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On f - prime radical in ordered semigroups

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Published/Copyright: June 7, 2018

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.

MSC 2010: 06F05; 20M10

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 “ ≀ ” such that

(∀a,b,x∈S) a≀b⇒xa≀xb and ax≀bx.

Let (S, ·, ≀) 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 âˆȘ IS ⊆ I; (2) a ∈ I and b ∈ S, b ≀ a implies b ∈ I. An ideal I of S is called weakly prime if AB ⊆ I implies A ⊆ I or B ⊆ I for any ideals A, B of S; an ideal I of S is called weakly semiprime if A2 ⊆ I implies A ⊆ I for any ideal A of S. For h ∈ S, we denote

(h]={t∈S∣t≀h};[h)={s∈S∣h≀s}.

A subset M of S is called an m-system of S, if for any a, b ∈ M, there exists x ∈ S such that (axb] ∩ M ≱ ∅. A subset N of S is said to be a n-system of S if for any a ∈ N, there exists x ∈ S such that [axa) ⊆ 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 → I(S) which satisfies the following conditions

  1. a ∈ f(a);

  2. x ∈ f(a) âˆȘ Iimplies thatf(x) ⊆ f(a) âˆȘ Ifor anyI ∈ I(S).

We call such mappingfa good mapping onS.

Example 2.2

LetSbe an ordered semigroup. Iff(a) = I(a) for alla ∈ 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:

⋅abca  a  a  ab  a  a  bc  a  b  b
≀:={(a,a),(a,b),(a,c),(b,b),(b,c),(c,c)}.

The ideals ofSare the sets:

{a}, {a,b} and S.

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-systemF∗, called the kernel ofF, such thatf(t) ∩ F∗ ≱ ∅ for anyt ∈ F.

Especially, ∅ is also defined to be anf-system.

Remark 2.5

  1. Everym-system is anf-system with kernel itself.

  2. IfFis anf-system with kernelF∗, thenF = ∅ if and only ifF∗ = ∅.

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 kernelF∗ = {b}. Hence, Iis anf-prime ideal. However, Iis not weakly prime. Indeed: {a, b} is an ideal ofSand {a, b}{a, b} ⊆ I, but {a, b} is not contained inI.

Proposition 2.9

LetPbe anf-prime ideal of an ordered semigroupSanda, b ∈ S. Iff(a)f(b) ⊆ P, then either a ∈ Por b ∈ P.

Proof

Suppose that a, b ∈ C(P). Since P is an f-prime ideal, C(P) is an f-system. Hence, f(a) ∩ [C(P)]∗ ≠ ∅ and f(b) ∩ [C(P)]∗ ≠ ∅, where [C(P)]∗ is the kernel of C(P). Let x1 ∈ f(a) ∩ [C(P)]∗ and x2 ∈ f(b) ∩ [C(P)]∗. Since [C(P)]∗ is an m-system, (x1rx2] ∩ [C(P)]∗ ≠ ∅ for some r ∈ S. Thus (x1rx2] ∩ C(P) ≠ ∅. Also x1rx2 ∈ f(a)f(b) ⊆ P. Thus (x1rx2] ⊆ P which is a contradiction. Therefore, either a ∈ P or b ∈ P. □

Corollary 2.10

LetPbe anf-primeideal of an ordered semigroupSandai ∈ S (i = 1, 2, ⋯, n). Iff(a1)f(a2) ⋯ f(an) ⊆ P, thenai ∈ 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 ∈ S|(∀ F ∈ fS(S)) a ∈ F → F ∩ I ≠ ∅} thef-prime radical ofI, denoted byrf(I).

Theorem 2.12

LetSbe an ordered semigroup andIbe an ideal ofS. Thenrf(I) = ∩P∈ΓPwhereΓ = {P ∈ fPI(S)|I ⊆ P}.

Proof

Let J = ∩P∈ΓP where Γ = {P ∈ fPI(S)|I ⊆ P}. If x ∉ J, then there exists an f-prime ideal P containing I such that x ∉ P. Thus C(P) is an f-system containing x but C(P) ∩ I = ∅. Hence x ∉ rf(I). Therefore, rf(I) ⊆ J. On the other hand, if y ∉ rf(I), then there exists an f-system F containing y such that F ∩ I = ∅. Thus y ∉ C(F) and C(F) is an f-prime ideal such that I ⊆ C(F). Hence y ∉ J and so J ⊆ rf(I). Therefore, rf(I) = J. □

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 ofSifA=⋃i∈ΓFiwhere {Fi|i ∈ Γ} ⊆ fS(S).

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. □

Proposition 3.4

LetPbe anf-semiprime ideal of an ordered semigroupSanda ∈ S. Iff(a)f(a) ⊆ P, thena ∈ P.

Proof

Suppose that a ∈ C(P). Since P is an f-semiprime ideal, C(P) is an fn-system. Thus C(P)=⋃i∈ΓFi, where {Fi| i ∈ Γ} ⊆ fS(S). Hence, a ∈ Fi for some i ∈ Γ. Therefore, f(a) ∩ Fi∗ ≠ ∅, where Fi∗ is the kernel of Fi. Let x ∈ f(a) ∩ Fi∗ . Since Fi∗ is an m-system, (xrx] ∩ Fi∗ ≠ ∅ for some r ∈ S. Thus (xrx] ∩ Fi ≠ ∅ and so (xrx] ∩ C(P) ≠ ∅. Also xrx ∈ f(a)f(a) ⊆ P. Therefore, (xrx] ⊆ P which is a contradiction. Hence a ∈ P. □

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) = ∩P∈ΓP where Γ = {P ∈ fPI(S)| I ⊆ P}. Thus C(rf(I)) = ⋃P∈ΓC(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. □

Proposition 3.6

LetSbe an ordered semigroup andIbe anf-semiprime ideal ofS. Thenrf(I) = I.

Proof

By Theorem 2.12, we have I ⊆ rf(I). Let x ∉ I. Since C(I) is an fn-system, C(I)=⋃i∈ΓFi where {Fi| i ∈ Γ} ⊆ fS(S). Thus x∈⋃i∈ΓFi and so x ∈ Fi for some i ∈ Γ, but Fi ∩ I = ∅. Hence Fi is an f-system containing x but not containing any element of I. By Definition 2.11, x ∉ rf(I). It follows that rf(I) ⊆ I. Therefore rf(I) = I. □

Definition 3.7

LetSbe an ordered semigroup andfa good mapping onS. Leta ∈ SandI ∈ I(S). The set {x ∈ S | f(a)f(x) ⊆ I}, denoted by I : a, is called the leftf-quotient ofIbya. Moreover, for any idealJofS, the leftf-quotient ofIbyJis defined to be⋂a∈J(I:a),denoted by I : J.

Remark 3.8

Similar to Definition 3.7, we call the set {x ∈ S| f(x)f(a) ⊆ I} rightf-quotientofIbya, denoted by a : I. Moreover, ⋂a∈J(a:I)is defined to berightf-quotient ofIbyJ. In what follows, unless otherwise mentioned, f-quotient means leftf-quotient.

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, Iâ€Č, I″, J, Jâ€Č, J″ ∈ I(S) and a ∈ S, then

  1. Iâ€Č ⊆ I″ ⇒ Iâ€Č : a ⊆ I″ : a and Iâ€Č : J ⊆ I″ : J;

  2. Jâ€Č ⊆ J″ ⇒ I : Jâ€Č ⊇ I : J″;

  3. (Iâ€Č ∩ I″) : a = (Iâ€Č : a) ∩ (I″ : a) and (Iâ€Č ∩ I″) : J = (Iâ€Č : J) ∩ (I″ : J).

Proposition 3.11

Let S be an ordered semigroup and f a good mapping on S. If I ∈ I(S) and a ∈ S, then I : a is either empty or an ideal containing I of S.

Proof

Suppose that I : a ≠ ∅. Let x ∈ I : a and r ∈ S. Then rx, xr ∈ f(x). Thus f(rx) ⊆ f(x) and f(xr) ⊆ f(x). Also f(a)f(x) ⊆ I. Therefore, f(a)f(rx) ⊆ I and f(a)f(xr) ⊆ I. Let z ≀ y ∈ I : a. Then z ∈ f(y) and f(a)f(y) ⊆ I. Thus f(z) ⊆ f(y). Therefore, f(a)f(z) ⊆ I, which implies that z ∈ I : a. Hence, I : a is an ideal of S. Next we prove that I ⊆ I : a.

Let b ∈ I and x ∈ I : a. Then b ∈ f(x) âˆȘ I. Thus f(b) ⊆ f(x) âˆȘ I. Also f(a)f(x) ⊆ I. It follows that f(a)f(b) ⊆ f(a)(f(x) âˆȘ I) = f(a)f(x) âˆȘ f(a)I ⊆ I. Hence b ∈ I : a and so I ⊆ I : a. □

Let S be an ordered semigroup and f a good mapping on S. Denote the following condition by (α):

(∀F∈fS(S)) (∀I∈I(S))  F∩I≠∅⇒F∗∩I≠∅.

If f(a) = I(a) for every a ∈ 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:

⋅abca  a  a  ab  a  a  bc  a  b  c
≀:={(a,a),(a,b),(a,c),(b,b),(b,c),(c,c)}.

It is easy to check that S is an ordered semigroup. The ideals of S are the sets:

{a}, {a,b} and S.

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 ∩ I ≠ ∅. However, F* ∩ I = ∅.

Proposition 3.13

Let S be an ordered semigroup and f a good mapping on S. If I, J ∈ I(S), then

  1. I ⊆ J ⇒ rf(I) ⊆ rf(J);

  2. rf(rf(I)) = rf(I);

  3. rf(I âˆȘ J) = rf(rf(I) âˆȘ rf(J));

  4. 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 IâˆȘJ ⊆ rf(I)âˆȘrf(J). By condition (1), we have rf(IâˆȘJ) ⊆ rf(rf(I)âˆȘrf(J)) and rf(I)âˆȘrf(J) ⊆ rf(IâˆȘJ). Combining conditions (1) and (2), we obtain rf(rf(I) âˆȘ rf(J)) ⊆ rf(rf(IâˆȘJ)) = rf(IâˆȘJ).

(4) It is obvious that rf(I ∩ J) ⊆ rf(I) ∩ rf(J) from condition (1). Next we prove the other inclusion. Let x ∈ rf(I) ∩ rf(J) and F be an f-system containing x. Suppose that a ∈ F ∩ I and b ∈ F ∩ J. By assumption (α), there exist a* ∈ F* ∩ I and b* ∈ F* ∩ J. Since F* is an m-system, (a*zb*] ∩ F* ≠ ∅ for some z ∈ S. Thus (a*zb*] ∩ F ≠ ∅. Moreover, a*zb* ∈ I ∩ J and so (a*zb*] ⊆ I ∩ J. Hence F ∩ (I ∩ J) ≠ ∅. It follows that x ∈ rf(I ∩ J). Therefore rf(I ∩ J) = rf(I) ∩ rf(J). □

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) ⊆ I implies that a ∈ rf(I) or b ∈ I.

Remark 4.2

By symmetry, we call an ideal I of S f-primary if f(a)f(b) ⊆ I implies that a ∈ I or b ∈ 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 Iâ€Čand I″are f-primary ideals of S such that rf(Iâ€Č) = rf(I″), then I = Iâ€Č ∩ I″is also an f-primary ideal of S such that rf(I) = rf(Iâ€Č) = rf(I″).

Let S be an ordered semigroup and f a good mapping on S. Denote the following condition by (ÎČ):

(∀I,J∈I(S))  J⊈rf(I)⇒I:J≠∅.

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 ⊈ 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 ≠ ∅. Thus I : b ≠ ∅ for all b ∈ J. Hence I ⊆ I : b for every b ∈ J and so I ⊆ I : J. Now we choose an element c ∈ J ∖ rf(I). By the condition (ÎČ), I : c ≠ ∅. Moreover, f(c)f(a) ⊆ I for any a ∈ I : c. Since I is f-primary and c ∉ rf(I), a ∈ I. Thus I : c ⊆ I. Therefore, I = I : c and so I : J ⊆ 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) ⊆ I and a ∉ rf(I). Since a ∈ f(a), f(a) is not a subset of rf(I) and so I : f(a) = I. For any aâ€Č ∈ f(a), f(aâ€Č) ⊆ f(a). Thus f(aâ€Č)f(b) ⊆ f(a)f(b) ⊆ I and thus b ∈ I : f(a) = I. It follows that I is f-primary. □

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 (γ): if for any f-primary ideal I of S, we have I : I = S. If f(a) = I(a) for all a ∈ S, then S satisfies the condition (γ). But the condition (γ) 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} ≠ 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 decompositionsA=⋂i=1nIi=⋂i=1mIiâ€Č,then n = m and rf(Ii) = rf( Iiâ€Č) for 1 ≀ i ≀ n = m by a suitable ordering.

Proof

It is easy to see that the result holds in the case A = S. Next we prove the case that A ≠ S, where all f-primary components I1, ⋯, In, I1â€Č,⋯,Imâ€Č are proper ideals. We may assume that rf(I1) is maximal in the set {rf(I1), ⋯, rf(In), rf( I1â€Č),

⋯, rf( Imâ€Č)}. Now we prove that rf(I1) = rf( Iiâ€Č) for some i. It is enough to show that I1 ⊆ rf( Iiâ€Č). Suppose that I1 ⊈ rf( Iiâ€Č) for all 1 ≀ i ≀ m. Then we have, by Theorem 4.4, Iiâ€Č:I1=Iiâ€Č for all 1 ≀ i ≀ m, and so

A:I1=(I1â€Č∩I2â€Čâˆ©â‹Żâˆ©Inâ€Č):I1=(I1â€Č:I1)∩(I2â€Č:I1)âˆ©â‹Żâˆ©(Inâ€Č:I1)=I1â€Č∩I2â€Čâˆ©â‹Żâˆ©Inâ€Č=A.

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 ⊈ rf(Ii) for all 2 ≀ i ≀ n, we have

A=A:I1=(I1∩I2âˆ©â‹Żâˆ©In):I1=(I1:I1)∩(I2:I1)âˆ©â‹Żâˆ©(In:I1)=I2âˆ©â‹Żâˆ©In.

This is also a contradiction. By a suitable ordering, we may have rf(I1) = rf( I1â€Č).

We use an induction on the number n of f-primary components. If n = 1, then A=I1=⋂i=1mIiâ€Č. Suppose that m > 1. Then I1 ⊈ rf(Ii) for all 2 ≀ i ≀ m. Since S=I1:I1=⋂i=1m(Iiâ€Č:I1), we have, by Theorem 4.4, S=I2â€Č=⋯=Imâ€Č which is a contradiction. Thus m = 1 = n. Now let us suppose that the conclusions hold for the ideals which are represented by fewer than nf-primary components. Let I=I1∩I1â€Č. Then I is an f-primary ideal such that rf(I) = rf(I1) = rf( I1â€Č) from Proposition 4.3. By the condition (Îł), we have S = I1 : I1 ⫅ I1 : I and thus I1 : I = S. From the fact that I ⊈ rf(Ii) for all 2 ≀ i ≀ n, we obtain Ii : I = Ii. Hence A:I=⋂i=2nIi. Similarly, we can show that A:I=⋂i=2mIiâ€Č. Consequently, A:I=⋂i=2nIi=⋂i=2mIiâ€Č and the decompositions are both normal. By the induction hypothesis, we have n – 1 = m – 1 and so n = m. Moreover, by a suitable ordering, we have rf(Ii) = rf( Iiâ€Č) for all 2 ≀ i ≀ n = m. □

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).

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Received: 2018-02-06
Accepted: 2018-04-25
Published Online: 2018-06-07

© 2018 Gu, published by De Gruyter

This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 License.

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  24. On algebraic characterization of SSC of the Jahangir’s graph 𝓙n,m
  25. A greedy algorithm for interval greedoids
  26. On nonlinear evolution equation of second order in Banach spaces
  27. A primal-dual approach of weak vector equilibrium problems
  28. On new strong versions of Browder type theorems
  29. A GerĆĄgorin-type eigenvalue localization set with n parameters for stochastic matrices
  30. Restriction conditions on PL(7, 2) codes (3 ≀ |𝓖i| ≀ 7)
  31. Singular integrals with variable kernel and fractional differentiation in homogeneous Morrey-Herz-type Hardy spaces with variable exponents
  32. Introduction to disoriented knot theory
  33. Restricted triangulation on circulant graphs
  34. Boundedness control sets for linear systems on Lie groups
  35. Chen’s inequalities for submanifolds in (Îș, ÎŒ)-contact space form with a semi-symmetric metric connection
  36. Disjointed sum of products by a novel technique of orthogonalizing ORing
  37. A parametric linearizing approach for quadratically inequality constrained quadratic programs
  38. Generalizations of Steffensen’s inequality via the extension of Montgomery identity
  39. Vector fields satisfying the barycenter property
  40. On the freeness of hypersurface arrangements consisting of hyperplanes and spheres
  41. Biderivations of the higher rank Witt algebra without anti-symmetric condition
  42. Some remarks on spectra of nuclear operators
  43. Recursive interpolating sequences
  44. Involutory biquandles and singular knots and links
  45. Constacyclic codes over đ”œpm[u1, u2,⋯,uk]/〈 ui2 = ui, uiuj = ujui〉
  46. Topological entropy for positively weak measure expansive shadowable maps
  47. Oscillation and non-oscillation of half-linear differential equations with coeffcients determined by functions having mean values
  48. On 𝓠-regular semigroups
  49. One kind power mean of the hybrid Gauss sums
  50. A reduced space branch and bound algorithm for a class of sum of ratios problems
  51. Some recurrence formulas for the Hermite polynomials and their squares
  52. A relaxed block splitting preconditioner for complex symmetric indefinite linear systems
  53. On f - prime radical in ordered semigroups
  54. Positive solutions of semipositone singular fractional differential systems with a parameter and integral boundary conditions
  55. Disjoint hypercyclicity equals disjoint supercyclicity for families of Taylor-type operators
  56. A stochastic differential game of low carbon technology sharing in collaborative innovation system of superior enterprises and inferior enterprises under uncertain environment
  57. Dynamic behavior analysis of a prey-predator model with ratio-dependent Monod-Haldane functional response
  58. The points and diameters of quantales
  59. Directed colimits of some flatness properties and purity of epimorphisms in S-posets
  60. Super (a, d)-H-antimagic labeling of subdivided graphs
  61. On the power sum problem of Lucas polynomials and its divisible property
  62. Existence of solutions for a shear thickening fluid-particle system with non-Newtonian potential
  63. On generalized P-reducible Finsler manifolds
  64. On Banach and Kuratowski Theorem, K-Lusin sets and strong sequences
  65. On the boundedness of square function generated by the Bessel differential operator in weighted Lebesque Lp,α spaces
  66. On the different kinds of separability of the space of Borel functions
  67. Curves in the Lorentz-Minkowski plane: elasticae, catenaries and grim-reapers
  68. Functional analysis method for the M/G/1 queueing model with single working vacation
  69. Existence of asymptotically periodic solutions for semilinear evolution equations with nonlocal initial conditions
  70. The existence of solutions to certain type of nonlinear difference-differential equations
  71. Domination in 4-regular Knödel graphs
  72. Stepanov-like pseudo almost periodic functions on time scales and applications to dynamic equations with delay
  73. Algebras of right ample semigroups
  74. Random attractors for stochastic retarded reaction-diffusion equations with multiplicative white noise on unbounded domains
  75. Nontrivial periodic solutions to delay difference equations via Morse theory
  76. A note on the three-way generalization of the Jordan canonical form
  77. On some varieties of ai-semirings satisfying xp+1 ≈ x
  78. Abstract-valued Orlicz spaces of range-varying type
  79. On the recursive properties of one kind hybrid power mean involving two-term exponential sums and Gauss sums
  80. Arithmetic of generalized Dedekind sums and their modularity
  81. Multipreconditioned GMRES for simulating stochastic automata networks
  82. Regularization and error estimates for an inverse heat problem under the conformable derivative
  83. Transitivity of the Δm-relation on (m-idempotent) hyperrings
  84. Learning Bayesian networks based on bi-velocity discrete particle swarm optimization with mutation operator
  85. Simultaneous prediction in the generalized linear model
  86. Two asymptotic expansions for gamma function developed by Windschitl’s formula
  87. State maps on semihoops
  88. 𝓜𝓝-convergence and lim-inf𝓜-convergence in partially ordered sets
  89. Stability and convergence of a local discontinuous Galerkin finite element method for the general Lax equation
  90. New topology in residuated lattices
  91. Optimality and duality in set-valued optimization utilizing limit sets
  92. An improved Schwarz Lemma at the boundary
  93. Initial layer problem of the Boussinesq system for Rayleigh-Bénard convection with infinite Prandtl number limit
  94. Toeplitz matrices whose elements are coefficients of Bazilevič functions
  95. Epi-mild normality
  96. Nonlinear elastic beam problems with the parameter near resonance
  97. Orlicz difference bodies
  98. The Picard group of Brauer-Severi varieties
  99. Galoisian and qualitative approaches to linear Polyanin-Zaitsev vector fields
  100. Weak group inverse
  101. Infinite growth of solutions of second order complex differential equation
  102. Semi-Hurewicz-Type properties in ditopological texture spaces
  103. Chaos and bifurcation in the controlled chaotic system
  104. Translatability and translatable semigroups
  105. Sharp bounds for partition dimension of generalized Möbius ladders
  106. Uniqueness theorems for L-functions in the extended Selberg class
  107. An effective algorithm for globally solving quadratic programs using parametric linearization technique
  108. Bounds of Strong EMT Strength for certain Subdivision of Star and Bistar
  109. On categorical aspects of S -quantales
  110. On the algebraicity of coefficients of half-integral weight mock modular forms
  111. Dunkl analogue of SzĂĄsz-mirakjan operators of blending type
  112. Majorization, “useful” Csiszár divergence and “useful” Zipf-Mandelbrot law
  113. Global stability of a distributed delayed viral model with general incidence rate
  114. Analyzing a generalized pest-natural enemy model with nonlinear impulsive control
  115. Boundary value problems of a discrete generalized beam equation via variational methods
  116. Common fixed point theorem of six self-mappings in Menger spaces using (CLRST) property
  117. Periodic and subharmonic solutions for a 2nth-order p-Laplacian difference equation containing both advances and retardations
  118. Spectrum of free-form Sudoku graphs
  119. Regularity of fuzzy convergence spaces
  120. The well-posedness of solution to a compressible non-Newtonian fluid with self-gravitational potential
  121. On further refinements for Young inequalities
  122. Pretty good state transfer on 1-sum of star graphs
  123. On a conjecture about generalized Q-recurrence
  124. Univariate approximating schemes and their non-tensor product generalization
  125. Multi-term fractional differential equations with nonlocal boundary conditions
  126. Homoclinic and heteroclinic solutions to a hepatitis C evolution model
  127. Regularity of one-sided multilinear fractional maximal functions
  128. Galois connections between sets of paths and closure operators in simple graphs
  129. KGSA: A Gravitational Search Algorithm for Multimodal Optimization based on K-Means Niching Technique and a Novel Elitism Strategy
  130. Ξ-type Calderón-Zygmund Operators and Commutators in Variable Exponents Herz space
  131. An integral that counts the zeros of a function
  132. On rough sets induced by fuzzy relations approach in semigroups
  133. 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
  134. The fourth order strongly noncanonical operators
  135. Topical Issue on Cyber-security Mathematics
  136. Review of Cryptographic Schemes applied to Remote Electronic Voting systems: remaining challenges and the upcoming post-quantum paradigm
  137. Linearity in decimation-based generators: an improved cryptanalysis on the shrinking generator
  138. On dynamic network security: A random decentering algorithm on graphs
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