The intention of this paper is to introduce and study some properties of the ideals of semirings
using the concept of Q-neutrosophic set

Mandal, Debabrata

English

The intention of this paper is to introduce and study some properties of the ideals of semirings using the concept of Q-neutrosophic set

Debabrata Mandal

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Neutrosophic Sets and Systems, Vol. 36, 2020University of New MexicoSome Properties of Q-Neutrosophic Ideals of SemiringsDebabrata MandalDepartment of Mathematics, Raja Peary Mohan College, Uttarpara, Hooghly-712258, Indiae-mail: dmandaljumath@gmail.com∗Correspondence: dmandaljumath@gmail.comAbstract. The intention of this paper is to introduce and study some properties of the ideals of semiringsusing the concept of Q-neutrosophic set.Keywords: Semiring; Q-neutrosophic ideal; Cartesian Product; Composition.—————————————————————————————————————————-1. IntroductionThe concept of fuzzy sets was introduced by Zadeh [21] in 1965 to overcome the uncertain-ties in various problems in environment, economics, engineering etc. As an extension of it,Atanassov [7] introduced intuitionistic fuzzy set in 1986, where a degree of non-membershipwas considered besides the degree of membership of each element with (membership value +non-membership value) ≤ 1.After that several generalizations such as, rough sets, vague sets, interval-valued sets etc. areconsidered as mathematical tools for dealing with uncertainties. In 2005, F. Smarandacheintroduced Neutrosophic set [19] in which he introduced the indeterminacy to intuitionisticfuzzy sets. So, the resultant can be taken as a tri-component logic which can be applied tonon-standard analysis such as decision making (for example, result of games (win/tie/defeat),votes, from no/yes/NA), control theory etc.. Since then several researchers has applied thisconcept in many practical fields such as multi-criteria decision making, signal processing, dis-aster management etc.. Some of its recent applications can be found in [1–5,9, 18,20].In 2011, Majumder [13] introduced and studied the concept of Q-fuzzifcation of ideals of Γ-semigroup. Akram et al [6], Lekkoksung [11, 12], Mandal [14], Qamar et al [15, 16] extendedthis concept in case of Γ-semigroup, ordered semigroups [10], ordered Γ-semiring, soft fields,group theory and investigated some important properties.Motivated by this idea and combining the concept with neutrosophic set, in the paper weDebabrata Mandal, Some Properties of Q-Neutrosophic Ideals of SemiringsNeutrosophic Sets and Systems, Vol. 36, 2020 369have studied the ideal theory of semirings since it has several applications in graph theory,automata theory, mathematical modelling etc. [8].2. PreliminariesAt first let us remember some definitions which will be used in the discussion of the paper.Definition 2.1. A semiring is a nonempty set S on which two operations + and · have beendefined such that (S,+) and (S, ·) form monoid where · distributes over + from any side.Definition 2.2. A nonempty subset X(6= S) of semiring S is said to be an ideal if for allx, y ∈ X and p ∈ S, x + y ∈ X, px ∈ X. Similarly we can define a right ideal also. An idealof S is a nonempty subset which satisfies both properties of left ideal and right ideal.Definition 2.3. A neutrosophic set N on the universe U is defined as N = {<u,AT (u), AI(u), AF (u) >, u ∈ U}, where AT , AI , AF : U →]−0, 1+[ and −0 ≤ AT (u)+AI(u)+AF (u) ≤ 3+ . For practical purposes, it is difficult to consider ]−0, 1+[. So, for studyingneutrosophic set we consider the set which takes the value from the subset of [0, 1].Definition 2.4. For a non-empty set Q, a mapping ν : S×Q −→ [0, 1] is said to be a Q-fuzzysubset of S and νl = {(s, q) ∈ S ×Q|ν(s, q) ≥ l} where l ∈ [0, 1] is its level subset.3. Main ResultsDefinition 3.1. Let ν = (νT , νI , νF ) be a non empty neutrosophic subset of a semiring S.Then ν is called a Q-neutrosophic left ideal of S if(i) νT (s1 + s2, p) ≥ min{νT (s1, p), νT (s2, p)}, νT (s1s2, p) ≥ νT (s2, p)(ii) νI(s1 + s2, p) ≥ νI(s1,p)+νI(s2,p)2 , νI(s1s2, p) ≥ νI(s2, p)(iii) νF (s1 + s2, p) ≤ max{νF (s1, p), νF (s2, p)}, νF (s1s2, p) ≤ νF (s2, p).for all s1, s2 ∈ S and p ∈ Q.Theorem 3.2. Any Q-neutrosophic set ν of a semiring S is a left ideal iff its level subsetsνTl := {(x, p) ∈ S × Q : νT (x, p) ≥ l, l ∈ [0, 1], p ∈ Q}, νIl := {(x, p) ∈ S × Q : νI(x, p) ≥l, l ∈ [0, 1]} and νFl := {(x, p) ∈ S ×Q : νF (x, p) ≤ l, l ∈ [0, 1]} are left ideals of S ×Q.Proof. Suppose ν of S is a Q-neutrosophic left ideal of S. Then anyone of νT , νI or νF is notequal to zero for some (s, p) ∈ S ×Q. Without loss of generality we consider, all of them arenot equal to zero.Debabrata Mandal, Some Properties of Q-Neutrosophic Ideals of SemiringsNeutrosophic Sets and Systems, Vol. 36, 2020 370Suppose a, b ∈ νl = (νTl , νIl , νFl ), s ∈ S and p ∈ Q. ThenνT (a+ b, p) ≥ min{νT (a, p), νT (b, p)} ≥ min{l, l} = lνI(a+ b, p) ≥ νI(a,p)+νI(b,p)2 ≥l+l2 = lνF (a+ b, p) ≤ max{νF (a, p), νF (b, p)} ≤ max{l, l} = lwhich implies (a+ b, p) ∈ νTl , νIl , νFl i.e., (a+ b, p) ∈ νl. AlsoνT (sa, p) ≥ νT (a, p) ≥ lνI(sa, p) ≥ νI(a, p) ≥ lνF (sa, p) ≤ νF (a, p) ≤ lHence (sa, p) ∈ νl.Therefore νl is a left ideal of S.Conversely, let νl( 6= φ) is a left ideal of S × Q and ν is not a Q-neutrosophic left ideal of S.Then for a, b ∈ S and p ∈ Q anyone of the following inequality will hold.νT (a+ b, p) < min{νT (a, p), νT (b, p)}νI(a+ b, p) < νI(a,p)+νI(b,p)2νF (a+ b, p) > max{νF (a, p), νF (b, p)}For the first inequality, choose l1 =12 [νT (a+b, p)+min{νT (a, p), νT (b, p)}]. Then νT (a+b, p) <l1 < min{νT (a, p), νT (b, p)} ⇒ (a, p), (b, p) ∈ νTl1 . but (a+ b, p) 6∈ νTl1- contradiction.For the second inequality, choose t2 =12 [νI(a + b, p) + min{νI(a, p), νI(b, p)}]. Then νI(a +b, p) < l2 <νI(a,p)+νI(b,p)2 ⇒ (a, p), (b, p) ∈ νIl2. But (a+ b, p) 6∈ νIl2 - contradiction.For the third inequality, choose t3 =12 [νF (a + b, p) + max{νF (a, p), νF (b, p)}]. Then νF (a +b, p) > l3 > max{νF (a, p), νF (b, p)} ⇒ (a, p), (b, p) ∈ νFt3 but (a+ b, p) 6∈ νFp3 - contradiction.Hence the theorem.Definition 3.3. For two Q-neutrosophic subsets ν and σ of S × Q, define their intersectionby(νT ∩ σT )(a, p) = min{νT (a, p), σT (a, p)}(νI ∩ σI)(a, p) = min{νI(a, p), σI(a, p)}(νF ∩ σF )(a, p) = max{νF (a, p), σF (a, p)}for all a ∈ S and p ∈ Q.Proposition 3.4. Intersection of any number of Q-neutrosophic left ideals of S is also aQ-neutrosophic left ideal.Debabrata Mandal, Some Properties of Q-Neutrosophic Ideals of SemiringsNeutrosophic Sets and Systems, Vol. 36, 2020 371Proof. Assume that {νi : c ∈ C} be a collection of Q-neutrosophic left ideals of S and a, b ∈ S,p ∈ Q. Then( ∩c∈CνTc )(a+ b, p) = infc∈CνTc (a+ b, p) ≥ infc∈C{min{νTc (a, p), νTc (b, p)}}= min{infc∈CνTc (a, p), infc∈CνTc (b, p)}= min{( ∩c∈CνTc )(a, p), ( ∩c∈CνTc )(b, p)}( ∩c∈CνIc )(a+ b, p) = infc∈CνIc (a+ b, p) ≥ infc∈CνIc (a,p)+νIc (b,p)2=infc∈CνIc (a,p)+infc∈CνIc (b,p)2=∩c∈CνIc (a,p)+ ∩c∈CνIc (b,p)2( ∩c∈CνFc )(a+ b, p) = supc∈CνFc (a+ b, p) ≤ supc∈C{max{νFc (a, p), νFc (b, p)}}= max{supc∈CνFc (a, p), supc∈CνFc (b, p)}= max{( ∩c∈CνFc )(a, p), ( ∩c∈CνFc )(b, p)}( ∩c∈CνTc )(ab, p) = infc∈CνTc (ab, p) ≥ infc∈CνTc (b, p) = ( ∩c∈CνTc )(b, p).( ∩c∈CνIc )(ab, p) = infc∈CνIc (ab, p) ≥ infc∈CνIc (b, p) = ( ∩c∈CνIc )(b, p).( ∩c∈CνFc )(ab, p) = supc∈CνFc (ab, p) ≤ supc∈CνFc (b, p) = ( ∩c∈CνFc )(b, p).Therefore ∩c∈Cνc is a Q-neutrosophic left ideal of S.Definition 3.5. For two Q-neutrosophic subsets ν and σ of S, define their cartesian productby(νT × σT )((a, b), p) = min{νT (a, p), σT (b, p)}(νI × σI)((a, b), p) = νI(a, p) + σI(b, p)2(νF × σF )((a, b), p) = max{νF ((a, p), σF (b, p)}∀a, b ∈ S, p ∈ Q.Theorem 3.6. For two Q-neutrosophic left ideals ν and σ of S, ν × σ is a Q-neutrosophicleft ideal of S × S.Proof. Let (a1, a2), (b1, b2) ∈ S × S and p ∈ Q. Now(νT × σT )((a1, a2) + (b1, b2), p) = (νT × σT )((a1 + b1, a2 + b2), p)= min{νT (a1 + b1, p), σT (a2 + b2, p)}≥ min{min{νT (a1, p), νT (b1, p)},min{σT (a2, p), σT (b2, p)}}= min{min{νT (a1, p), σT (a2, p)},min{νT (b1, p), σT (b2, p)}}= min{(νT × σT )((a1, a2), p), (νT × σT )((b1, b2), p)}.Debabrata Mandal, Some Properties of Q-Neutrosophic Ideals of SemiringsNeutrosophic Sets and Systems, Vol. 36, 2020 372(νI × σI)((a1, a2) + (b1, b2), p) = (νI × σI)((a1 + b1, a2 + b2), p)= νI(a1+b1,p)+σI(a2+b2,p)2≥ 12{νI(a1,p)+νI(b1,p)2 +σI(a2,p)+σI(b2,p)2 }= 12{νI(a1,p)+σI(a2,p)2 +νI(b1,p)+σI(b2,p)2 }= 12{(νI × σI)((a1, a2), p) + (νI × σI)((b1, b2), p)}.(νF × σF )((a1, a2) + (b1, b2), p) = (νF × σF )((a1 + b1, a2 + b2), p)= max{νF (a1 + b1, p), νF (a2 + b2, p)}≤ max{max{νF (a1, p), νF (b1, p)},max{σF (a2, p), σF (b2, p)}}= max{max{νF (a1, p), σF (a2, p)},max{νF (b1, p), σF (b2, p)}}= max{(νF × σF )((a1, a2), p), (νF × σF )((b1, b2), p)}.(νT × σT )((a1, a2)(b1, b2), p) = (νT × σT )((a1b1, a2b2), p) = min{νT (a1b1, p), σT (a2b2, p)}≥ min{νT (b1, p), σT (b2, p)} = (νT × σT )((b1, b2), p).(νI × σI)((a1, a2)(b1, b2), p) = (νI × σI)((a1b1, a2b2), p) = νI(a1b1,p)+σI(a2b2,p)2≥ νI(b1,p)+σI(b2,p)2 = (νI × σI)((b1, b2), p).(νF × σF )((a1, a2)(b1, b2), p) = (νF × σF )((a1b1, a2b2), p) = max{νF (a1b1, p), σF (a2b2, p)}≤ max{νF (b1, p), νF (b2, p)} = (νF × σF )(b1, b2, p).Therefore ν × σ is a Q-neutrosophic left ideal of S × S.Theorem 3.7. A Q-neutrosophic set ν of S is a Q-neutrosophic left ideal iff ν × ν is aQ-neutrosophic left ideal of S × S.Proof. If a Q-neutrosophic subset ν of S is a Q-neutrosophic left ideal then by Theorem 3.6,ν × ν is a Q-neutrosophic left ideal of S × S.Conversely, suppose ν × ν is a Q-neutrosophic left ideal of S × S and a1, a2, b1, b2 ∈ S, p ∈ Q.Thenmin{νT (a1 + b1, p), νT (a2 + b2, p)} = (νT × νT )((a1 + b1, a2 + b2), p)= (νT × νT )((a1, a2) + (b1, b2), p)≥ min{(νT × νT )((a1, a2), p), (νT × νT )((b1, b2), p)}= min{min{νT (a1, p), νT (a2, p)},min{νT (b1, p), νT (b2, p)}}.νI(a1+b1,p)+νI(a2+b2,p)2 = (νI × νI)((a1 + b1, a2 + b2), p)= (νI × νI)((a1, a2) + (b1, b2), p)≥ (νI×νI)((a1,a2),p)+(νI×νI)((b1,b2),p)2= 12 [νI(a1,p)+νI(a2,p)2 +νI(b1,p)+νI(b2,p)2 ].Debabrata Mandal, Some Properties of Q-Neutrosophic Ideals of SemiringsNeutrosophic Sets and Systems, Vol. 36, 2020 373max{νF (a1 + b1, p), νF (a2 + b2, p)} = (νF × νF )((a1 + b1, a2 + b2), p)= (νF × νF )((a1, a2) + (b1, b2), p)≤ max{(νF × νF )((a1, a2), p), (νF × νF )((b1, b2), p)}= min{max{νF (a1, p), νF (a2, p)},max{νF (b1, p), νF (b2, p)}}.Now, putting a1 = a, a2 = 0, b1 = b and b2 = 0, in the above inequalities and noting thatνT (0) ≥ νT (x), νI(0) = 0 and νF (0) ≤ νF (x) for all a ∈ S we obtainνT (a+ b, p) ≥ min{νT (a, p), νT (b, p)}νI(a+ b, p) ≥ νI(a,p)+νI(b,p)2νF (a+ b, p) ≤ max{νF (a, p), νF (b, p)}.Next, we havemin{νT (a1b1), νT (a2b2)} = (νT × νT )(a1b1, a2b2) = (νT × νT )((a1, a2)(b1, b2))≥ (νT × νT )(b1, b2) = min{νT (b1), νT (b2)}.νI(a1b1,q)+νI(a2b2,q)2 = (νI × νI)((a1, a2)(b1, b2), q)≥ (νI × νI)((b1, b2), q)= νI(b1,q)+νI(b2,q)2 .max{νF (a1b1, q), νF (a2b2, q)} = (νF × νF )((a1b1, a2b2), q) = (νF × νF )((a1, a2)(b1, b2), q)≤ (νF × νF )((b1, b2), q) = max{νF (b1, q), νF (b2, q)}.Taking a1 = a, b1 = b and b2 = 0, we obtainνT (ab, p) ≥ νT (b, p)νI(ab, p) ≥ νI(b, p)νF (ab, p) ≤ νF (b, p).Hence ν becomes a Q-neutrosophic left ideal of S.Definition 3.8. For two Q-neutrosophic sets ν and σ of a semiring S, define their compositionbyνT oσT (a, p) = supa=m∑c=1acbc{minc{νT (ac, p), σT (bc, p)}}= 0, otherwiseνIoσI(a, p) = supa=m∑c=1acbc∑mc=1νI(ac,p)+σI(bc,p)2m= 0, otherwiseDebabrata Mandal, Some Properties of Q-Neutrosophic Ideals of SemiringsNeutrosophic Sets and Systems, Vol. 36, 2020 374νF oσF (a, p) = infa=m∑c=1acbc{maxc{νF (ac, p), σF (bc, p)}}= 0, otherwisewhere p ∈ Q, a, ac, bc ∈ S , m ∈ N -the set of natural number.Theorem 3.9. For two Q-neutrosophic left ideals ν and σ of S, νoσ also forms a Q-neutrosophic left ideal of S.Proof. Consider two Q-neutrosophic left ideals ν, σ of S with a, b ∈ S, p ∈ Q. If (a+ b, p) hasthe expression (∑mi=1 acbc, p), where ac, bc ∈ S and p ∈ Q, then the proof is immediate fromthe definition. So, assume that a+ b can be expressed in the said form. Then(νT oσT )(a+ b, p)= supa+b=m∑c=1acbc{minc{νT (ac, p), σT (bc, p)}}≥ sup{minc{νT (cc, p), σT (dc, p),a=m∑c=1ccdc, b =m∑c=1ecfcνT (ec, p), σT (fc, p)}}= min{ sup{minc{a=m∑c=1ccdcνT (cc, p), σT (dc, p)}}, sup{minc{b=m∑c=1ecfcνT (ec, p), σT (fc, p)}}}= min{(νT oσT )(a, p), (νT oσT )(b, p)}(νIoσI)(a+ b, p)= supa+b=m∑c=1acbc∑mc=1νI(ac,p)+σI(bc,p)2m≥ supa=m∑c=1ccdc, b =m∑c=1ecfc∑mc=1νI(cc,p)+σI(dc,p)+νI(ec,p)+σI(fc,p)2m≥ 12 [ supa=m∑c=1ccdc∑mc=1νI(cc,p)+σI(dc,p)2m , supb=m∑c=1ecfc∑mc=1νI(ec,p)+σI(fc,p)2m ]= (νIoσI)(a,p)+(νIoσI)(b,p)2Debabrata Mandal, Some Properties of Q-Neutrosophic Ideals of SemiringsNeutrosophic Sets and Systems, Vol. 36, 2020 375(νF oσF )(a+ b, p)= infa+b=m∑c=1acbc{maxc{νF (ac, p), σF (bc, p)}}≤ infa=m∑c=1ccdc, b =m∑c=1ecfc{maxc{νF (cc, p), σF (dc, p), νF (ec, p), σF (fc, p)}}= max{inf{maxc{νF (cc, p),a=m∑c=1ccdcσF (dcp)}}, inf{maxc{νF (ec, p),b=m∑c=1ecfcσF (fc, p)}}}= max{(νF oσF )(a, p), (νF oσF )(b, p)}(νT oσT )(ab, p) = supab=m∑c=1acbc{minc{νT (ac, p), σT (bc, p)}}≥ supab=m∑c=1aecfc{minc{νT (aec, p), σT (fc, p)}}≥ supb=m∑c=1ecfc{minc{νT (ec, p), σT (fc, p)}} = (νT oσT )(b, p)(νIoσI)(ab, p) = supab=m∑c=1acbc∑mc=1νI(ac,p)+σI(bc,p)2m≥ supab=m∑c=1aecfc∑mc=1νI(aec,p)+σI(fc,p)2m≥ supb=m∑c=1ecfc∑mc=1νI(ec,p)+σI(fc,p)2m = (νIoσI)(b, p)(νF oσF )(ab, p) = infab=m∑c=1acbc{maxc{νF (ac, p), σF (bc, p)}}≤ infab=m∑c=1aecfc{maxc{νF (aec, p), σF (fc, p)}}≤ infb=m∑c=1ecfc{maxc{νF (ec, p), σF (fc, p)}} = (νF oνF )(b, p)Debabrata Mandal, Some Properties of Q-Neutrosophic Ideals of SemiringsNeutrosophic Sets and Systems, Vol. 36, 2020 376Therefore νoσ is a Q-neutrosophic left ideal of S.Conclusion: In this paper, we have defined Q-neutrosophic ideals of a semiring and studiedsome its elementary properties. Here also we obtain its characterizations by label subsetcriteria, cartesian product and composition of two Q-neutrosophic ideals. Our next aim toextend the idea in case of Q-neutrosophic bi-ideals, Q-neutrosophic quasi-ideals and investigatesome properties of regular semirings.Acknowledgement: The author is thankful to the referees for their valuable comments toimprove the paper.References1. Abdel-Basset, M.; Mohamed, R.; Elhoseny, M.; Chang, V. Evaluation framework for smart disaster responsesystems in uncertainty environment. Mechanical Systems and Signal Processing, 2020, 145, 106941.2. Abdel-Basset, M.; Ali, M.; Atef, A. Uncertainty assessments of linear time-cost tradeoffs using neutrosophicset. Computers and Industrial Engineering, 2020, 141, 106286.3. Abdel-Basset, M.; Ali, M.; Atef, A. Resource levelling problem in construction projects under neutrosophicenvironment. The Journal of Supercomputing,2020, Vol 76, No. 2, pp. 964 - 988.4. Abdel-Basset, M.; Gamal, A.; Son, L. 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Fuzzy sets, Information and Control, 1965, 8, pp.338 - 353.Received: May 05, 2020/ Accepted: September 30, 2020Debabrata Mandal, Some Properties of Q-Neutrosophic Ideals of Semirings

Some Properties of Q-Neutrosophic Ideals of Semirings

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Some Properties of Q-Neutrosophic Ideals of Semirings 
Debabrata Mandal 
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Neutrosophic Sets and Systems, Vol. 36, 2020
University of New Mexico
Some Properties of Q-Neutrosophic Ideals of Semirings
Debabrata Mandal
Department of Mathematics, Raja Peary Mohan College, Uttarpara, Hooghly-712258, India
e-mail: dmandaljumath@gmail.com
∗Correspondence: dmandaljumath@gmail.com
Abstract. The intention of this paper is to introduce and study some properties of the ideals of semirings
using the concept of Q-neutrosophic set.
Keywords: Semiring; Q-neutrosophic ideal; Cartesian Product; Composition.
—————————————————————————————————————————-
1. Introduction
The concept of fuzzy sets was introduced by Zadeh [21] in 1965 to overcome the uncertain-
ties in various problems in environment, economics, engineering etc. As an extension of it,
Atanassov [7] introduced intuitionistic fuzzy set in 1986, where a degree of non-membership
was considered besides the degree of membership of each element with (membership value +
non-membership value) ≤ 1.
After that several generalizations such as, rough sets, vague sets, interval-valued sets etc. are
considered as mathematical tools for dealing with uncertainties. In 2005, F. Smarandache
introduced Neutrosophic set [19] in which he introduced the indeterminacy to intuitionistic
fuzzy sets. So, the resultant can be taken as a tri-component logic which can be applied to
non-standard analysis such as decision making (for example, result of games (win/tie/defeat),
votes, from no/yes/NA), control theory etc.. Since then several researchers has applied this
concept in many practical fields such as multi-criteria decision making, signal processing, dis-
aster management etc.. Some of its recent applications can be found in [1–5,9, 18,20].
In 2011, Majumder [13] introduced and studied the concept of Q-fuzzifcation of ideals of Γ-
semigroup. Akram et al [6], Lekkoksung [11, 12], Mandal [14], Qamar et al [15, 16] extended
this concept in case of Γ-semigroup, ordered semigroups [10], ordered Γ-semiring, soft fields,
group theory and investigated some important properties.
Motivated by this idea and combining the concept with neutrosophic set, in the paper we
Debabrata Mandal, Some Properties of Q-Neutrosophic Ideals of Semirings
Neutrosophic Sets and Systems, Vol. 36, 2020 369
have studied the ideal theory of semirings since it has several applications in graph theory,
automata theory, mathematical modelling etc. [8].
2. Preliminaries
At first let us remember some definitions which will be used in the discussion of the paper.
Definition 2.1. A semiring is a nonempty set S on which two operations + and · have been
defined such that (S,+) and (S, ·) form monoid where · distributes over + from any side.
Definition 2.2. A nonempty subset X(6= S) of semiring S is said to be an ideal if for all
x, y ∈ X and p ∈ S, x + y ∈ X, px ∈ X. Similarly we can define a right ideal also. An ideal
of S is a nonempty subset which satisfies both properties of left ideal and right ideal.
Definition 2.3. A neutrosophic set N on the universe U is defined as N = {<
u,AT (u), AI(u), AF (u) >, u ∈ U}, where AT , AI , AF : U →]−0, 1+[ and −0 ≤ AT (u)+AI(u)+
AF (u) ≤ 3+ . For practical purposes, it is difficult to consider ]−0, 1+[. So, for studying
neutrosophic set we consider the set which takes the value from the subset of [0, 1].
Definition 2.4. For a non-empty set Q, a mapping ν : S×Q −→ [0, 1] is said to be a Q-fuzzy
subset of S and νl = {(s, q) ∈ S ×Q|ν(s, q) ≥ l} where l ∈ [0, 1] is its level subset.
3. Main Results
Definition 3.1. Let ν = (νT , νI , νF ) be a non empty neutrosophic subset of a semiring S.
Then ν is called a Q-neutrosophic left ideal of S if
(i) νT (s1 + s2, p) ≥ min{νT (s1, p), νT (s2, p)}, νT (s1s2, p) ≥ νT (s2, p)
(ii) νI(s1 + s2, p) ≥ ν
I(s1,p)+νI(s2,p)
2 , ν
I(s1s2, p) ≥ νI(s2, p)
(iii) νF (s1 + s2, p) ≤ max{νF (s1, p), νF (s2, p)}, νF (s1s2, p) ≤ νF (s2, p).
for all s1, s2 ∈ S and p ∈ Q.
Theorem 3.2. Any Q-neutrosophic set ν of a semiring S is a left ideal iff its level subsets
νTl := {(x, p) ∈ S × Q : νT (x, p) ≥ l, l ∈ [0, 1], p ∈ Q}, νIl := {(x, p) ∈ S × Q : νI(x, p) ≥
l, l ∈ [0, 1]} and νFl := {(x, p) ∈ S ×Q : νF (x, p) ≤ l, l ∈ [0, 1]} are left ideals of S ×Q.
Proof. Suppose ν of S is a Q-neutrosophic left ideal of S. Then anyone of νT , νI or νF is not
equal to zero for some (s, p) ∈ S ×Q. Without loss of generality we consider, all of them are
not equal to zero.
Debabrata Mandal, Some Properties of Q-Neutrosophic Ideals of Semirings
Neutrosophic Sets and Systems, Vol. 36, 2020 370
Suppose a, b ∈ νl = (νTl , νIl , νFl ), s ∈ S and p ∈ Q. Then
νT (a+ b, p) ≥ min{νT (a, p), νT (b, p)} ≥ min{l, l} = l
νI(a+ b, p) ≥ ν
I(a,p)+νI(b,p)
2 ≥
l+l
2 = l
νF (a+ b, p) ≤ max{νF (a, p), νF (b, p)} ≤ max{l, l} = l
which implies (a+ b, p) ∈ νTl , νIl , νFl i.e., (a+ b, p) ∈ νl. Also
νT (sa, p) ≥ νT (a, p) ≥ l
νI(sa, p) ≥ νI(a, p) ≥ l
νF (sa, p) ≤ νF (a, p) ≤ l
Hence (sa, p) ∈ νl.
Therefore νl is a left ideal of S.
Conversely, let νl( 6= φ) is a left ideal of S × Q and ν is not a Q-neutrosophic left ideal of S.
Then for a, b ∈ S and p ∈ Q anyone of the following inequality will hold.
νT (a+ b, p) < min{νT (a, p), νT (b, p)}
νI(a+ b, p) < ν
I(a,p)+νI(b,p)
2
νF (a+ b, p) > max{νF (a, p), νF (b, p)}
For the first inequality, choose l1 =
1
2 [ν
T (a+b, p)+min{νT (a, p), νT (b, p)}]. Then νT (a+b, p) <
l1 < min{νT (a, p), νT (b, p)} ⇒ (a, p), (b, p) ∈ νTl1 . but (a+ b, p) 6∈ ν
T
l1
- contradiction.
For the second inequality, choose t2 =
1
2 [ν
I(a + b, p) + min{νI(a, p), νI(b, p)}]. Then νI(a +
b, p) < l2 <
νI(a,p)+νI(b,p)
2 ⇒ (a, p), (b, p) ∈ ν
I
l2
. But (a+ b, p) 6∈ νIl2 - contradiction.
For the third inequality, choose t3 =
1
2 [ν
F (a + b, p) + max{νF (a, p), νF (b, p)}]. Then νF (a +
b, p) > l3 > max{νF (a, p), νF (b, p)} ⇒ (a, p), (b, p) ∈ νFt3 but (a+ b, p) 6∈ ν
F
p3 - contradiction.
Hence the theorem.
Definition 3.3. For two Q-neutrosophic subsets ν and σ of S × Q, define their intersection
by
(νT ∩ σT )(a, p) = min{νT (a, p), σT (a, p)}
(νI ∩ σI)(a, p) = min{νI(a, p), σI(a, p)}
(νF ∩ σF )(a, p) = max{νF (a, p), σF (a, p)}
for all a ∈ S and p ∈ Q.
Proposition 3.4. Intersection of any number of Q-neutrosophic left ideals of S is also a
Q-neutrosophic left ideal.
Debabrata Mandal, Some Properties of Q-Neutrosophic Ideals of Semirings
Neutrosophic Sets and Systems, Vol. 36, 2020 371
Proof. Assume that {νi : c ∈ C} be a collection of Q-neutrosophic left ideals of S and a, b ∈ S,
p ∈ Q. Then
( ∩
c∈C
νTc )(a+ b, p) = inf
c∈C
νTc (a+ b, p) ≥ inf
c∈C
{min{νTc (a, p), νTc (b, p)}}
= min{inf
c∈C
νTc (a, p), inf
c∈C
νTc (b, p)}
= min{( ∩
c∈C
νTc )(a, p), ( ∩
c∈C
νTc )(b, p)}
( ∩
c∈C
νIc )(a+ b, p) = inf
c∈C
νIc (a+ b, p) ≥ inf
c∈C
νIc (a,p)+ν
I
c (b,p)
2
=
inf
c∈C
νIc (a,p)+inf
c∈C
νIc (b,p)
2
=
∩
c∈C
νIc (a,p)+ ∩
c∈C
νIc (b,p)
2
( ∩
c∈C
νFc )(a+ b, p) = sup
c∈C
νFc (a+ b, p) ≤ sup
c∈C
{max{νFc (a, p), νFc (b, p)}}
= max{sup
c∈C
νFc (a, p), sup
c∈C
νFc (b, p)}
= max{( ∩
c∈C
νFc )(a, p), ( ∩
c∈C
νFc )(b, p)}
( ∩
c∈C
νTc )(ab, p) = inf
c∈C
νTc (ab, p) ≥ inf
c∈C
νTc (b, p) = ( ∩
c∈C
νTc )(b, p).
( ∩
c∈C
νIc )(ab, p) = inf
c∈C
νIc (ab, p) ≥ inf
c∈C
νIc (b, p) = ( ∩
c∈C
νIc )(b, p).
( ∩
c∈C
νFc )(ab, p) = sup
c∈C
νFc (ab, p) ≤ sup
c∈C
νFc (b, p) = ( ∩
c∈C
νFc )(b, p).
Therefore ∩
c∈C
νc is a Q-neutrosophic left ideal of S.
Definition 3.5. For two Q-neutrosophic subsets ν and σ of S, define their cartesian product
by
(νT × σT )((a, b), p) = min{νT (a, p), σT (b, p)}
(νI × σI)((a, b), p) = ν
I(a, p) + σI(b, p)
2
(νF × σF )((a, b), p) = max{νF ((a, p), σF (b, p)}
∀a, b ∈ S, p ∈ Q.
Theorem 3.6. For two Q-neutrosophic left ideals ν and σ of S, ν × σ is a Q-neutrosophic
left ideal of S × S.
Proof. Let (a1, a2), (b1, b2) ∈ S × S and p ∈ Q. Now
(νT × σT )((a1, a2) + (b1, b2), p) = (νT × σT )((a1 + b1, a2 + b2), p)
= min{νT (a1 + b1, p), σT (a2 + b2, p)}
≥ min{min{νT (a1, p), νT (b1, p)},min{σT (a2, p), σT (b2, p)}}
= min{min{νT (a1, p), σT (a2, p)},min{νT (b1, p), σT (b2, p)}}
= min{(νT × σT )((a1, a2), p), (νT × σT )((b1, b2), p)}.
Debabrata Mandal, Some Properties of Q-Neutrosophic Ideals of Semirings
Neutrosophic Sets and Systems, Vol. 36, 2020 372
(νI × σI)((a1, a2) + (b1, b2), p) = (νI × σI)((a1 + b1, a2 + b2), p)
= ν
I(a1+b1,p)+σI(a2+b2,p)
2
≥ 12{
νI(a1,p)+νI(b1,p)
2 +
σI(a2,p)+σI(b2,p)
2 }
= 12{
νI(a1,p)+σI(a2,p)
2 +
νI(b1,p)+σI(b2,p)
2 }
= 12{(ν
I × σI)((a1, a2), p) + (νI × σI)((b1, b2), p)}.
(νF × σF )((a1, a2) + (b1, b2), p) = (νF × σF )((a1 + b1, a2 + b2), p)
= max{νF (a1 + b1, p), νF (a2 + b2, p)}
≤ max{max{νF (a1, p), νF (b1, p)},max{σF (a2, p), σF (b2, p)}}
= max{max{νF (a1, p), σF (a2, p)},max{νF (b1, p), σF (b2, p)}}
= max{(νF × σF )((a1, a2), p), (νF × σF )((b1, b2), p)}.
(νT × σT )((a1, a2)(b1, b2), p) = (νT × σT )((a1b1, a2b2), p) = min{νT (a1b1, p), σT (a2b2, p)}
≥ min{νT (b1, p), σT (b2, p)} = (νT × σT )((b1, b2), p).
(νI × σI)((a1, a2)(b1, b2), p) = (νI × σI)((a1b1, a2b2), p) = ν
I(a1b1,p)+σI(a2b2,p)
2
≥ ν
I(b1,p)+σI(b2,p)
2 = (ν
I × σI)((b1, b2), p).
(νF × σF )((a1, a2)(b1, b2), p) = (νF × σF )((a1b1, a2b2), p) = max{νF (a1b1, p), σF (a2b2, p)}
≤ max{νF (b1, p), νF (b2, p)} = (νF × σF )(b1, b2, p).
Therefore ν × σ is a Q-neutrosophic left ideal of S × S.
Theorem 3.7. A Q-neutrosophic set ν of S is a Q-neutrosophic left ideal iff ν × ν is a
Q-neutrosophic left ideal of S × S.
Proof. If a Q-neutrosophic subset ν of S is a Q-neutrosophic left ideal then by Theorem 3.6,
ν × ν is a Q-neutrosophic left ideal of S × S.
Conversely, suppose ν × ν is a Q-neutrosophic left ideal of S × S and a1, a2, b1, b2 ∈ S, p ∈ Q.
Then
min{νT (a1 + b1, p), νT (a2 + b2, p)} = (νT × νT )((a1 + b1, a2 + b2), p)
= (νT × νT )((a1, a2) + (b1, b2), p)
≥ min{(νT × νT )((a1, a2), p), (νT × νT )((b1, b2), p)}
= min{min{νT (a1, p), νT (a2, p)},min{νT (b1, p), νT (b2, p)}}.
νI(a1+b1,p)+νI(a2+b2,p)
2 = (ν
I × νI)((a1 + b1, a2 + b2), p)
= (νI × νI)((a1, a2) + (b1, b2), p)
≥ (ν
I×νI)((a1,a2),p)+(νI×νI)((b1,b2),p)
2
= 12 [
νI(a1,p)+νI(a2,p)
2 +
νI(b1,p)+νI(b2,p)
2 ].
Debabrata Mandal, Some Properties of Q-Neutrosophic Ideals of Semirings
Neutrosophic Sets and Systems, Vol. 36, 2020 373
max{νF (a1 + b1, p), νF (a2 + b2, p)} = (νF × νF )((a1 + b1, a2 + b2), p)
= (νF × νF )((a1, a2) + (b1, b2), p)
≤ max{(νF × νF )((a1, a2), p), (νF × νF )((b1, b2), p)}
= min{max{νF (a1, p), νF (a2, p)},max{νF (b1, p), νF (b2, p)}}.
Now, putting a1 = a, a2 = 0, b1 = b and b2 = 0, in the above inequalities and noting that
νT (0) ≥ νT (x), νI(0) = 0 and νF (0) ≤ νF (x) for all a ∈ S we obtain
νT (a+ b, p) ≥ min{νT (a, p), νT (b, p)}
νI(a+ b, p) ≥ ν
I(a,p)+νI(b,p)
2
νF (a+ b, p) ≤ max{νF (a, p), νF (b, p)}.
Next, we have
min{νT (a1b1), νT (a2b2)} = (νT × νT )(a1b1, a2b2) = (νT × νT )((a1, a2)(b1, b2))
≥ (νT × νT )(b1, b2) = min{νT (b1), νT (b2)}.
νI(a1b1,q)+νI(a2b2,q)
2 = (ν
I × νI)((a1, a2)(b1, b2), q)
≥ (νI × νI)((b1, b2), q)
= ν
I(b1,q)+νI(b2,q)
2 .
max{νF (a1b1, q), νF (a2b2, q)} = (νF × νF )((a1b1, a2b2), q) = (νF × νF )((a1, a2)(b1, b2), q)
≤ (νF × νF )((b1, b2), q) = max{νF (b1, q), νF (b2, q)}.
Taking a1 = a, b1 = b and b2 = 0, we obtain
νT (ab, p) ≥ νT (b, p)
νI(ab, p) ≥ νI(b, p)
νF (ab, p) ≤ νF (b, p).
Hence ν becomes a Q-neutrosophic left ideal of S.
Definition 3.8. For two Q-neutrosophic sets ν and σ of a semiring S, define their composition
by
νT oσT (a, p) = sup
a=
m∑
c=1
acbc
{min
c
{νT (ac, p), σT (bc, p)}}
= 0, otherwise
νIoσI(a, p) = sup
a=
m∑
c=1
acbc
∑m
c=1
νI(ac,p)+σI(bc,p)
2m
= 0, otherwise
Debabrata Mandal, Some Properties of Q-Neutrosophic Ideals of Semirings
Neutrosophic Sets and Systems, Vol. 36, 2020 374
νF oσF (a, p) = inf
a=
m∑
c=1
acbc
{max
c
{νF (ac, p), σF (bc, p)}}
= 0, otherwise
where p ∈ Q, a, ac, bc ∈ S , m ∈ N -the set of natural number.
Theorem 3.9. For two Q-neutrosophic left ideals ν and σ of S, νoσ also forms a Q-
neutrosophic left ideal of S.
Proof. Consider two Q-neutrosophic left ideals ν, σ of S with a, b ∈ S, p ∈ Q. If (a+ b, p) has
the expression (
∑m
i=1 acbc, p), where ac, bc ∈ S and p ∈ Q, then the proof is immediate from
the definition. So, assume that a+ b can be expressed in the said form. Then
(νT oσT )(a+ b, p)
= sup
a+b=
m∑
c=1
acbc
{min
c
{νT (ac, p), σT (bc, p)}}
≥ sup{min
c
{νT (cc, p), σT (dc, p),
a=
m∑
c=1
ccdc, b =
m∑
c=1
ecfc
νT (ec, p), σ
T (fc, p)}}
= min{ sup{min
c
{
a=
m∑
c=1
ccdc
νT (cc, p), σ
T (dc, p)}}, sup{min
c
{
b=
m∑
c=1
ecfc
νT (ec, p), σ
T (fc, p)}}}
= min{(νT oσT )(a, p), (νT oσT )(b, p)}
(νIoσI)(a+ b, p)
= sup
a+b=
m∑
c=1
acbc
∑m
c=1
νI(ac,p)+σI(bc,p)
2m
≥ sup
a=
m∑
c=1
ccdc, b =
m∑
c=1
ecfc
∑m
c=1
νI(cc,p)+σI(dc,p)+νI(ec,p)+σI(fc,p)
2m
≥ 12 [ sup
a=
m∑
c=1
ccdc
∑m
c=1
νI(cc,p)+σI(dc,p)
2m , sup
b=
m∑
c=1
ecfc
∑m
c=1
νI(ec,p)+σI(fc,p)
2m ]
= (ν
IoσI)(a,p)+(νIoσI)(b,p)
2
Debabrata Mandal, Some Properties of Q-Neutrosophic Ideals of Semirings
Neutrosophic Sets and Systems, Vol. 36, 2020 375
(νF oσF )(a+ b, p)
= inf
a+b=
m∑
c=1
acbc
{max
c
{νF (ac, p), σF (bc, p)}}
≤ inf
a=
m∑
c=1
ccdc, b =
m∑
c=1
ecfc
{max
c
{νF (cc, p), σF (dc, p), νF (ec, p), σF (fc, p)}}
= max{inf{max
c
{νF (cc, p),
a=
m∑
c=1
ccdc
σF (dcp)}}, inf{max
c
{νF (ec, p),
b=
m∑
c=1
ecfc
σF (fc, p)}}}
= max{(νF oσF )(a, p), (νF oσF )(b, p)}
(νT oσT )(ab, p) = sup
ab=
m∑
c=1
acbc
{min
c
{νT (ac, p), σT (bc, p)}}
≥ sup
ab=
m∑
c=1
aecfc
{min
c
{νT (aec, p), σT (fc, p)}}
≥ sup
b=
m∑
c=1
ecfc
{min
c
{νT (ec, p), σT (fc, p)}} = (νT oσT )(b, p)
(νIoσI)(ab, p) = sup
ab=
m∑
c=1
acbc
∑m
c=1
νI(ac,p)+σI(bc,p)
2m
≥ sup
ab=
m∑
c=1
aecfc
∑m
c=1
νI(aec,p)+σI(fc,p)
2m
≥ sup
b=
m∑
c=1
ecfc
∑m
c=1
νI(ec,p)+σI(fc,p)
2m = (ν
IoσI)(b, p)
(νF oσF )(ab, p) = inf
ab=
m∑
c=1
acbc
{max
c
{νF (ac, p), σF (bc, p)}}
≤ inf
ab=
m∑
c=1
aecfc
{max
c
{νF (aec, p), σF (fc, p)}}
≤ inf
b=
m∑
c=1
ecfc
{max
c
{νF (ec, p), σF (fc, p)}} = (νF oνF )(b, p)
Debabrata Mandal, Some Properties of Q-Neutrosophic Ideals of Semirings
Neutrosophic Sets and Systems, Vol. 36, 2020 376
Therefore νoσ is a Q-neutrosophic left ideal of S.
Conclusion: In this paper, we have defined Q-neutrosophic ideals of a semiring and studied
some its elementary properties. Here also we obtain its characterizations by label subset
criteria, cartesian product and composition of two Q-neutrosophic ideals. Our next aim to
extend the idea in case of Q-neutrosophic bi-ideals, Q-neutrosophic quasi-ideals and investigate
some properties of regular semirings.
Acknowledgement: The author is thankful to the referees for their valuable comments to
improve the paper.
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Received: May 05, 2020/ Accepted: September 30, 2020
Debabrata Mandal, Some Properties of Q-Neutrosophic Ideals of Semirings


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Some Properties of Q-Neutrosophic Ideals of Semirings

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