Research article

Existence and compatibility of positive solutions for boundary value fractional differential equation with modified analytic kernel

  • Received: 11 September 2022 Revised: 30 December 2022 Accepted: 16 January 2023 Published: 28 January 2023
  • MSC : 34B18, 45B05

  • In this article, a Green's function for a fractional boundary value problem in connection with modified analytic kernel has been constructed to study the existence of multiple solutions of a type of characteristic fractional boundary value problems. It is done here by using a well-known result: Krasnoselskii fixed point theorem. Moreover, a practical example is created to understand the importance of main results regarding the existence of solution of a boundary value fractional differential problem with homogeneous conditions. This example analytically and graphically, explains circumstances under which the Green's functions with different types of differential operator are compatible.

    Citation: Amna Kalsoom, Sehar Afsheen, Akbar Azam, Faryad Ali. Existence and compatibility of positive solutions for boundary value fractional differential equation with modified analytic kernel[J]. AIMS Mathematics, 2023, 8(4): 7766-7786. doi: 10.3934/math.2023390

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  • In this article, a Green's function for a fractional boundary value problem in connection with modified analytic kernel has been constructed to study the existence of multiple solutions of a type of characteristic fractional boundary value problems. It is done here by using a well-known result: Krasnoselskii fixed point theorem. Moreover, a practical example is created to understand the importance of main results regarding the existence of solution of a boundary value fractional differential problem with homogeneous conditions. This example analytically and graphically, explains circumstances under which the Green's functions with different types of differential operator are compatible.



    A semiring is a natural generalization of ring and it has many applications to various fields such as optimization theory, max/min algebra, algebra of formal process, combinatorial optimization, automata theory, etc. Semirings were introduced explicitly by Vandiverin [21] in 1934. By considering the ring of rational numbers (Q,+,) that plays an important role in ring theory, we get the subset (Q+{0},+) of all non-negative rational numbers is an abelian additive semigroup which is closed under the usual multiplication of rational numbers, i.e., (Q+{0},+,) forms a semiring.

    Fuzzy set, a generalization of the crisp set, was introduced in 1965 by Zadeh [23] where he assigned to each set's element a degree of belongingness (truth value: "t") with 0t1. Intuitionistic fuzzy set, as an extension of the fuzzy set, was introduced in 1986 by Atanassov [8] where he assigned to each set's element a degree of belongingness (truth value: "t") and a degree of non-belongingness (falsity value: "f") with 0t,f1,0t+f1. Neutrosophic set, as an extension of intuitionistic fuzzy set, was introduced in 1998 by Smarandache [18] where he assigned to each set's element a truth value "T", indeterminacy value "I", and falsity value "F" with 0T,I,F1. For detailed information about neutrosophic sets, we refer to [19,20].

    The connection between algebraic structures (hyperstructures) and each of fuzzy, intuitionistic fuzzy, and neutrosophic sets have been of great interest to many algebraists. Different concepts were introduced and studied. For more details, we refer to [3,4,5,6,7] for neutrosophic algebraic structures (hyperstructures) and to [1,9,10,12,15,16] for fuzzy algebraic structures (hyperstructures).

    In [3], Al-Tahan et al. discussed single valued neutrosophic (SVN) subsets of ordered groupoids and in [7], Akara et al. studied SVN subsets of ordered semigroups. In 2019, Mahboub et al. [15] studied fuzzy (m,n)-ideals in semigroups. Our paper is concerned about SVN-(m,n)-ideals in ordered semirings and it is structured as follows: In Section 2, we present basic definitions and examples on single valued neutrosophic sets and their operations. In Section 3, we discuss some results on (m,n)-ideals in ordered semirings. Finally in Section 4, we define SVN-(m,n)-ideals in ordered semirings, study their properties, present non-trivial examples on them, and find a relationship between (m,n)-ideals of ordered semirings and level sets.

    In this section, we present some preliminaries related to single valued neutrosophic sets (SVNSs) and some of their operations.

    Definition 2.1. [22] Let U be a non-empty space of elements. A single valued neutrosophic set (SVNS) A on U is characterized by the functions: TA (truth-membership), IA (indeterminacy-membership), and FA (falsity-membership). Here, the range of each of these functions is a subset of the unit interval and A is denoted as follows:

    A={x(TA(x),IA(x),FA(x)):xU}.

    Definition 2.2. [6] Let U be a non-empty set, A an SVNS over U, and 0α1,α2,α31. Then the (α1,α2,α3)-level set of A is defined as follows:

    L(α1,α2,α3)={xU:TA(x)α1,IA(x)α2,FA(x)α3}.

    Definition 2.3. [22] Let U be a non-empty set and A,B be SVNSs over U defined as follows.

    A={x(TA(x),IA(x),FA(x)):xU},B={x(TB(x),IB(x),FB(x)):xU}.

    Then

    (1) A is called a single valued neutrosophic subset of B (AB) if for every xU, we have TA(x)TB(x), IA(x)IB(x), and FA(x)FB(x).

    Moreover, A and B are said to be equal single valued neutrosophic sets (A=B) if A is a single valued neutrosophic subset of B and B is a single valued neutrosophic subset of A.

    (2) The intersection of A and B is defined to be the SVNS over U:

    AB={x(TAB(x),IAB(x),FAB(x)):xU}.

    Here, TAB(x)=TA(x)TB(x), IAB(x)=IA(x)IB(x), and FAB(x)=FA(x)FB(x) for all xU.

    (3) The union of A and B is defined to be the SVNS over U:

    AB={x(TAB(x),IAB(x),FAB(x)):xU}.

    Here, TAB(x)=TA(x)TB(x), IAB(x)=IA(x)IB(x), and FAB(x)=FA(x)FB(x) for all xU.

    Example 1. Let U={m0,m1,m2} and A1,A2 be SVNS over U defined as follows.

    A1={m0(0.67,0.64,0.45),m1(0.83,0.54,0.32),m2(0.12,0.75,1)},
    A2={m0(0.89,0.01,0.7),m1(1,0,0.76),m2(0.99,0.53,0.763)}.

    Then the SVNSs A1A2 and A1A2 over U are as follows.

    A1A2={m0(0.89,0.01,0.45),m1(1,0,0.32),m2(0.99,0.53,0.763)},
    A1A2={m0(0.67,0.64,0.7),m1(0.83,0.54,0.76),m2(0.12,0.75,1)}.

    Definition 2.4. Let X,Y be non-empty sets, f:XY a function, and A be an SVNS over Y defined as follows.

    A={y(TA(y),IA(y),FA(y)):yY}.

    Then f1(A) is a single valued neutrosophic set over X defined as follows.

    f1(A)={x(Tf1(A)(x),If1(A)(x),Ff1(A)(x)):xX}.

    Here, Tf1(A)(x)=TA(f(x)), If1(A)(x)=IA(f(x)), and Ff1(A)(x)=FA(f(x)).

    Definition 2.5. Let X,Y be non-empty sets, A,B be SVNSs over X,Y respectively defined as follows.

    A={x(TA(x),IA(x),FA(x):xX},B={y(TA(y),IA(y),FA(y)):yY}.

    Then A×B is an SVNS over X×Y defined as follows.

    A×B={(x,y)(TA×B((x,y)),IA×B((x,y)),FA×B((x,y))):xX,yY}.

    Here, TA×B((x,y))=TA(x)TB(y), IA×B((x,y))=IA(x)IB(y), and FA×B((x,y))=FA(x)FB(y).

    In this section, we deal with ordered semirings by presenting some results related to their (m,n)-ideals. For more details about (ordered) semirings, we refer to the books [13,14].

    Definition 3.1 [14] Let R be a non-empty set with binary operations "+" and "". Then (R,+,) is called a semiring if the following conditions hold for all a,b,cR:

    (1) (R,+) is an abelian semigroup with identity "0";

    (2) (R,) is a semigroup with "0" as bilaterally absorbing element (i.e. a0=0a=0);

    (3) a(b+c)=ab+ac and (a+b)c=ac+bc.

    A semiring (R,+,) is called semifield if (R,) is a commutative semigroup with unity "1", and for every non-zero element r1R, there exists r2R with r1r2=1.

    Definition 3.2. [13] Let (R,+,) be a semiring with 0R and "" be a partial order on R. Then (R,+,,) is called an ordered semiring if the following conditions are satisfied for all x,y,aR.

    (1) If xy then a+xa+y;

    (2) If xy then axay and xaya for all 0a.

    Remark 1. Every semiring R is an ordered semiring under the trivial order (i.e., uv if and only if u=v for all u,vR).

    Example 2. [11] Let Q+{0} be the set of non-negative rational numbers, R be the set of real numbers, and "" be the usual order of real numbers. Then (Q+{0},+,,) and (R{±},,,) are examples of infinite ordered semirings. Here "" and "" denote the minimum and maximum respectively.

    Example 3. Let R be the set of real numbers. Then (R{},,+,) is an ordered semifield.

    Example 4. [16] Let "" and "" be the operations defined on the unit interval [0,1] as follows: For all x,y[0,1],

    xy=xy      and      xy=(x+y1)0.

    Then ([0,1],,,) is an ordered semiring (with "0" as a zero element) under the usual order of real numbers. Moreover, it is not an ordered semifield.

    We can present finite semirings by means of Cayley's tables.

    Example 5. Let M={0,t1,t2} and define (M,,) by the following tables.

    |         0      t1      t2   |00t1t2t1t1t1t2t2t2t2t2|         0      t1      t2   |0000t1000t2000

    Then (M,,) is a semiring. By defining the partial order "" on M as follows

    ≤={(0,0),(0,t1),(0,t2),(t1,t1),(t1,t2),(t2,t2)},

    we get that (M,,) is a commutative ordered semiring.

    Example 6. Let N={0,1,2,3} and define (N,,) by the following tables.

    |         0      1      2      3   |00123111332232233322|         0      1      2      3   |00000101232012330123

    Then (N,,) is a semiring. By defining the partial order "" on N as follows

    ≤={(0,1),(0,1),(1,1),(2,2),(2,3),(3,3)},

    we get that (N,,,) is an ordered semiring.

    Definition 3.3. Let (R,+,,) be an ordered semiring and SR. Then S is a subsemiring of R if the following conditions hold.

    (1) 0S;

    (2) S+SS and S2S;

    (3) (S]={xR:xs for some sS}S.

    Example 7. Let (N,,,) be the ordered semiring of Example 6. Then {0,1} is a subsemiring of N.

    Ideals play an important role in semiring theory. Some generalizations of ideals were established so that one can study a semiring by the properties of its (generalized) ideals.

    Definition 3.4. Let (R,+,,) be an ordered semiring, I a subsemiring of R, and m,n be non-negative integers with (m,n)(0,0). Then I is:

    (1) a left ideal of R if RII.

    (2) a right ideal of R if IRI.

    (3)m a bi-ideal of R if IRII.

    (4) an (m,n)- ideal of R if ImRInI.

    Remark 2. Let (R,+,,) be an ordered semiring and IR. Then the following statements hold.

    (1) If I is an ideal of R then I is a left (right) ideal, bi-ideal, and an (m,n)-ideal of R for all m,n1.

    (2) If I is a left ideal of R and n1 then I is a (0,n)-ideal of R.

    (3) m If I is a right ideal of R and m1 then I is an (m,0)-ideal of R.

    (4) m If I is an (m,n)-ideal of R and sm,tn then I is an (s,t)-ideal of R.

    Proposition 3.5. Let (R,+,,) be an ordered semifield, IR, and m,n be non-negative integers with (m,n)(0,0). If I is an (m,n)-ideal of R then I={0} or I=R. Moreover, I={0} is an (m,n)-ideal of R if and only if ({0}]={0}.

    Proof. Let I{0} be an (m,n)-ideal of R. Since R is commutative, it follows that I is an (m+n,0)-ideal of R. Let x0I. Then 1=x1x1m+n timesxm+nI. For any rR, we have r=11m+n timesrI. Therefore, I=R.

    It is clear that I={0} is an (m,n)-ideal of R if and only if ({0}]={0}.

    Example 8. Let (R{},,+,) be the ordered semifield in Example 3 and m,n be non-negative integers with (m,n)(0,0). Proposition 3.5 asserts that {} and R are the only (m,n)-ideals of R{}.

    Definition 3.6. Let (R1,+1,1,1) and (R2,+2,2,2) be ordered semirings and ϕ:R1R2 be a function. Then ϕ is an ordered semiring homomorphism if the following conditions hold for all r,sR1.

    (1) ϕ(r+1s)=ϕ(r)+2ϕ(s);

    (2) ϕ(r1s)=ϕ(r)2ϕ(s);

    (3) If r1s then ϕ(r)2ϕ(s).

    If ϕ is a bijective ordered semiring homomorphism then ϕ is an ordered semiring isomorphism and R1 and R2 are said to be isomorphic ordered semirings.

    Theorem 3.7. Let (R1,+1,1,1) and (R2,+2,2,2) be ordered semirings. Then (R1×R2,+,,) is an ordered semiring, where "+", "", and "" is defined as follows for all (r,s),(r,s)R1×R2.

    ((r,s)+(r,s)=(r+1r,s+s),(r,s)(r,s)=(r1r,s2s),
    (r,s)(r,s)if and only ifr1rands2s.

    In this section, we define SVN-(m,n)-ideals of ordered semirings and present some related non-trivial examples. Furthermore, we study various properties of them and find a relationship between their level sets and the (m,n)-ideals.

    Notation 1. Let U be a non-empty set, x,yU, and A be an SVNS over U. Then NA(x)NA(y) is equivalent to TA(x)TA(y), IA(x)IA(y), FA(x)FA(y).

    Definition 4.1. Let (R,+,,) be an ordered semiring and A an SVNS on R. Then A is an SVN-subsemiring of R if the following conditions hold for all x,yR.

    (1) NA(x+y)NA(x)NA(y);

    (2) NA(xy)NA(x)NA(y);

    (3) If xy then NA(x)NA(y).

    Definition 4.2. Let (R,+,,) be an ordered semiring and A an SVNS on R. Then A is an SVN-left (right) ideal of R if the following conditions hold for all x,yR.

    (1) NA(x+y)NA(x)NA(y);

    (2) NA(xy)NA(y) (respectively NA(xy)NA(x));

    (3) If xy then NA(x)NA(y).

    Moreover, A is an SVN-ideal of R if it is both: an SVN-left ideal of R and an SVN-right ideal of R.

    Example 9. Let ([0,1],,,) be the ordered semiring of Example 4 and A be the SVNS on [0,1] defined as follows:

    NA(x)={(0.6,0.3,0.2) if 0x0.5;(0.3,0.4,0.7) otherwise.

    Then A is an SVN-ideal of [0,1].

    Next, we give an example on an SVN-right ideal of R that is not an SVN-ideal of R.

    Example 10. Let O={0,k1,k2,k3} and define (O,+,) by the following tables.

    |   +      0      k1      k2      k3|00k1bk3k1k1k1k1k1k2k2k1k2k3k3k3k1k3k3|         0      k1      k2      k3|00000k10k1k1k1k20k2k2k2k30k1k1k1

    Then (O,+,) is a semiring [17]. By considering the trivial order "" on O, we get that (O,+,,) is an ordered semiring. Let A be the SVNS on R defined as follows:

    A={0(0.7,0.2,0.35),k1(0.65,0.32,0.4),k2(0.1,0.6,0.9),k3(0.1,0.6,0.9)}.

    Then A is an SVN-right ideal of R. Moreover, it is not an SVN-left ideal of R as NA(b)=NA(ab)NA(a). Thus, it is not an SVN-ideal of R.

    Definition 4.3. Let (R,+,,) be an ordered semiring, m,n be non-negative integers with (m,n)(0,0), and A an SVNS on R. Then A is an SVN-(m,n)-ideal of R if the following conditions hold for all x,y,x1,,xm,y1,,yn,rR.

    (1) NA(x+y)NA(x)NA(y);

    (2) NA(x1xmry1yn)NA(x1)NA(xm)NA(y1)NA(yn);

    (3) If xy then NA(x)NA(y).

    Moreover, if m=n=1 then A is an SVN-bi-ideal of R.

    Next, we give an example on an SVN-bi-ideal of an ordered semiring that is not an SVN-left ideal nor an SVN-right ideal.

    Example 11. Let M be the set of non-negative real numbers and (M2(M),+,) be the semiring of all 2×2 matrices with non-negative real entries under usual addition and multiplication of matrices. By defining "" on M2(M) to be the trivial order, we get that (M2(M),+,,) is an ordered semiring.

    Let A be the SVNS on M2(M) defined as follows:

    NA((a11a12a21a22))={(0.9,0.2,0.204) if a12=a21=a22=0;(0.54,0.367,0.359) otherwise.

    Then A is an SVN-bi-ideal of M2(S).

    Since NA((1111)(1000))=(0.54,0.367,0.359)(0.9,0.2,0.204)=NA((1000)), it follows that A is not an SVN-left ideal of M2(M). Furthermore, because NA((1000)(1111))=NA((1100))=(0.54,0.367,0.359)(0.9,0.2,0.204)=NA((1000)), it follows that A is not an SVN-right ideal of M2(M).

    Remark 3. Let (R,+,,) be an ordered semiring, m,n be non-negative integers with (m,n)(0,0), and A an SVNS on R. Then the following are true.

    (1) If A is an SVN-ideal of R then A is an SVN left (right)-SVN-ideal, SVN-bi-ideal, and an SVN-(m,n)-ideal of R for all m,n1.

    (2) If A is an SVN-left ideal of R then A is an SVN-(0,n)-ideal of R for all n1.

    (3) If A is an SVN-right ideal of R then A is an SVN-(m,0)-ideal of R for all m1.

    (4) If A is an SVN-(m,n)-ideal of R then A is an SVN-(s,t)-ideal of R for all sm,tn.

    Next, we give an example on an SVN-(m,n)-ideal of an ordered semiring that is not an SVN-bi-ideal.

    Example 12. Let T=Q+{0} and (M3(T),+,) be the semiring of all 3×3 matrices with non-negative rational entries under usual addition and multiplication of matrices. By defining "" on M3(T) to be the trivial order, we get that (M3(T),+,,) is an ordered semiring.

    Let A be the SVNS on M3(T) defined as follows:

    NA((a11a12a13a21a22a23a31a32a33))={(0.889,0.12,0.0322) if a11=a21=a22=a31=a32=a33=0;(0.687,0.597,0.549) otherwise.

    Then A is an SVN-(2,1)-ideal of M3(T) that is not an SVN-bi-ideal of M3(T). Moreover, A is an SVN-(m,0)-ideal of M3(T) and an SVN-(0,n)-ideal of M3(T) for all m,n3.

    Proposition 4.4. Let (R,+,,) be an ordered semiring, m,n be non-negative integers with (m,n)(0,0), and A an SVNS on R. If A is an SVN-(m,n)-ideal of R then NA(0)NA(x) for all xR.

    Proof. Having A an SVN-(m,n)-ideal of R implies that NA(0)=NA(xm0xn)NA(x) for all xR.

    Proposition 4.5. Let (R,+,,) be an ordered semifield, m,n be non-negative integers with (m,n)(0,0), and A an SVNS on R. If A is an SVN-(m,n)-ideal of R then NA(r)=NA(1) for all rR{0}.

    Proof. Having A an SVN-(m,n)-ideal of R and R commutative implies that NA(r)=NA(1m+nr)NA(1) for all rR. Moreover, we have NA(1)=NA(rmnrm+n)NA(r) for all rR{0}. Therefore, NA(r)=NA(1) for all xR{0}.

    Example 13. Let (R{},,+,) be the ordered semifield in Example 3, m,n be non-negative integers with (m,n)(0,0), and A be the SVNS on R{} defined as follows.

    NA(x)={(0.687,0.34,0.28) if x[1,[;(0.956,0.14,0.08) otherwise.

    Proposition 4.5 asserts that A is not an SVN-(m,n)-ideal of R{}.

    Next, we deal with some operations involving SVN-(m,n)-ideals of ordered semirings.

    Lemma 4.6. Let (R1,+1,1,1) and (R2,+2,2,2) be ordered semirings, m,n be non-negative integers with (m,n)(0,0), and A,B be SVN-(m,0)-ideal and SVN-(0,n)-ideal of R1,R2 respectively. Then A×B is an SVN-(m,n)-ideal of R1×R2.

    Proof. Let (x,y),(z,w)R1×R2. Then NA×B(x+1z,y+2w)=NA(x+1z)NA(y+2w). Having A,B SVN-(m,0)-ideal and SVN-(0,n)-ideal of R1,R2 respectively implies that NA×B(x+1z,y+2w)NA(x)NA(z)NB(y)NB(w)=NA×B(x,y)NA×B(z,w). Let (x1,y1),,(xm,ym),(r1,r2),(z1,w1),, (zn,wn)R1×R2,

    v=(x1,y1)(xm,ym)(r1,r2)(z1,w1)(zn,wn), v1=x1xmr1z1zn, and v2=y1ymr2w1wn. Then NA×B(v)=NA×B((v1,v2))=NA(v1)NB(v2). Having A,B SVN-(m,0)-ideal and SVN-(0,n)-ideal of R1,R2 respectively implies that NA(v1)=NA(x1xm(r1z1zn))NA(x1)NA(xm) and NB(v2)=NB((y1ymr2)w1wn)NB(w1)NB(wn). The latter implies that NA×B(v)NA(x1)NA(xm)NB(w1)NB(wn)NA(x1)NA(xm)NA(z1)NA(zn)NB(y1)NB(ym)NB(w1)NB(wn) and hence,

    NA×B(v)NA×B((x1,y1))NA×B((xm,ym))NA×B((z1,w1))NA×B((zn,wn)).

    Let (x,y)(z,w)R1×R2. Having x1z,y2w and A,B SVN-(m,0)-ideal and SVN-(0,n)-ideal of R1,R2 respectively implies that NA(x)NA(z) and NB(y)NB(w). The latter implies that NA×B((x,y))NA×B((z,w)). Therefore, A×B is an SVN-(m,n)-ideal of R1×R2.

    Corollary 4.7. Let (R1,+1,1,1) and (R2,+2,2,2) be ordered semirings and A,B be SVN-right-ideal and SVN-left-ideal of R1,R2 respectively. Then A×B is an SVN-bi-ideal of R1×R2.

    Example 14. Let O={0,p1,p2,p3} and define (O,+,) by the following tables.

    |   +      0      p1      p2      p3|00p1p2p3p1p1p1p1p1p2p2p1p2p3p3p3p1p3p3|         0      p1      p2      p3|00000p10p1p2p3p20p1p2p3p30p1p2p3

    Then (O,+,) is a semiring. By defining the trivial order "" on O, we get that (O,+,,) is an ordered semiring [2]. Let A be the SVNS on O defined as follows:

    A={0(0.7,0.2,0.35),p1(0.65,0.32,0.4),p2(0.1,0.6,0.9),p3(0.1,0.6,0.9)}.

    Then A is an SVN-left ideal of O. By taking the ordered semiring (O,+,,) in Example 5 and its SVN-right ideal A, we get by using Corollary 4.7 that A×B is an SVN-bi-ideal of O×O. Here, A×A is defined as follows.

    NA((x,y))={(0.7,0.2,0.35) if (x,y)=(0,0);(0.65,0.32,0.4) if x{0,k1},y{0,p1}, and (x,y)(0,0);(0.1,0.6,0.9) otherwise.

    Proposition 4.8. Let (R,+,,) be an ordered semiring, m,n be non-negative integers with (m,n)(0,0), and A1,A2 be SVN-(m,n)-ideals of R. Then A1A2 is an SVN-(m,n)-ideal of R.

    Proof. The proof is straightforward.

    Remark 4. Let (R,+,,) be an ordered semiring, m,n be non-negative integers with (m,n)(0,0), and A,B be SVN-(m,n)-ideals of R. Then AB is not necessary an SVN-(m,n)-ideal of R.

    Example 15. Let (N,,,) be the ordered semiring of Example 6 and A,B be the SVNS on R defined as follows.

    A={0(0.67,0.02,0.135),1(0.55,0.25,0.24),2(0.21,0.46,0.89),3(0.21,0.46,0.89)},
    B={0(0.63,0.01,0.335),1(0.321,0.56,0.91),2(0.55,0.15,0.44),3(0.321,0.56,0.91)}.

    It is easy to see that A,B are SVN-(0,1)-ideals of N. Having

    AB={0(0.67,0.01,0.135),1(0.55,0.25,0.24),2(0.55,0.15,0.44),3(0.321,0.46,0.89)},

    3=1+2, and NAB(3)NAB(1)NAB(2) implies that AB is not an SVN-(0,1)-ideal of N.

    Lemma 4.9. Let (R1,+1,1,1) and (R2,+2,2,2) be ordered semirings, m,n be non-negative integers with (m,n)(0,0), and A,B be SVN-(m,n)-ideals of R1,R2 respectively. Then A×B is an SVN-(m,n)-ideal of R1×R2.

    Proof. The proof is similar to that of Lemma 4.6.

    Proposition 4.10. Let (R,+,,) be an ordered semiring with a2=a for all aR, m,n be positive integers, and A an SVNS on R. Then A is an SVN-(m,n)-ideal of R if and only if A is an SVN-bi-ideal of R.

    Proof. The proof is straightforward as xmryn=xry for all x,y,rR.

    Proposition 4.11. Let (R,+,,) be a commutative ordered semiring with a2=a for all aR, m,n be non-negative integers with (m,n)(0,0), and A an SVNS on R. Then A is an SVN-(m,n)-ideal of R if and only if A is an SVN-ideal of R.

    Proof. The proof is straightforward as xm+n=x for all xR.

    Next, we show that the pre-image of an SVN-(m,n)-ideal under ordered semiring homomorphism is an SVN-(m,n)-ideal.

    Proposition 4.12. Let (R1,+1,1,1) and (R2,+2,2,2) be ordered semirings, A an SVNS on R2, m,n be non-negative integers with (m,n)(0,0), and f:R1R2 an ordered semiring homomorphism. If A is an SVN-(m,n)-ideal of R2 then f1(A) is an SVN-(m,n)-ideal of R1.

    Proof. Let x,yR1. Then Nf1(A)(x+1y)=NA(f(x+1y))=NA(f(x)+2f(y)). Having A an SVN-(m,n)-ideal of R2 implies that NA(f(x)+2f(y))NA(f(x))NA(f(y))=Nf1(A)(x)Nf1(A)(y) and hence, Nf1(A)(x+1y)Nf1(A)(x)Nf1(A)(y). Let x1,,xm,r,y1,,ynR1. Then

    Nf1(A)(x111xm1r1y111ym)=NA(f(x111xm1r1y111ym)).

    Since f is a homomorphism, it follows that

    Nf1(A)(x111xm1r1y111ym)=NA(f(x1)22f(xm)2f(r)2f(y1)22f(ym)).

    The latter and having A an SVN-(m,n)-ideal of R2 implies that for z=x111xm1r1y111ym,

    Nf1(A)(z)Nf1(A)(x1)Nf1(A)(xm)Nf1(A)(y1)Nf1(A)(yn).

    Let x1yR1. Having f(x)2f(y)R2 and A an SVN-(m,n)-ideal of R2 implies that NA(f(x))NA(f(y)). The latter implies that Nf1(A)(x)Nf1(A)(y). Therefore, f1(A) is an SVN-(m,n)-ideal of R1.

    Corollary 4.13. Let (R,+,,) be an ordered semiring, m,n be non-negative integers with (m,n)(0,0), and A={x(TA(x),IA(x),FA(x)):xR} an SVNS on R, and S a subsemiring of R. If A is an SVN-(m,n)-ideal of R then A={x(TA(x),IA(x),FA(x)):xS} is an SVN-(m,n)-ideal of S.

    Proof. Let f:SR be the inclusion map (i.e. f(x)=x for all xS). Proposition 4.12 asserts that f1(A)=A is an SVN-(m,n)-ideal of S.

    Next, we find a relationship between (m,n)-ideals of ordered semirings and level sets.

    Lemma 4.14. Let (R,+,,) be an ordered semiring, m,n be non-negative integers with (m,n)(0,0), and IR an (m,n)-ideal of R. Then I is a level set of an SVN-(m,n)-ideal of R.

    Proof. Let A be the SVNS on R defined as follows:

    N(x)={(1,0,0) if xI;(0,1,1) otherwise.

    One can easily see that A is an SVN-(m,n)-ideal of R and that A(1,0,0)=I.

    Theorem 4.15. Let (R,+,,) be an ordered semiring, m,n be non-negative integers with (m,n)(0,0), 0α,β,γ1, and A an SVNS on R. Then A is an SVN-(m,n)-ideal of R if and only if A(α,β,γ) is either the empty set or an (m,n)-ideal of R.

    Proof. Let A be an SVN-(m,n)-ideal of R and x,yA(α,β,γ). Then NA(x),NA(y)(α,β,γ). Having A an SVN-(m,n)-ideal of R implies that NA(x+y)NA(x)NA(y)(α,β,γ) and hence x+yA(α,β,γ). Let x1,,xm,y1,,ynA(α,β,γ) and rR. Having A an SVN-(m,n)-ideal of R implies that NA(x1xmry1yn)NA(x1)NA(xm)NA(y1)NA(yn)(α,β,γ) and hence, x1xmry1ynA(α,β,γ). Let yA(α,β,γ) and xy. Having A an SVN-(m,n)-ideal of R implies that NA(y)NA(x)(α,β,γ) and hence, yA(α,β,γ). Thus, A(α,β,γ) is an (m,n)-ideal of R.

    Conversely, let x,yR with NA(x)NA(y)=(α,β,γ). Having A(α,β,γ) an (m,n)-ideal of R and x,yA(α,β,γ) implies that x+yA(α,β,γ) and hence NA(x+y)(α,β,γ)=NA(x)NA(y). Let x1,,xm,r,y1,,ynR with NA(x1)NA(xm)NA(y1)NA(yn)=(α,β,γ). Having A(α,β,γ) an (m,n)-ideal of R implies that x1xmry1ynA(α,β,γ) and hence, NA(x1xmry1yn)(α,β,γ)=NA(x1)NA(xm)NA(y1)NA(yn). Let xy and NA(y)=(α0,β0,γ0). Having A(α0,β0,γ0) an (m,n)-ideal of R and yA(α0,β0,γ0) implies that xA(α0,β0,γ0) and hence, NA(x)(α0,β0,γ0)=NA(y). Thus, A is an SVN-(m,n)-ideal of R.

    This paper added to the neutrosophic algebraic structures theory by considering SVN subsets of ordered semirings. The concept of SVN-(m,n)-ideals of ordered semirings was defined and studied. Having every SVN-ideal (bi-ideal) of ordered semiring an SVN-(m,n)-ideal makes our results a generalization of SVN-ideals (bi-ideals) of ordered semirings. Also, having SVNSs a generalization of both: fuzzy sets and intuitionistic fuzzy sets makes the study of fuzzy-(m,n)-ideals and intuitionistic fuzzy-(m,n)-ideals of ordered semirings special cases of our study.

    It is well known that the concept of single valued neutrosophic sets and that of ordered semirings are well established in dealing with many real life problems. So, the new defined concepts in this paper would help to approach these problems with a different perspective. For future research, it would be interesting to consider other types of SVN substructures in ordered semirings.

    The authors declare that they have no conflict of interest.



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