki [16] put up the notion of rough ideals in semigroups. Also, in [17], Jun investigated the roughness of Γ-subsemigroups and ideals in Γ-semigroups. The notions of rough prime ideals and rough fuzzy prime ideals in semigroups were introduced in [18].
It is obvious that congruence relations in rough sets are rather important. In Pawlak rough sets, congruence relations are necessary in the constructions of the lower and upper approximations. Therefore, if we try to investigate the rough sets, we should define a congruence relation. In recent years, some researchers try to find something to serve as a congruence relation in some algebraic structures. For examples, Kuroki and Wang [19] made use of a normal subgroup of groups as a congruence relation to discuss the properties of the lower and upper approximations. In [20], Davvaz proved that an ideal of rings could be regarded as a congruence relation. Further, Davvaz studied rough subrings and rough ideals in rings. Especially, in [21] Davvaz constructed the t-level relation based on a fuzzy ideal μ and he proved that U(μ; t) is a congruence relation in rings.
Based on the above idea, in this paper we will study the properties of rough subsemigroups and rough fuzzy subsemigroups of semigroups. We construct the t-level relation U(μ; t) of a fuzzy ideal μ of semigroups and prove that U(μ; t) is a congruence relation. We divide this paper into four parts. In section 2, we recall some basic results. In section 3, based on this congruence relation, we construct rough subsemigroup and rough (prime) ideals of semigroups. Some examples are also presented. Finally, we investigate rough fuzzy semigroups of semigroups based on this congruence relation in section 4.
2 Preliminaries
In this section we will recall some basic definitions that can be used in this paper.
A semigroup S is a nonempty set with a binary operation “*” such that
a * b ∈ S, for all a, b ∈ S.
(a * b)*c = a*(b * c), for all a, b, c ∈ S.
([18]). Let S be a semigroup, a subset A of S is called a subsemigroup of S if A is closed under “*” and SA ⊆ A(AS ⊆ A); If A is not only a left ideal but also is a right ideal, it is a bi-ideal; A is called a prime ideal if A is ideal, and if a * b ∈ A, we can get a ∈ A or b ∈ A, for all a, b ∈ A.
([18]). A fuzzy set μ of S is called a fuzzy subsemigroup of S if it satisfies μ(x * y) ≥ μ(x)∧ μ(y); μis called a fuzzy let (right) ideal if it satisfies (1) μ(x * y) ≥ μ(x)∧ μ(y)(2)μ(x * y) ≥ μ(y)(μ(x * y) ≥ μ(x)), for all x, y ∈ S; μ is called a fuzzy ideal if it is not only a fuzzy let ideal but also is a fuzzy right ideal; μ is called a fuzzy prime ideal if μ(x * y) = μ(x)orμ(x * y) = μ(y) ,for all x,y ∈ S.
([3]). Let R be an equivalence relation in a set A, if R is a binary relation and it satisfies
Reflexive:for any a ∈ A,(a, a) ∈ R.
Symmetry:for all a,b ∈ A,(a, b) ∈ R, then (b, a) ∈ R.
Transitivity:for all a, b, c ∈ A,(a, b) ∈ R,(b, c) ∈ R, then (a, c) ∈ R.
An equivalence relation R is called a congruence relation if (a, b) ∈ R, then (ax, bx) ∈ Rand (xa, xb) ∈ R, for all x ∈ A.
3 Rough semigroups based on fuzzy ideals
In the section, we introduce a new congruence relation of semigroups. Referring to [18], we study rough semigroups and, at the same time, we provide some examples. Throughout this paper, S is a semigroup.
Let μ be a fuzzy ideal of S. For each t ∈ [0, μ(0)], the set U(μ, t) = {(x, y) ∈ × S|(μ(x) ∧ μ(y)) ∨ I dS(x, y) ≥ t} is called a t-level relation of μ.
For I ds(x, y), we know x = y, then I ds(x, y) = 1; x ≠ y, then I ds(x, y) = 0. Next, we prove U(μ, t)is a congruence relation.
Let μ be a fuzzy ideal of S and t ∈ [0, μ(0)].Then U(μ, t) is a congruence relation on S.
First of all, we show U(μ, t) is an equivalence relation.
Reflexive: For any element x ∈ S. (μ(x)∧ μ(x))∨ I ds(x, x) = I ds(x, x) = 1 ≥ t.
Symmetry: Obviously, U(μ, t) is symmetric.
Transitivity: Let (x, y) ∈ U(μ, t) and (y, z) ∈ U(μ, t), and then we have (μ(x) ∧ μ(y)) ∨ I ds(x, y) ≥ t,(μ (y) ∧ μ (z))∨ I ds(y, z) ≥ t. If x = y = z, it is clear that (x, z) ∈ U(μ, t); If x = y ≠ z, then μ(y, z) ≥ t and ((μ(x) ∧ μ(z))∨ I ds(x, z) = μ (x) ∧ μ(z) = μ (y) ∧ μ (z) ≥ t, therefore, (x, z) ∈ U(μ, t); if x ≠ y = z, we have μ(x) ∧ μ (y) ≥ t and (μ (x) ∧ μ(z))∨ I ds(x, z) = μ(x) ∧ μ(z) = μ (x) ∧ μ (y) ≥ t, so (x, z) ∈ U(μ, t); if x ≠ y ≠ z, then we have μ(x) ∧ μ(y) ≥ t, μ(y) ∧ μ (z) ≥ t and (μ(x) ∧ μ(z))∨ I ds(x, z) = μ (x) ∧ μ(z) ≥ μ(x) ∧ μ(y) ∧ μ(z) ≥ t, so (x, z) ∈ U(μ, t) . Conclusion, U(μ, t) is an equivalence relation.
Next, we prove that U(μ, t) is a congruence relation. For (x, y) ∈ U(μ, t), we prove (ax, ay) ∈ U(μ, t) and (xa, ya) ∈ U(μ, t) is also right. In the following, we only prove the former, the latter is the same. In other words, we only prove (μ (ax) ∧ μ(ay))∨ I ds(ax, ay) ≥ t. If ax = ay, clearly, 1 ≥ t. If ax ≠ ay, then x ≠ y, then
Thus, U(μ, t) is a congruence relation.
For any fuzzy ideal μ of S, we know that μ(0) ≥ μ(x) and μ(0) ≤ 1, so when t ∈ [0, μ(0)], the above lemma is proper. We say x is congruent to y model μ, written x ≡ty(mod μ). If for elements x, y ∈ S, t ∈ [0,1], (μ(x) ∧ μ(y)) ∨ I ds(x, y) ≥ t, we use [x](μ,t) as the equivalence class of x. However, U(μ, t) is not a complete congruence relation. Through our research, we can obtain Lemma 3.3 as follows.
Letμbe a fuzzy ideal ofSandt ∈ [0, 1], then [x](μ, t)[y](μ, t) ⊆ [xy](μ, t).
Let m ∈ [x](μ, t)[y](μ, t), then m = x′y′, where x′ ∈ [x](μ, t) and y′ ∈ [y](μ, t). Hence μ(x, x′) ≥ t and μ(y, y′) ≥ t. Since U(μ, t) is a congruence relation in S, we have μ(xy, x′y′) ≥ t, and so x′y′ ∈ [xy](μ, t). This means that m ∈ [xy](μ, t). Thus [x](μ, t)[y](μ, t) ⊆ [xy](μ, t).
Now, we give an example to prove that we can not use “=” to replace “ ⊆ ” in Lemma 3.3.
LetS = {a, b, c, d} be a semigroup with the following “*” table.
| * | a | b | c | d |
|---|---|---|---|---|
| a | a | b | b | d |
| b | b | b | b | d |
| c | b | b | b | d |
| d | d | d | d | d |
Assume that
From the above example, we see we can not write [x](μ, t)[y](μ, t) = [xy](μ, t), it is only a containment relation. However there exists special U(μ, t) such that [x](μ, t)[y](μ, t) = [xy](μ, t). So we give the definition to make U(μ, t) a complete congruence relation as follows.
U(μ, t) is called a complete congruence relation if it satisfies: for anyx, y ∈ S, [x](μ, t)[y](μ, t) = [xy](μ, t).
LetS = {0, a, b, c} be a semigroup with the following “*” table.
| * | 0 | a | b | c |
|---|---|---|---|---|
| 0 | 0 | a | b | c |
| a | a | a | b | c |
| b | b | b | b | c |
| c | c | c | c | b |
Assume that
Let μ be a fuzzy ideal of S and t ∈ [0, 1]. We see that U(μ, t) is a congruence relation. According to Pawlak rough sets, we can obtain the approximation space (S, μ, t), just like (U, θ) in Pawlak rough sets.
Letμbe a fuzzy ideal ofSandU(μ, t) be at-level set. ForX ⊆ SandX ≠ ∅, we define the upper and lower approximations over (S, μ, t) as follows:
If
If
Letμbe any fuzzy ideal ofS, A andBbe any nonempty subsets ofS, then the following hold:
The proofs are trivial.
Letμbe a fuzzy ideal ofSandt ∈ [0, 1]. IfA1andA2are nonempty subsets ofS, then
Suppose that
Hence,
Next, we give an example to prove that the converse of containment in Proposition 3.9 does not hold.
Consider the subsets {a} and {c} ofSas in Example 3.4. We have
Let U(μ, t) be a complete congruence relation on S. IfA1A2are non-empty subsets ofS, then we have
Suppose that
Hence,
Here, we prove that if U(μ,t) is not a complete congruence relation on S, the containment in Proposition 3.11 may not be true as in the following example.
Consider the subsets {b, c} and {b, cofSin Example 3.4. We have
In the following, we use the above conclusions to study the properties of rough subsemigroups and rough (prime) ideals.
Letμbe a fuzzy ideal ofSandt ∈ [0, 1], then
If A is a subsemigroup ofS, A is an upper rough subsemigroup ofS;
If A is an ideal ofS, A is an upper rough ideal ofS.
Let A be a subsemigroup of S, we have AA ⊆ A, according to Propositions 3.8 and 3.9, we have
Let A be an ideal of S, then SAS ⊆ A. By Proposition 3.9, we have
The above theorem shows that the notion of an upper rough subsemigroup (ideal) is an extended notion of a usual subsemigroup (ideal) of a semigroup. The following example shows that the converse of Theorem 3.13 does not hold in general.
Consider the subset {c, d} as in Example 3.4. We have
Let U(μ, t) be a complete congruence relation onS, then
If A is a subsemigroup ofS, A is a lower rough subsemigroup ofS;
If A is an ideal ofS, A is a lower rough ideal ofS.
It is similar to the proof of Theorem 3.13. The following example shows that even if A is not a subsemigroup of S, then
Consider the semigroupSand the congruence class as in Example 3.6. ThenA = {0, a, c} is not a subsemigroup ofSbut
Letμbe a fuzzy ideal ofSandt ∈ [0, 1]. Then [0](μ,t)is an ideal ofS.
For all x1, x2 ∈ S and a ∈ [0](μ, t), then we have x1ax2 ∈ [x1](μ, t)[0](μ, t)[x2](μ, t) ⊆ [x1}0x2](μ, t)= [0](μ, t). This means x1ax2 ∈ [0](μ, t). Hence [0](μ, t) is an ideal of S.
Let μ be a fuzzy ideal of S and
By Proposition 3.8, we have
For every x∈[0](μt), then(0, x) ∈U(μ,t). Lety ∈[x](μ, t), then (x,y) ∈ U(μ,t) . Since U(μ,t) is acongruence relation, we have (0, y)∈ U(μ,t), this implies y ∈[0]μ,t) Hence
Let μ be a fuzzy ideal of S and
Let μ be a fuzzy ideal of S and t ∈ [0, μ(0)].Then μt=[0](μ,t).
For all x ∈μt, then μ(x)≥t and μ(0)≥μ(x)≥t. So (μ(x)∧μ(0))∨ IdS(x,0)≥μ(x)∧μ(0)∧t, by Definition 3.1, we have (x,O)∈ U(μ,t). It implies x ∈[0](μ,t). This means
On the other hand, for all x ∈ [0]μ,t), then (x,O)∈ U(μ, t) and(μ x)∧μ(0))∨ IdS(x,0)≥t. If x = 0μ(x) = μ(0) ≥ t is obvious, so x ∈ μt, it implies [0](μ,t) ⊆ μt. If x ≠ 0, then μ(x)∧ μ(0)≥ t, μ(x) ≥ t, this means x ∈ μt. Hence [0](μ, t) ⊆ μt. Therefore μt = [0](μ,t).
Let U(μ, t) be a complete congruence relation on S. If A is a prime ideal of S, then
Since A is an ideal of S, by Proposition 3.16, we know that
Let U(μ, t) be a complete congruence relation on S. If A is a prime ideal of S, then A is an upper rough prime ideal of S.
Since A is aprime ideal of S, by Theorem 3.13, we know that
4 Rough fuzzy semigroups based on fuzzy ideals
In Section 3, we discuss rough semigroups and their properties of semigroups. Now, in this section, we introduce roughness of fuzzy semigroups based on a fuzzy ideal of semigroups. First of all, we give the definition of rough fuzzy semigroups of semigroups. Here, we also use U(μ, t) as a congruence relation.
LetU(μ, t) be a complete congruence relation on S and β be a fuzzy subset of S, we can define a new approximation space (μ,t,β) and the lower and upper approximations of β, respectively
Let μ be any fuzzy ideal of S and β and γ be any two fuzzy subsets, then
Here we only prove that (12) and (13) hold and others are trivial.
then
there exists y ∈[x](μ,t), and β(y)≥ α and there exists y ∈ [x](μ,t), and
Let μ be a fuzzy ideal of S and t ∈ [0, 1], then
If β is a fuzzy subsemigroup of S, β is an upper rough fuzzy subsemigroup of S;
If β is a fuzzy left (right, two-sided) ideal of S, β is an upper rough fuzzy left ( right, two-sided) ideal of S.
Let β be a fuzzy subsemigroup of S, for every α∈[0, 1], βα is a subsemigroup of S. According to Propositions 3.9 and 4.2, we have
If β is a fuzzy left ideal of S, for every α∈ [0, 1], βα is a left ideal of S, that is, S · βα ⊆ βα, since
From this theorem, we see that upper rough fuzzy subsemigroups (left ideals, right ideals, two-sided ideals) are the generalizations of the fuzzy subsemigroups (left ideals, right ideals, two-sided ideals).
Let U(μ,t) be a complete congruence relation on S, then
If β is a fuzzy subsemigroup of S, β is a lower rough fuzzy subsemigroup of S;
If β is a fuzzy left (right, two-sided) ideal of S, β is a lower rough fuzzy left ( right, two-sided) ideal of S.
Let β be a fuzzy subsemigroup of S, for every α ∈ [0, 1], βα is a subsemigroup of S. According to Propositions 3.9 and 4.2, we have
If β is a fuzzy left ideal of S, for every α ∈ [0, 1], βα is a left ideal of S, that is S βα ⊆ βα, since
Let U(μt) be a complete congruence relation on S. If β is a fuzzy prime ideal of S, then we have
If
If
Since β is a fuzzy prime ideal of S, we know βα(α ∈ [0,1]) is a prime ideal of S. By Theorem 3.22 we have
It can be proven as (1).
Acknowledgement
The authors are extremely grateful to the editor and the referees for their valuable comments and helpful suggestions which help to improve the presentation of this paper.
This research was supported by NNSFC (11561023).
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- Semi-quotient mappings and spaces
- Regular Article
- Fractional multilinear integrals with rough kernels on generalized weighted Morrey spaces
- Regular Article
- A family of singular functions and its relation to harmonic fractal analysis and fuzzy logic
- Regular Article
- Solution to Fredholm integral inclusions via (F, δb)-contractions
- Regular Article
- An Ulam stability result on quasi-b-metric-like spaces
- Regular Article
- On the arrowhead-Fibonacci numbers
- Regular Article
- Rough semigroups and rough fuzzy semigroups based on fuzzy ideals
- Regular Article
- The general solution of impulsive systems with Riemann-Liouville fractional derivatives
- Regular Article
- A remark on local fractional calculus and ordinary derivatives
- Regular Article
- Elastic Sturmian spirals in the Lorentz-Minkowski plane
- Topical Issue: Metaheuristics: Methods and Applications
- Bias-variance decomposition in Genetic Programming
- Topical Issue: Metaheuristics: Methods and Applications
- A novel generalized oppositional biogeography-based optimization algorithm: application to peak to average power ratio reduction in OFDM systems
- Special Issue on Recent Developments in Differential Equations
- Modeling of vibration for functionally graded beams
- Special Issue on Recent Developments in Differential Equations
- Decomposition of a second-order linear time-varying differential system as the series connection of two first order commutative pairs
- Special Issue on Recent Developments in Differential Equations
- Differential equations associated with generalized Bell polynomials and their zeros
- Special Issue on Recent Developments in Differential Equations
- Differential equations for p, q-Touchard polynomials
- Special Issue on Recent Developments in Differential Equations
- A new approach to nonlinear singular integral operators depending on three parameters
- Special Issue on Recent Developments in Differential Equations
- Performance and stochastic stability of the adaptive fading extended Kalman filter with the matrix forgetting factor
- Special Issue on Recent Developments in Differential Equations
- On new characterization of inextensible flows of space-like curves in de Sitter space
- Special Issue on Recent Developments in Differential Equations
- Convergence theorems for a family of multivalued nonexpansive mappings in hyperbolic spaces
- Special Issue on Recent Developments in Differential Equations
- Fractional virus epidemic model on financial networks
- Special Issue on Recent Developments in Differential Equations
- Reductions and conservation laws for BBM and modified BBM equations
- Special Issue on Recent Developments in Differential Equations
- Extinction of a two species non-autonomous competitive system with Beddington-DeAngelis functional response and the effect of toxic substances
Articles in the same Issue
- Regular Article
-
A metric graph satisfying
- Regular Article
- On the Riemann-Hilbert problem in multiply connected domains
- Regular Article
- Hamilton cycles in almost distance-hereditary graphs
- Regular Article
- Locally adequate semigroup algebras
- Regular Article
- Parabolic oblique derivative problem with discontinuous coefficients in generalized weighted Morrey spaces
- Corrigendum
- Corrigendum to: parabolic oblique derivative problem with discontinuous coefficients in generalized weighted Morrey spaces
- Regular Article
- Some new bounds of the minimum eigenvalue for the Hadamard product of an M-matrix and an inverse M-matrix
- Regular Article
- Integral inequalities involving generalized Erdélyi-Kober fractional integral operators
- Regular Article
- Results on the deficiencies of some differential-difference polynomials of meromorphic functions
- Regular Article
- General numerical radius inequalities for matrices of operators
- Regular Article
- The best uniform quadratic approximation of circular arcs with high accuracy
- Regular Article
- Multiple gcd-closed sets and determinants of matrices associated with arithmetic functions
- Regular Article
- A note on the rate of convergence for Chebyshev-Lobatto and Radau systems
- Regular Article
- On the weakly(α, ψ, ξ)-contractive condition for multi-valued operators in metric spaces and related fixed point results
- Regular Article
- Existence of a common solution for a system of nonlinear integral equations via fixed point methods in b-metric spaces
- Regular Article
- Bounds for the Z-eigenpair of general nonnegative tensors
- Regular Article
- Subsymmetry and asymmetry models for multiway square contingency tables with ordered categories
- Regular Article
- End-regular and End-orthodox generalized lexicographic products of bipartite graphs
- Regular Article
- Refinement of the Jensen integral inequality
- Regular Article
- New iterative codes for 𝓗-tensors and an application
- Regular Article
- A result for O2-convergence to be topological in posets
- Regular Article
- A fixed point approach to the Mittag-Leffler-Hyers-Ulam stability of a fractional integral equation
- Regular Article
- Uncertainty orders on the sublinear expectation space
- Regular Article
- Generalized derivations of Lie triple systems
- Regular Article
- The BV solution of the parabolic equation with degeneracy on the boundary
- Regular Article
- Malliavin method for optimal investment in financial markets with memory
- Regular Article
- Parabolic sublinear operators with rough kernel generated by parabolic calderön-zygmund operators and parabolic local campanato space estimates for their commutators on the parabolic generalized local morrey spaces
- Regular Article
- On annihilators in BL-algebras
- Regular Article
- On derivations of quantales
- Regular Article
-
On the closed subfields of
- Regular Article
- A class of tridiagonal operators associated to some subshifts
- Regular Article
- Some notes to existence and stability of the positive periodic solutions for a delayed nonlinear differential equations
- Regular Article
- Weighted fractional differential equations with infinite delay in Banach spaces
- Regular Article
- Laplace-Stieltjes transform of the system mean lifetime via geometric process model
- Regular Article
- Various limit theorems for ratios from the uniform distribution
- Regular Article
- On α-almost Artinian modules
- Regular Article
- Limit theorems for the weights and the degrees in anN-interactions random graph model
- Regular Article
- An analysis on the stability of a state dependent delay differential equation
- Regular Article
- The hybrid mean value of Dedekind sums and two-term exponential sums
- Regular Article
- New modification of Maheshwari’s method with optimal eighth order convergence for solving nonlinear equations
- Regular Article
- On the concept of general solution for impulsive differential equations of fractional-order q ∈ (2,3)
- Regular Article
- A Riesz representation theory for completely regular Hausdorff spaces and its applications
- Regular Article
- Oscillation of impulsive conformable fractional differential equations
- Regular Article
- Dynamics of doubly stochastic quadratic operators on a finite-dimensional simplex
- Regular Article
- Homoclinic solutions of 2nth-order difference equations containing both advance and retardation
- Regular Article
- When do L-fuzzy ideals of a ring generate a distributive lattice?
- Regular Article
- Fully degenerate poly-Bernoulli numbers and polynomials
- Commentary
- Commentary to: Generalized derivations of Lie triple systems
- Regular Article
- Simple sufficient conditions for starlikeness and convexity for meromorphic functions
- Regular Article
- Global stability analysis and control of leptospirosis
- Regular Article
- Random attractors for stochastic two-compartment Gray-Scott equations with a multiplicative noise
- Regular Article
- The fuzzy metric space based on fuzzy measure
- Regular Article
- A classification of low dimensional multiplicative Hom-Lie superalgebras
- Regular Article
- Structures of W(2.2) Lie conformal algebra
- Regular Article
- On the number of spanning trees, the Laplacian eigenvalues, and the Laplacian Estrada index of subdivided-line graphs
- Regular Article
- Parabolic Marcinkiewicz integrals on product spaces and extrapolation
- Regular Article
- Prime, weakly prime and almost prime elements in multiplication lattice modules
- Regular Article
- Pochhammer symbol with negative indices. A new rule for the method of brackets
- Regular Article
- Outcome space range reduction method for global optimization of sum of affine ratios problem
- Regular Article
- Factorization theorems for strong maps between matroids of arbitrary cardinality
- Regular Article
- A convergence analysis of SOR iterative methods for linear systems with weak H-matrices
- Regular Article
- Existence theory for sequential fractional differential equations with anti-periodic type boundary conditions
- Regular Article
- Some congruences for 3-component multipartitions
- Regular Article
- Bound for the largest singular value of nonnegative rectangular tensors
- Regular Article
- Convolutions of harmonic right half-plane mappings
- Regular Article
- On homological classification of pomonoids by GP-po-flatness of S-posets
- Regular Article
- On CSQ-normal subgroups of finite groups
- Regular Article
- The homogeneous balance of undetermined coefficients method and its application
- Regular Article
- On the saturated numerical semigroups
- Regular Article
- The Bruhat rank of a binary symmetric staircase pattern
- Regular Article
- Fixed point theorems for cyclic contractive mappings via altering distance functions in metric-like spaces
- Regular Article
- Singularities of lightcone pedals of spacelike curves in Lorentz-Minkowski 3-space
- Regular Article
- An S-type upper bound for the largest singular value of nonnegative rectangular tensors
- Regular Article
- Fuzzy ideals of ordered semigroups with fuzzy orderings
- Regular Article
- On meromorphic functions for sharing two sets and three sets in m-punctured complex plane
- Regular Article
- An incremental approach to obtaining attribute reduction for dynamic decision systems
- Regular Article
- Very true operators on MTL-algebras
- Regular Article
- Value distribution of meromorphic solutions of homogeneous and non-homogeneous complex linear differential-difference equations
- Regular Article
- A class of 3-dimensional almost Kenmotsu manifolds with harmonic curvature tensors
- Regular Article
- Robust dynamic output feedback fault-tolerant control for Takagi-Sugeno fuzzy systems with interval time-varying delay via improved delay partitioning approach
- Regular Article
- New bounds for the minimum eigenvalue of M-matrices
- Regular Article
- Semi-quotient mappings and spaces
- Regular Article
- Fractional multilinear integrals with rough kernels on generalized weighted Morrey spaces
- Regular Article
- A family of singular functions and its relation to harmonic fractal analysis and fuzzy logic
- Regular Article
- Solution to Fredholm integral inclusions via (F, δb)-contractions
- Regular Article
- An Ulam stability result on quasi-b-metric-like spaces
- Regular Article
- On the arrowhead-Fibonacci numbers
- Regular Article
- Rough semigroups and rough fuzzy semigroups based on fuzzy ideals
- Regular Article
- The general solution of impulsive systems with Riemann-Liouville fractional derivatives
- Regular Article
- A remark on local fractional calculus and ordinary derivatives
- Regular Article
- Elastic Sturmian spirals in the Lorentz-Minkowski plane
- Topical Issue: Metaheuristics: Methods and Applications
- Bias-variance decomposition in Genetic Programming
- Topical Issue: Metaheuristics: Methods and Applications
- A novel generalized oppositional biogeography-based optimization algorithm: application to peak to average power ratio reduction in OFDM systems
- Special Issue on Recent Developments in Differential Equations
- Modeling of vibration for functionally graded beams
- Special Issue on Recent Developments in Differential Equations
- Decomposition of a second-order linear time-varying differential system as the series connection of two first order commutative pairs
- Special Issue on Recent Developments in Differential Equations
- Differential equations associated with generalized Bell polynomials and their zeros
- Special Issue on Recent Developments in Differential Equations
- Differential equations for p, q-Touchard polynomials
- Special Issue on Recent Developments in Differential Equations
- A new approach to nonlinear singular integral operators depending on three parameters
- Special Issue on Recent Developments in Differential Equations
- Performance and stochastic stability of the adaptive fading extended Kalman filter with the matrix forgetting factor
- Special Issue on Recent Developments in Differential Equations
- On new characterization of inextensible flows of space-like curves in de Sitter space
- Special Issue on Recent Developments in Differential Equations
- Convergence theorems for a family of multivalued nonexpansive mappings in hyperbolic spaces
- Special Issue on Recent Developments in Differential Equations
- Fractional virus epidemic model on financial networks
- Special Issue on Recent Developments in Differential Equations
- Reductions and conservation laws for BBM and modified BBM equations
- Special Issue on Recent Developments in Differential Equations
- Extinction of a two species non-autonomous competitive system with Beddington-DeAngelis functional response and the effect of toxic substances
