Abstract
In this paper, we introduce the concepts of f-prime ideals, f-semiprime ideals and f-prime radicals in ordered semigroups. Furthermore, some results on f-prime radicals and f-primary decomposition of an ideal in an ordered semigroup are obtained.
1 Introduction and preliminaries
Prime radical theorem is an important result in commutative ring theory and commutative semigroup theory (see [1,2]). In [3], Hoo and Shum gave a similar result in a residuated negatively ordered semigroup. Wu and Xie extended the result to a commutative ordered semigroup by using m-systems in [4]. In [5], Tang and Xie characterized in detail the radicals of ideals in ordered semigroups. In 1969, Murata, Kurata and Marubayashi introduced the notions of f-prime ideals and f-prime radicals in ring theory (see [6]), which generalized the concepts of prime ideals and prime radicals. In [7], Sardar and Goswami extended these concepts and results of ring theory to semirings. In this paper, we introduce the concepts of f-prime ideals and f-prime radicals in ordered semigroups and extend some results of rings and semirings to ordered semigroups. We also introduce the notion of f-semiprime ideals in ordered semigroups and obtain the result that the f-prime radical of an ideal I in an ordered semigroup is the least f-semiprime ideal containing I.
Next we list some basic concepts and notations on ordered semigroups (see [8]). An ordered semigroup is a semigroup (S, ·) endowed with an order relation â †â such that
Let (S, ·, â€) be an ordered semigroup. A non-empty subset I of S is called an ideal of S if it satisfies the following conditions: (1) SI âȘ IS â I; (2) a â I and b â S, b †a implies b â I. An ideal I of S is called weakly prime if AB â I implies A â I or B â I for any ideals A, B of S; an ideal I of S is called weakly semiprime if A2 â I implies A â I for any ideal A of S. For h â S, we denote
A subset M of S is called an m-system of S, if for any a, b â M, there exists x â S such that (axb] â© M âą â . A subset N of S is said to be a n-system of S if for any a â N, there exists x â S such that [axa) â N.
2 f-prime ideals and f-prime radical of an ideal
Definition 2.1
LetSbe an ordered semigroup. Denote byI(S) the set of all ideals ofS. Define a mapping f: S â I(S) which satisfies the following conditions
a â f(a);
x â f(a) âȘ Iimplies thatf(x) â f(a) âȘ Ifor anyI â I(S).
We call such mappingfa good mapping onS.
Example 2.2
LetSbe an ordered semigroup. Iff(a) = I(a) for alla â S, whereI(a) is the principal ideal generated bya, then it is easy to see thatfsatisfies the above conditions.
Example 2.3
We consider the ordered semigroupS = {a, b, c} defined by multiplication and the order below:
The ideals ofSare the sets:
If we definef(a) = {a}, f(b) = {a, b} andf(c) = S, then it is easy to see thatfsatisfies the above conditions.
Definition 2.4
LetSbe an ordered semigroup andfa good mapping onS. A subsetFofSis called anf-system ofSifFcontains anm-systemFâ, called the kernel ofF, such thatf(t) â© Fâ âą â for anyt â F.
Especially, â is also defined to be anf-system.
Remark 2.5
Everym-system is anf-system with kernel itself.
IfFis anf-system with kernelFâ, thenF = â if and only ifFâ = â .
Definition 2.6
LetSbe an ordered semigroup andfa good mapping onS. An idealIofSis calledf-prime if its complementC(I) inSis anf-system.
Remark 2.7
As we know, the complement of a weakly prime ideal of an ordered semigroup is anm-system. Furthermore, everym-system is anf-system. Hence, every weakly prime ideal of an ordered semigroup is anf-prime ideal. But the converse is not true in general.
Example 2.8
We consider the ordered semigroupSin Example 2.3. LetI = {a}. ThenC(I) = {b, c} is anf-system with kernelFâ = {b}. Hence, Iis anf-prime ideal. However, Iis not weakly prime. Indeed: {a, b} is an ideal ofSand {a, b}{a, b} â I, but {a, b} is not contained inI.
Proposition 2.9
LetPbe anf-prime ideal of an ordered semigroupSanda, b â S. Iff(a)f(b) â P, then either a â Por b â P.
Proof
Suppose that a, b â C(P). Since P is an f-prime ideal, C(P) is an f-system. Hence, f(a) â© [C(P)]â â â and f(b) â© [C(P)]â â â , where [C(P)]â is the kernel of C(P). Let x1 â f(a) â© [C(P)]â and x2 â f(b) â© [C(P)]â. Since [C(P)]â is an m-system, (x1rx2] â© [C(P)]â â â for some r â S. Thus (x1rx2] â© C(P) â â . Also x1rx2 â f(a)f(b) â P. Thus (x1rx2] â P which is a contradiction. Therefore, either a â P or b â P. âĄ
Corollary 2.10
LetPbe anf-primeideal of an ordered semigroupSandai â S (i = 1, 2, âŻ, n). Iff(a1)f(a2) ⯠f(an) â P, thenai â Pfor somei.
Let S be an ordered semigroup. Denote by fPI(S) and fS(S) the set of all f-prime ideals of S and the set of all f-systems of S respectively.
Definition 2.11
LetSbe an ordered semigroup andIbe an ideal ofS. We call the set {a â S|(â F â fS(S)) a â F â F â© I â â } thef-prime radical ofI, denoted byrf(I).
Theorem 2.12
LetSbe an ordered semigroup andIbe an ideal ofS. Thenrf(I) = â©PâÎPwhereÎ = {P â fPI(S)|I â P}.
Proof
Let J = â©PâÎP where Î = {P â fPI(S)|I â P}. If x â J, then there exists an f-prime ideal P containing I such that x â P. Thus C(P) is an f-system containing x but C(P) â© I = â . Hence x â rf(I). Therefore, rf(I) â J. On the other hand, if y â rf(I), then there exists an f-system F containing y such that F â© I = â . Thus y â C(F) and C(F) is an f-prime ideal such that I â C(F). Hence y â J and so J â rf(I). Therefore, rf(I) = J. âĄ
Corollary 2.13
LetSbe an ordered semigroup andIbe an ideal ofS. Thenrf(I) is an ideal ofS.
3 f-semiprime ideals
In this section, we introduce the concept of f-semiprime ideals in ordered semigroups and obtain that the f-prime radical of an ideal I is the least f-semiprime ideal containing I.
Definition 3.1
LetSbe an ordered semigroup. A subsetAofSis said to be anfn-system ofSif
Definition 3.2
An idealPof an ordered semigroupSis said to bef-semiprime if its complementC(P) inSis anfn-system.
Clearly, every f-system is an fn-system. Therefore, every f-prime ideal is an f-semiprime ideal.
Proposition 3.3
LetSbe an ordered semigroup andIa weakly semiprime ideal ofS. ThenIis anf-semiprime ideal ofS.
Proof
Since I is a weakly semiprime ideal, C(I) is a n-system. Thus C(I) is the union of some m-systems of S. Since every m-system is an f-system, C(I) is the union of some f-systems of S. Hence C(I) is an fn-system. Therefore, I is an f-semiprime ideal. âĄ
Proposition 3.4
LetPbe anf-semiprime ideal of an ordered semigroupSanda â S. Iff(a)f(a) â P, thena â P.
Proof
Suppose that a â C(P). Since P is an f-semiprime ideal, C(P) is an fn-system. Thus
Proposition 3.5
LetSbe an ordered semigroup andIbe an ideal ofS. Thenrf(I) is anf-semiprime ideal ofS.
Proof
By Theorem 2.12, we know that rf(I) = â©PâÎP where Î = {P â fPI(S)| I â P}. Thus C(rf(I)) = âPâÎC(P). Since every C(P) is an f-system, C(rf(I)) is an fn-system. Therefore, rf(I) is an f-semiprime ideal of S. âĄ
Proposition 3.6
LetSbe an ordered semigroup andIbe anf-semiprime ideal ofS. Thenrf(I) = I.
Proof
By Theorem 2.12, we have I â rf(I). Let x â I. Since C(I) is an fn-system,
Definition 3.7
LetSbe an ordered semigroup andfa good mapping onS. Leta â SandI â I(S). The set {x â S
| f(a)f(x) â I}, denoted by I : a, is called the leftf-quotient ofIbya. Moreover, for any idealJofS, the leftf-quotient ofIbyJis defined to be
Remark 3.8
Similar to Definition 3.7, we call the set {x â S| f(x)f(a) â I} rightf-quotientofIbya, denoted by a : I. Moreover,
We note that I : a may be empty. See the following example.
Example 3.9
We consider the ordered semigroup S of Example 2.3. If we define f(a) = f(b) = f(c) = S, then the mapping f is a good mapping. Let I = {a}. Then I : a, I : b and I : c are all empty.
The following result can be easily obtained from the above definition.
Proposition 3.10
Let S be an ordered semigroup and f a good mapping on S. If I, IâČ, Iâł, J, JâČ, Jâł â I(S) and a â S, then
IâČ â Iâł â IâČ : a â Iâł : a and IâČ : J â Iâł : J;
JâČ â Jâł â I : JâČ â I : Jâł;
(IâČ â© Iâł) : a = (IâČ : a) â© (Iâł : a) and (IâČ â© Iâł) : J = (IâČ : J) â© (Iâł : J).
Proposition 3.11
Let S be an ordered semigroup and f a good mapping on S. If I â I(S) and a â S, then I : a is either empty or an ideal containing I of S.
Proof
Suppose that I : a â â . Let x â I : a and r â S. Then rx, xr â f(x). Thus f(rx) â f(x) and f(xr) â f(x). Also f(a)f(x) â I. Therefore, f(a)f(rx) â I and f(a)f(xr) â I. Let z †y â I : a. Then z â f(y) and f(a)f(y) â I. Thus f(z) â f(y). Therefore, f(a)f(z) â I, which implies that z â I : a. Hence, I : a is an ideal of S. Next we prove that I â I : a.
Let b â I and x â I : a. Then b â f(x) âȘ I. Thus f(b) â f(x) âȘ I. Also f(a)f(x) â I. It follows that f(a)f(b) â f(a)(f(x) âȘ I) = f(a)f(x) âȘ f(a)I â I. Hence b â I : a and so I â I : a.ââĄ
Let S be an ordered semigroup and f a good mapping on S. Denote the following condition by (α):
If f(a) = I(a) for every a â S, then S satisfies the condition (α). But this is not true for any good mapping f. See the following example.
Example 3.12
We consider the ordered semigroup S = {a, b, c} defined by multiplication and the order below:
It is easy to check that S is an ordered semigroup. The ideals of S are the sets:
If we define f(a) = {a}, f(b) = f(c) = S, then it is easy to see that f is a good mapping. Let I = {a, b} and F = {b, c}. Then F is an f-system with kernel F* = {c} and F â© I â â . However, F* â© I = â .
Proposition 3.13
Let S be an ordered semigroup and f a good mapping on S. If I, J â I(S), then
I â J â rf(I) â rf(J);
rf(rf(I)) = rf(I);
rf(I âȘ J) = rf(rf(I) âȘ rf(J));
rf(I ⩠J) = rf(I) ⩠rf(J), if S satisfies the condition (α).
Proof
(1) and (2) are obvious.
(3) From Theorem 2.12, we have IâȘJ â rf(I)âȘrf(J). By condition (1), we have rf(IâȘJ) â rf(rf(I)âȘrf(J)) and rf(I)âȘrf(J) â rf(IâȘJ). Combining conditions (1) and (2), we obtain rf(rf(I) âȘ rf(J)) â rf(rf(IâȘJ)) = rf(IâȘJ).
(4) It is obvious that rf(I â© J) â rf(I) â© rf(J) from condition (1). Next we prove the other inclusion. Let x â rf(I) â© rf(J) and F be an f-system containing x. Suppose that a â F â© I and b â F â© J. By assumption (α), there exist a* â F* â© I and b* â F* â© J. Since F* is an m-system, (a*zb*] â© F* â â for some z â S. Thus (a*zb*] â© F â â . Moreover, a*zb* â I â© J and so (a*zb*] â I â© J. Hence F â© (I â© J) â â . It follows that x â rf(I â© J). Therefore rf(I â© J) = rf(I) â© rf(J).ââĄ
Combining Proposition 3.5, 3.6 and 3.13 (1), we have the following result.
Theorem 3.14
Let I be an ideal of an ordered semigroup S. Then rf(I) is the least f-semiprime ideal containing I.
4 f-primary decomposition of an ideal
In this section, we introduce the concepts of f-primary ideals and f-primary decomposition of an ideal in ordered semigroups, and conclude that the number of f-primary components and the f-prime radicals of f-primary components of a normal decomposition of an ideal I depend only on I under some assumptions.
Definition 4.1
Let S be an ordered semigroup and f a good mapping on S. An ideal I of S is called left f-primary if f(a)f(b) â I implies that a â rf(I) or b â I.
Remark 4.2
By symmetry, we call an ideal I of S f-primary if f(a)f(b) â I implies that a â I or b â rf(I). In what follows unless otherwise mentioned, f-primary means left f-primary.
By Proposition 2.9, we note that f-prime ideals must be f-primary ideals.
From Definition 4.1, we obtain easily the following result.
Proposition 4.3
Let S be an ordered semigroup satisfying the condition (α). If IâČand Iâłare f-primary ideals of S such that rf(IâČ) = rf(Iâł), then I = IâČ â© Iâłis also an f-primary ideal of S such that rf(I) = rf(IâČ) = rf(Iâł).
Let S be an ordered semigroup and f a good mapping on S. Denote the following condition by (ÎČ):
Theorem 4.4
Let S be an ordered semigroup satisfying the condition (ÎČ). An ideal I of S is f-primary if and only if I : J = I for all ideals J â rf(I).
Proof
Suppose that I is an f-primary ideal of S and J is an ideal of S not contained in rf(I). Since S satisfies the condition (ÎČ), I : J â â . Thus I : b â â for all b â J. Hence I â I : b for every b â J and so I â I : J. Now we choose an element c â J â rf(I). By the condition (ÎČ), I : c â â . Moreover, f(c)f(a) â I for any a â I : c. Since I is f-primary and c â rf(I), a â I. Thus I : c â I. Therefore, I = I : c and so I : J â I : c = I. Consequently, I = I : J.
Conversely, suppose that I : J = I for all ideals J not contained in rf(I). Let f(a)f(b) â I and a â rf(I). Since a â f(a), f(a) is not a subset of rf(I) and so I : f(a) = I. For any aâČ â f(a), f(aâČ) â f(a). Thus f(aâČ)f(b) â f(a)f(b) â I and thus b â I : f(a) = I. It follows that I is f-primary.ââĄ
Definition 4.5
If an ideal I of an ordered semigroup S can be written as I = I1 ⩠I2 ⩠⯠⩠In where each Ii is an f-primary ideal, then this is called an f-primary decomposition of I and each Ii is called the f-primary component of the decomposition.
A f-primary decomposition in which no Ii contains the intersection of the remaining Ij is called irredundant. Moreover, an irredundant f-primary decomposition in which the radicals of the various f-primary components are all different is called a normal decomposition.
From Proposition 4.3, we note that each f-primary decomposition can be refined into one which is normal.
Let S be an ordered semigroup and f a good mapping on S. Denote the following condition by (Îł): if for any f-primary ideal I of S, we have I : I = S. If f(a) = I(a) for all a â S, then S satisfies the condition (Îł). But the condition (Îł) need not be satisfied for any good mapping f.
Example 4.6
From Example 2.8, P = {a} is an f-prime ideal of S and so P is f-primary. However P : P = {a} â S.
Theorem 4.7
Let S be an ordered semigroup satisfying the conditions (α), (ÎČ) and (Îł). If an ideal A of S has two normal f-primary decompositions
Proof
It is easy to see that the result holds in the case A = S. Next we prove the case that A â S, where all f-primary components I1, âŻ, In,
âŻ, rf(
If n = 1, then, by the condition (Îł), S = I1 : I1 = A : I1 = A which is a contradiction. If n > 1, then, by the condition (Îł) and the fact that I1 â rf(Ii) for all 2 †i †n, we have
This is also a contradiction. By a suitable ordering, we may have rf(I1) = rf(
We use an induction on the number n of f-primary components. If n = 1, then
Acknowledgement
This work is supported by the National Natural Science Foundation of China (No. 11701504), the Young Innovative Talent Project of Department of Education of Guangdong Province (No. 2016KQNCX180) and the University Natural Science Project of Anhui Province (No. KJ2018A0329).
References
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Articles in the same Issue
- Regular Articles
- Algebraic proofs for shallow water biâHamiltonian systems for three cocycle of the semi-direct product of KacâMoody and Virasoro Lie algebras
- On a viscous two-fluid channel flow including evaporation
- Generation of pseudo-random numbers with the use of inverse chaotic transformation
- Singular Cauchy problem for the general Euler-Poisson-Darboux equation
- Ternary and n-ary f-distributive structures
- On the fine Simpson moduli spaces of 1-dimensional sheaves supported on plane quartics
- Evaluation of integrals with hypergeometric and logarithmic functions
- Bounded solutions of self-adjoint second order linear difference equations with periodic coeffients
- Oscillation of first order linear differential equations with several non-monotone delays
- Existence and regularity of mild solutions in some interpolation spaces for functional partial differential equations with nonlocal initial conditions
- The log-concavity of the q-derangement numbers of type B
- Generalized state maps and states on pseudo equality algebras
- Monotone subsequence via ultrapower
- Note on group irregularity strength of disconnected graphs
- On the security of the Courtois-Finiasz-Sendrier signature
- A further study on ordered regular equivalence relations in ordered semihypergroups
- On the structure vector field of a real hypersurface in complex quadric
- Rank relations between a {0, 1}-matrix and its complement
- Lie n superderivations and generalized Lie n superderivations of superalgebras
- Time parallelization scheme with an adaptive time step size for solving stiff initial value problems
- Stability problems and numerical integration on the Lie group SO(3) Ă R3 Ă R3
- On some fixed point results for (s, p, α)-contractive mappings in b-metric-like spaces and applications to integral equations
- On algebraic characterization of SSC of the Jahangirâs graph đn,m
- A greedy algorithm for interval greedoids
- On nonlinear evolution equation of second order in Banach spaces
- A primal-dual approach of weak vector equilibrium problems
- On new strong versions of Browder type theorems
- A GerĆĄgorin-type eigenvalue localization set with n parameters for stochastic matrices
- Restriction conditions on PL(7, 2) codes (3 †|đi| †7)
- Singular integrals with variable kernel and fractional differentiation in homogeneous Morrey-Herz-type Hardy spaces with variable exponents
- Introduction to disoriented knot theory
- Restricted triangulation on circulant graphs
- Boundedness control sets for linear systems on Lie groups
- Chenâs inequalities for submanifolds in (Îș, ÎŒ)-contact space form with a semi-symmetric metric connection
- Disjointed sum of products by a novel technique of orthogonalizing ORing
- A parametric linearizing approach for quadratically inequality constrained quadratic programs
- Generalizations of Steffensenâs inequality via the extension of Montgomery identity
- Vector fields satisfying the barycenter property
- On the freeness of hypersurface arrangements consisting of hyperplanes and spheres
- Biderivations of the higher rank Witt algebra without anti-symmetric condition
- Some remarks on spectra of nuclear operators
- Recursive interpolating sequences
- Involutory biquandles and singular knots and links
- Constacyclic codes over đœpm[u1, u2,âŻ,uk]/ă ui2 = ui, uiuj = ujuiă
- Topological entropy for positively weak measure expansive shadowable maps
- Oscillation and non-oscillation of half-linear differential equations with coeffcients determined by functions having mean values
- On đ -regular semigroups
- One kind power mean of the hybrid Gauss sums
- A reduced space branch and bound algorithm for a class of sum of ratios problems
- Some recurrence formulas for the Hermite polynomials and their squares
- A relaxed block splitting preconditioner for complex symmetric indefinite linear systems
- On f - prime radical in ordered semigroups
- Positive solutions of semipositone singular fractional differential systems with a parameter and integral boundary conditions
- Disjoint hypercyclicity equals disjoint supercyclicity for families of Taylor-type operators
- A stochastic differential game of low carbon technology sharing in collaborative innovation system of superior enterprises and inferior enterprises under uncertain environment
- Dynamic behavior analysis of a prey-predator model with ratio-dependent Monod-Haldane functional response
- The points and diameters of quantales
- Directed colimits of some flatness properties and purity of epimorphisms in S-posets
- Super (a, d)-H-antimagic labeling of subdivided graphs
- On the power sum problem of Lucas polynomials and its divisible property
- Existence of solutions for a shear thickening fluid-particle system with non-Newtonian potential
- On generalized P-reducible Finsler manifolds
- On Banach and Kuratowski Theorem, K-Lusin sets and strong sequences
- On the boundedness of square function generated by the Bessel differential operator in weighted Lebesque Lp,α spaces
- On the different kinds of separability of the space of Borel functions
- Curves in the Lorentz-Minkowski plane: elasticae, catenaries and grim-reapers
- Functional analysis method for the M/G/1 queueing model with single working vacation
- Existence of asymptotically periodic solutions for semilinear evolution equations with nonlocal initial conditions
- The existence of solutions to certain type of nonlinear difference-differential equations
- Domination in 4-regular Knödel graphs
- Stepanov-like pseudo almost periodic functions on time scales and applications to dynamic equations with delay
- Algebras of right ample semigroups
- Random attractors for stochastic retarded reaction-diffusion equations with multiplicative white noise on unbounded domains
- Nontrivial periodic solutions to delay difference equations via Morse theory
- A note on the three-way generalization of the Jordan canonical form
- On some varieties of ai-semirings satisfying xp+1 â x
- Abstract-valued Orlicz spaces of range-varying type
- On the recursive properties of one kind hybrid power mean involving two-term exponential sums and Gauss sums
- Arithmetic of generalized Dedekind sums and their modularity
- Multipreconditioned GMRES for simulating stochastic automata networks
- Regularization and error estimates for an inverse heat problem under the conformable derivative
- Transitivity of the Δm-relation on (m-idempotent) hyperrings
- Learning Bayesian networks based on bi-velocity discrete particle swarm optimization with mutation operator
- Simultaneous prediction in the generalized linear model
- Two asymptotic expansions for gamma function developed by Windschitlâs formula
- State maps on semihoops
- đđ-convergence and lim-infđ-convergence in partially ordered sets
- Stability and convergence of a local discontinuous Galerkin finite element method for the general Lax equation
- New topology in residuated lattices
- Optimality and duality in set-valued optimization utilizing limit sets
- An improved Schwarz Lemma at the boundary
- Initial layer problem of the Boussinesq system for Rayleigh-Bénard convection with infinite Prandtl number limit
- Toeplitz matrices whose elements are coefficients of BazileviÄ functions
- Epi-mild normality
- Nonlinear elastic beam problems with the parameter near resonance
- Orlicz difference bodies
- The Picard group of Brauer-Severi varieties
- Galoisian and qualitative approaches to linear Polyanin-Zaitsev vector fields
- Weak group inverse
- Infinite growth of solutions of second order complex differential equation
- Semi-Hurewicz-Type properties in ditopological texture spaces
- Chaos and bifurcation in the controlled chaotic system
- Translatability and translatable semigroups
- Sharp bounds for partition dimension of generalized Möbius ladders
- Uniqueness theorems for L-functions in the extended Selberg class
- An effective algorithm for globally solving quadratic programs using parametric linearization technique
- Bounds of Strong EMT Strength for certain Subdivision of Star and Bistar
- On categorical aspects of S -quantales
- On the algebraicity of coefficients of half-integral weight mock modular forms
- Dunkl analogue of SzĂĄsz-mirakjan operators of blending type
- Majorization, âusefulâ CsiszĂĄr divergence and âusefulâ Zipf-Mandelbrot law
- Global stability of a distributed delayed viral model with general incidence rate
- Analyzing a generalized pest-natural enemy model with nonlinear impulsive control
- Boundary value problems of a discrete generalized beam equation via variational methods
- Common fixed point theorem of six self-mappings in Menger spaces using (CLRST) property
- Periodic and subharmonic solutions for a 2nth-order p-Laplacian difference equation containing both advances and retardations
- Spectrum of free-form Sudoku graphs
- Regularity of fuzzy convergence spaces
- The well-posedness of solution to a compressible non-Newtonian fluid with self-gravitational potential
- On further refinements for Young inequalities
- Pretty good state transfer on 1-sum of star graphs
- On a conjecture about generalized Q-recurrence
- Univariate approximating schemes and their non-tensor product generalization
- Multi-term fractional differential equations with nonlocal boundary conditions
- Homoclinic and heteroclinic solutions to a hepatitis C evolution model
- Regularity of one-sided multilinear fractional maximal functions
- Galois connections between sets of paths and closure operators in simple graphs
- KGSA: A Gravitational Search Algorithm for Multimodal Optimization based on K-Means Niching Technique and a Novel Elitism Strategy
- Ξ-type Calderón-Zygmund Operators and Commutators in Variable Exponents Herz space
- An integral that counts the zeros of a function
- On rough sets induced by fuzzy relations approach in semigroups
- Computational uncertainty quantification for random non-autonomous second order linear differential equations via adapted gPC: a comparative case study with random Fröbenius method and Monte Carlo simulation
- The fourth order strongly noncanonical operators
- Topical Issue on Cyber-security Mathematics
- Review of Cryptographic Schemes applied to Remote Electronic Voting systems: remaining challenges and the upcoming post-quantum paradigm
- Linearity in decimation-based generators: an improved cryptanalysis on the shrinking generator
- On dynamic network security: A random decentering algorithm on graphs