In general, ordered algebraic structures, particularly ordered semigroups, play an important role in fuzzification in many applied areas, such as computer science, formal languages, coding theory, error correction, etc. Nowadays, the concept of ambiguity is important in dealing with a variety of issues related to engineering modeling problems, network theory, decision-making problems in real-life situations, and so on. Several theories have been developed by various researchers to overcome the difficulties that arise from uncertainty, including fuzzy sets, intuitionistic fuzzy sets, probability, soft sets, neutrosophic sets, and many more. In this paper, we focus solely on neutrosophic set theory. In ordered semigroups, we define and investigate the properties of neutrosophic ϰ-ideals and neutrosophic ϰ-interior ideals. We also use neutrosophic ϰ-ideals and neutrosophic ϰ-interior ideals to characterize ordered semigroups.
In [1], Zadeh proposed the theory of fuzzy sets to model vague notions in the universe. In [2], Atanassov generalized the fuzzy set theory concepts and renamed as Intuitionistic fuzzy set theory. According to his view, there are the two kinds of degrees of freedom in a globe such as membership to a vague subset and non-membership to that given subset. In [3], Rosenfeld introduced the concepts of fuzziness in groups and obtained several results. Recently, many researchers pursue their research in this area and these concepts have been applied to different algebraic structures such as semigroups, ordered semigroups, rings (see [4 –10]).
Smarandache proposed the notions of neutrosophic sets to handle uncertainty that arises everywhere. It is the generalization of fuzzy sets and intuitionistic fuzzy sets. Using these three attributes such as a truth (T), an indeterminacy (I) and a falsity (F) membership functions, neutrosophic sets are characterized. These sets have numerous applications in various disciplines to deal with the complexities that arise primarily from ambiguity data. A neutrosophic set can differentiate between relative and absolute membership functions. Smarandache used these sets in non-standard analysis, namely decision making theory, control theory, decision of sports (winning/defeating/tie), etc.
In [11], Muhiuddin et al. presented the notion of implicative neutrosophic quadruple-algebras, and various properties were investigated. In [12], Muhiuddin et al. found the relationship between (ɛ, ɛ)-neutrosophic ideal and (ɛ, ɛ)-neutrosophic subalgebra in a BCK-algebra. Also, they provided conditions under which an (ɛ, ɛ)-neutrosophic subalgebraic structure to be an (ɛ, ɛ)-neutrosophic ideal structure. In [13], Muhiuddin et al. introduced the theory of neutrosophic implicative ϰ-ideal in BCK-algebras, and examined its properties. In addition, the relationship between different kinds of neutrosophic implicative ϰ-ideals were discussed.
In [14], Khan et al. defined and discussed various properties of neutrosophic ϰ-subsemigroup and ɛ-neutrosophic ϰ-subsemigroup in a semigroup. As a motivation from [14], we delve into different types of notions of neutrosophic ϰ-structures, namely neutrosophic ϰ-ideals, neutrosophic ϰ-bi-ideals, neutrosophic ϰ-interior ideals and investigated various properties. In [15], Elavarasan et al. proposed the notions of neutrosophic ϰ-ideals and characteristic neutrosophic ϰ-structure in semigroup and discussed its properties. Further, the equivalent assertions for characteristic neutrosophic ϰ-structure were provided. In [16], Porselvi et al. defined and obtained various properties of neutrosophic ϰ-bi-ideal structure in a semigroup. We have shown that both neutrosophic ϰ-bi-ideals and neutrosophic ϰ-right ideals were the same if the semigroup is regular left duo. Moreover, we have obtained equivalent conditions for regular semigroup in terms of neutrosophic ϰ-product. In [17], Porselvi et al. defined and discussed the notions of neutrosophic ϰ-interior ideal structures and neutrosophic ϰ-simple in semigroup. Also, we explored equivalent assertions for a simple semigroup, and neutrosophic ϰ-interior ideal structures. For more concepts related to this work, we refer the readers to [18 –23].
The aim of this paper is to define the concepts of neutrosophic ϰ-ideals, neutrosophic ϰ-bi-ideals and neutrosophic ϰ-interior ideals in ordered semigroup, and discuss its properties. We obtain an equivalent assertion of an interior ideal in ordered semigroup in terms of characteristic neutrosophic ϰ-structures. Further, we define the notion of ɛ-neutrosophic ϰ-ideals and ɛ-neutrosophic ϰ-interior ideals in ordered semigroup, and explore its properties. Moreover we show that the preimage of neutrosophic ϰ-right (resp., left, ideal) ideal is neutrosophic ϰ-right (resp., left, ideal) ideal under a homomorphism of an ordered semigroup.
Preliminaries
An non-empty set N together with two operations “.” and “≤”, denoted by (N,.,≤), is said to be an ordered semigroup if the given assertions are valid:
(N,.) is a semigroup,
(N,≤) is a poset,
For k1,k2∈N,k1≤k2⇒kk1≤kk2 and k1k ≤ k2k for all k∈N.
For any A,B⊆N,(A]={k∈N:k≤h for some h ∈ A} and AB = {k1k2: for all k1∈ A and k2∈ B }.
Definition 2.1. [24] Let (N,.,≤) be an ordered semigroup and ϕ≠K⊆N. Then K is a subsemigroup of N if K2⊆K.
Definition 2.2. [24] Let (N,.,≤) be an ordered semigroup. A non-empty subset K in N is a left (resp., right) ideal of N if
NK⊆K (resp., KN⊆K),
For k1∈ K and k2∈N;k2≤k1⇒k2∈K.
K is an ideal of N if it is a right and a left ideal of N.
Definition 2.3. [24] Let (N,.,≤) be an ordered semigroup. A subsemigroup K in N is a bi-ideal of N if
KNK⊆K,
For all k1,k2∈N,k1∈K and k2≤ k1 imply k2∈ K.
Defintion 2.4. [5] Let (N,.,≤) be an ordered semigroup. N is called regular if for each s∈N,∃k∈N∋s≤sks.
Defintion 2.5. [5] Let (N,.,≤) be an ordered semigroup. N is called intra-regular if for each s∈N,∃k1,k2∈N∋s≤k1s2k2.
Let (N,.,≤) be an ordered semigroup. Then the ϰ-function on N is a function h:N→[−1,0] and the collection of all the ϰ-functions is denoted by F(N,[−1,0]). A ϰ-structure is an ordered pair (N,h) of N and an ϰ-function h on N.
Definition 2.6. Let (N,.,≤) be an ordered semigroup. A neutrosophic ϰ-structure in N is of the form:
NP:=N(TP,IP,FP)={k(TP(k),IP(k),FP(k))|k∈N}, where TP,IP and FP are the negative truth, negative indeterminacy and negative falsity membership functions respectively in N (ϰ-functions). Obviously −3≤TP(k)+IP(k)+FP(k)≤0∀k∈N.
Definition 2.7. Let (N,.,≤) be an ordered semigroup. A neutrosophic ϰ-structure NP in N is a neutrosophic ϰ-subsemigroup in N if the below assertion is valid:
(∀k1,k2∈N)(TP(k1k2)≤TP(k1)∨TP(k2)IP(k1k2)≥IP(k1)∧IP(k2)FP(k1k2)≤FP(k1)∨FP(k2)).
Let NP be a neutrosophic ϰ −structure and μ, δ, ν ∈ [ −1, 0]. Consider the sets:
TPμ={k∈N|TP(k)≤μ},IPδ={k∈N|IP(k)≥δ},FPν={k∈N|FP(k)≤ν}.
The set NP(μ,δ,ν):={k∈N|TP(k)≤μ,IP(k)≥δ,FP(k)≤ν} is known as (μ, δ, ν)-level set on NP. It is evident that NP(μ,δ,ν)=TPμ∩IPδ∩FPν.
We refer to [14–17] for basic definitions of neutrosophic ϰ-structures in a semigroup, such as neutrosophic ϰ-ideals, neutrosophic ϰ-bi-ideals, and neutrosophic ϰ-interior ideal. We define the neutrosophic ϰ-structures in an ordered semigroup N as follows:
Definition 2.8. Let (N,.,≤) be an ordered semigroup. A neutrosophic ϰ-structure NP in N is a neutrosophic ϰ-ideal in N if the below assertions are valid:
A neutrosophic ϰ-structure NP in N is a neutrosophic ϰ-left ideal in N if the conditions (i) and (iii) are true. A neutrosophic ϰ-structure NP in N is a neutrosophic ϰ-right ideal in N if the conditions (ii) and (iii) are true.
Definition 2.9. Let (N,.,≤) be an ordered semigroup. A neutrosophic ϰ-subsemigroup NP in N is a neutrosophic ϰ-bi-ideal in N if the below assertions are valid:
It is evident that all the neutrosophic ϰ-ideals are neutrosophic ϰ-bi-ideals, but neutrosophic ϰ-bi-ideal need not be a neutrosophic ϰ-ideal, as given by an example.
Example 2.1. Consider the ordered semigroup N:={k1,k2,k3,k4,k5} with the binary operation “.” and the partial order “≤” as follows:
.
k1
k2
k3
k4
k5
k1
k1
k4
k1
k4
k4
k2
k1
k2
k1
k4
k4
k3
k1
k4
k3
k4
k5
k4
k1
k4
k1
k4
k4
k5
k1
k4
k3
k4
k5
and ≤:={(k1,k1),(k1,k3),(k1,k4),(k1,k5),(k2,k2),(k2,k4),(k2,k5),(k3,k3),(k3,k5),(k4,k4),(k4,k5),(k5,k5)}.
Let NP={k1(−0.8,−0.3,−0.7),k2(−0.6,−0.6,−0.3),k3(−0.4,−0.4,−0.5),k4(−0.8,−0.3,−0.7),k5(−0.1,−0.5,−0.3)}. Then NP is a neutrosophic ϰ-bi-ideal, but not a neutrosophic ϰ-ideal as TN(k3k5)=−0.1>TN(k3),IN(k3k5)=−0.5<IN(k3) and FN(k3k5)=−0.3>FN(k3).
Definition 2.10. Let (N,.,≤) be an ordered semigroup. A neutrosophic ϰ-subsemigroup NP in N is a neutrosophic ϰ-interior ideal in N if the given assertions are valid:
It is evident that neutrosophic ϰ-ideals are always neutrosophic ϰ-interior ideals, but not vice versa, as given by an example.
Example 2.2. Let N be the set of all non-negative integers except 1. Then (N,.,≤) is an ordered semigroup under usual multiplication and the relation ≤.
Let NP={0(−0.8,−0.1,−0.7),3(−0.2,−0.4,−0.5),7(−0.5,−0.6,−0.6),21(−0.1,−0.5,−0.3),otherwise(−0.8,−0.1,−0.7)}. Then NP is a neutrosophic ϰ-interior ideal in N, but not neutrosophic ϰ-ideal as TN(3.7)=−0.1>TN(3) and TN(3.7)=−0.1>TN(7).
Definition 2.11. Let (N,.,≤) be an ordered semigroup. For any R⊆N, the characteristic neutrosophic χ-structure in N is defined as
χR(NP)=N(χR(T)P,χR(I)P,χR(F)P) where
χR(T)P:N→[−1,0],k↦{−1ifk∈R0otherwise,χR(I)P:N→[−1,0],k↦{0ifk∈R−1otherwise,χR(F)P:N→[−1,0],k↦{−1ifk∈R0otherwise.
Definition 2.12. Let (N,.,≤) be an ordered semigroup and let NK:=N(TK,IK,FK) and NP:=N(TP,IP,FP) be neutrosophic ϰ-structures in N. Then (i) NK is called a neutrosophic ϰ-substructure in NP, denote by NP⊆NK, if TP(k)≥TK(k),IP(k)≤IK(k),FP(k)≥FK(k) for all k∈N.
If NK⊆NP and NP⊆NK, then we say that NK=NP. (ii) The union of two neutrosophic ϰ-structures NK and NP over N is defined as
NK∪NP=NK∪P=(N;TK∪P,IK∪P,FK∪P), where ∀k1∈N,(TK∪TP)(k1)=TK∪P(k1)=TK(k1)∧TP(k1),(IK∪IP)(k1)=IK∩P(k1)=IK(k1)∨IP(k1),(FK∪FP)(k1)=FK∪P(k1)=FK(k1)∧FP(k1). (iii) The intersection of two neutrosophic ϰ-structures NK and NP over N is defined as
NK∩NP=NK∩P=(N;TK∩P,IK∩P,FK∩P), where ∀k1∈N,(TK∩TP)(k1)=TK∩P(k1)=TK(k1)∨TP(k1),(IK∩IP)(k1)=IK∩P(k1)=IK(k1)∧IP(k1),(FK∩FP)(k1)=FK∩P(k1)=FK(k1)∨FP(k1).
Neutrosophic <italic>ϰ</italic>-Structures in Ordered Semigroups
In this section, we study some properties of neutrosophic ϰ-ideals and neutrosophic ϰ-interior ideal structure in an ordered semigroup N. It is evident that neutrosophic ϰ-ideals are neutrosophic ϰ-interior ideals in N, but the converse part need not be true in general. Further, we show that all neutrosophic ϰ-interior ideals are neutrosophic ϰ-ideals under certain conditions. Unless otherwise stated, we assume that NP and NK are neutrosophic ϰ-structures in N throughout this section.
Theorem 3.1. [15] Let N be a semigroup. Then for any K⊆N, the given assertions are equivalent:
K is left ideal (resp., right ideal),
χK(NK) is neutrosophic ϰ-left ideal(resp., right ideal).
Theorem 3.2. Let (N,.,≤) be an ordered semigroup. Then for any K⊆N, the given assertions are equivalent:
K is left ideal (resp., right ideal),
χK(NK) is neutrosophic ϰ-left ideal(resp., right ideal).
Proof. (i)⇒(ii) Suppose K is left ideal and let k1,k2∈N with k1≤ k2. Then TK(k1)≤TK(k2),IK(k1)≥IK(k2) and FK(k1)≤FK(k2).
If k2∈ K, then χK(T)K(k2)=−1,χK(I)K(k2)=0 and χK(F)K(k2)=−1 and so χK(T)K(k1)≤χK(T)K(k2)=−1,χK(I)K(k1)≥χK(I)K(k2)=0 and χK(F)K(k1)≤χK(F)K (k2) = −1.
If k2∉ K, then χK(T)K(k2)=0,χK(I)K(k2)=−1 and χK(F)K(k2)=0 and so χK(T)K(k1)≤0=χK(T)K(k2),χK(I)K(k1)≥−1=χK(I)K(k2) and χK(F)K(k1)≤0=χK(F)K(k2). Thus χK(NK) is neutrosophic ϰ-left ideal. By Theorem 3.1, χK(NK) is neutrosophic ϰ-left ideal.
(ii)⇒(i) Assume χK(NK) is neutrosophic ϰ-left ideal. Let k1∈K,k2∈N and k2≤ k1. Then χK(T)K(k2)≤χK(T)K(k1)=−1,χK(I)K(k2)≥χK(I)K(k1)=0 and χK(F)K(k2)≤χK(F)K(k1)=−1 which imply k2∈ K. Hence by Theorem 3.1, K is left ideal.
Theorem 3.3. [17] Suppose N is a semigroup and for any K⊆N, the equivalent assertions are:
K is interior ideal,
χK(NK) is neutrosophic ϰ-interior ideal.
Theorem 3.4. Let (N,.,≤) be an ordered semigroup. Then for any K⊆N, the given assertions are equivalent:
K is interior ideal,
χK(NK) is neutrosophic ϰ-interior ideal.
Proof. (i)⇒(ii) Suppose K is interior ideal and let k1,k2∈N with k1≤ k2. Then TK(k1)≤TK(k2),IK(k1)≥IK(k2) and FK(k1)≤FK(k2).
If k2∈ K, then χK(T)K(k2)=−1,χK(I)K(k2)=0 and χK(F)K(k2)=−1 and so χK(T)K(k1)=−1,χK(I)K(k1)=0 and χK(F)K(k1)=−1.
If k2∉ K, then χK(T)K(k2)=0,χK(I)K(k2)=−1 and χK(F)K(k2)=0 and so χK(T)K(k1)≤χK(T)K(k2)=0,χK(I)K(k1)≥χK(I)K(k2)=−1 and χK(F)K(k1)≤χK(F)K(k2)=0. Thus χK(NK) is neutrosophic ϰ-interior ideal by Theorem 3.3.
(ii)⇒(i) Assume χK(NK) is neutrosophic ϰ-interior ideal. Let k1∈K,k2∈N and k2≤ k1. Then χK(T)K(k2)≤χK(T)K(k1)=−1,χK(I)K(k2)≥χK(I)K(k1)=0 and χK(F)K(k2)≤χK(F)K(k1)=−1 which imply k2∈ K. By Theorem 3.3, K is interior ideal.
Theorem 3.5. Let (N,.,≤) be an ordered semigroup. Then the arbitrary intersection (resp., union) of neutrosophic ϰ-interior ideals in N is a neutrosophic ϰ-interior ideal in N.
Proof. The proof is a routine procedure.
Theorem 3.6. Let (N,.,≤) be an ordered semigroup. If N is regular, then neutrosophic ϰ-interior ideals in N are neutrosophic ϰ-ideals.
Proof. Assume NP is neutrosophic ϰ-interior ideal and let k1,k2∈N. As k1∈N and N is regular, there is r∈N such that k1≤ k1rk1. Now, TP(k2k1)≤TP(k2k1rk1)≤TP(k1),IP(k2k1)≥IP(k2k1rk1)≥IP(k1) and FP(k2k1)≤FP(k2k1rk1)≤FP(k1). Therefore NP is neutrosophic ϰ-left ideal. In a similar way, we can claim that NP is neutrosophic ϰ-right ideal.
Theorem 3.7. Let (N,.,≤) be an ordered semigroup. If N is intra-regular, then neutrosophic ϰ-interior ideals in N are neutrosophic ϰ-ideals.
Proof. Let NP be neutrosophic ϰ-interior ideal and k1,k2∈N. As k1∈N and N is intra regular, ∃s,t∈N ∋ k1≤sk12t. Now,
TP(k2k1)≤TP(k2sk12t)≤TP(k1),IP(k2k1)≥IP(k2sk12t)≥IP(k1),FP(k2k1)≤FP(k2sk12t)≤FP(k1).
Therefore NP is neutrosophic ϰ-left ideal. In the same way, we can claim that NP is neutrosophic ϰ-right ideal and hence NP is neutrosophic ϰ-ideal.
Definition 3.1. An ordered semigroup N is said to be
left (resp., right) simple if it does not contain any proper left (resp., right) ideal of N.
simple if it does not contain any proper ideal of N.
Definition 3.2. An ordered semigroup N is known as neutrosophic ϰ-simple if all the neutrosophic ϰ-ideals are constant functions i.e., for any neutrosophic ϰ-ideal NP in N, we can have TP(k1)=TP(k2),IP(k1)=IP(k2) and FP(k1)=FP(k2) for all k1,k2∈N.
Notation 3.1. Let (N,.,≤) be an ordered semigroup. Then for any k∈N, we define Jk⊆N as follows:
Jk:={m∈N|TK(m)≤TK(k),IK(m)≥IK(k)andFK(m)≤FK(k)}.
Theorem 3.8. Let (N,.,≤) be an ordered semigroup. If NK is a neutrosophic ϰ-right (resp., ϰ-left, ϰ-ideal) ideal in N, then for any k∈N,Jk is a right ideal (resp., left ideal, ideal) of N.
Proof. Let k∈N. Then clearly ϕ≠Jk⊆N. Let k1∈ Jk and k2∈N. Then k1k2∈ Jk. Indeed; Since k1,k2∈N and NK is neutrosophic ϰ-right ideal, we get TK(k1k2)≤TK(k1),IK(k1k2)≥IK(k1) and FK(k1k2)≤FK(k1). Since k1∈ Jk, we get TK(k1)≤TK(k),IK(k1)≥IK(k) and FK(k1)≤FK(k) which imply k1k2∈ Jk. Let a1∈Jk,a2∈N with a2≤ a1. Then TK(a2)≤TK(a1),IK(a2)≥IK(a1) and FK(a2)≤FK(a1). Since a1∈ Jk, we have TK(a1)≤TK(k),IK(a1)≥IK(k) and FK(a1)≤FK(k). So TK(a2)≤TK(k),IK(a2)≥IK(k) and FK(a2)≤FK(k) which imply a2∈ Jk. Therefore Jk is a right ideal in N.
Theorem 3.9. If N is an ordered semigroup, then N is neutrosophic ϰ-simple if and only if N is simple.
Proof. Suppose N is neutrosophic ϰ-simple. Let J be an ideal in N. Then by Theorem 3.2, χJ(NK) is neutrosophic ϰ-ideal. We now prove that N=J. Let k∈N. Since N is neutrosophic ϰ-simple, χJ(NK) is constant and χJ(NK)(k)=χJ(NK)(k′) for every k′∈N. In particular, we have χJ(TK)(k)=χJ(TK)(d)=−1, χJ(IK)(k)=χJ(IK)(d)=0 and χJ(FK)(k)=χJ(FK)(d)=−1 for any d ∈ J which gives k ∈ J. Thus N⊆J and hence N=J.
Conversely, let NK be neutrosophic ϰ-ideal with k1,k2∈N. Then by Theorem 3.8, Jk1 is an ideal. As N is simple, we have Jk1=N. Since k2∈ Jk1, we have TK(k2)≤TK(k1),IK(k2)≥IK(k1) and FK(k2)≤FK(k1). Similarly, we can prove that TK(k1)≤TK(k2),IK(k1)≥IK(k2) and FK(k1)≤FK(k2). So TK(k2)=TK(k1),IK(k2)=IK(k1) and FK(k2)=FK(k1). Hence N is neutrosophic ϰ-simple.
Lemma 3.1. [25] An ordered semigroup N is simple if and only if N=(NtN] for all t∈N.
Theorem 3.10. For any ordered semigroup N,N is simple if and only if all the neutrosophic ϰ-interior ideals in N are constant functions.
Proof. Suppose k1,k2∈N and N is simple. Let NK be neutrosophic ϰ-interior ideal. Then by Lemma 3.1, we get N=(Nk1N]=(Nk2N]. Since k1∈(Nk1N], we get k1≤ t k2s for t,s∈N. Since NK is neutrosophic ϰ-interior ideal, we can have TK(k1)≤TK(tk2s)≤TK(k2),IK(k1)≥IK(tk2s)≥IK(k2) and FK(k1)≤FK(tk2s)≤FK(k2). Similarly, we can prove that TK(k2)≤TK(k1), IK(k2)≥IK(k1) and FK(k2)≤FK(k1). So NK is constant.
Conversely, suppose NK is a neutrosophic ϰ-ideal in N. Then NK is neutrosophic ϰ-interior ideal. By assumption, NK is constant and hence NK is neutrosophic ϰ-simple. Therefore N is simple, by Theorem 3.9.
Theorem 3.11. Let (N,.,≤) be an ordered semigroup. If NP is neutrosophic ϰ-interior ideal with μ, δ, ν ∈ [ −1, 0] and −3≤ μ + δ + ν ≤ 0, then (μ, δ, ν)-level set in NP is neutrosophic ϰ-interior ideal provided NP(μ,δ,ν)≠ϕ.
Proof. Suppose NP(μ,δ,ν)≠ϕ for μ, δ, ν ∈ [ −1, 0]. Let NP be neutrosophic ϰ-interior ideal and k1,k2∈N with k1≤ k2. Then TP(k1)≤TP(k2),IP(k1)≥IP(k2) and FP(k1)≤FP(k2). If k2∈TPμ, then TP(k2)≤μ which implies TP(k1)≤μ and so k1∈TPμ. If k2∈IPδ, then IP(k2)≥δ which implies IP(k1)≥δ and so k1∈IPδ. If k2∈FPν, then FP(k2)≤ν which implies FP(k1)≤ν and so k1∈FPν. Hence k1∈NP(μ,δ,ν). By [[17], Theorem 3.16], NP(μ,δ,ν) is a neutrosophic ϰ-interior ideal in N.
Theorem 3.12. ([17], Theorem 3.17) Let N be a semigroup and NK be neutrosophic ϰ-structure in N with α, β, γ ∈ [ −1, 0] such that −3 ≤ α + β + γ ≤ 0. If TKα,IKβ and FKγ are interior ideals in N, then NK is neutrosophic ϰ-interior ideal in N whenever TKα≠ϕ,IKβ≠ϕ and FKγ≠ϕ.
Theorem 3.13. Let (N,.,≤) be an ordered semigroup. Let NK be neutrosophic ϰ-structure with α, β, γ ∈ [ −1, 0] and −3 ≤ α + β + γ ≤ 0. If TKα,IKβ and FKγ are interior ideals, then NK is neutrosophic ϰ-interior ideal whenever TKα≠ϕ,IKβ≠ϕ and FKγ≠ϕ.
Proof. Let k1∈N,k2∈TKα with k1≤ k2. Then k1∈TKα as TKα is an interior ideal of N. Suppose TK(k1)>TK(k2). Then TK(k1)>tα≥TK(k2) for some tα∈ [ −1, 0). So k2∈TKtα but k1∉TKtα, a contradiction. Thus TK(k1)≤TK(k2).
Let k1∈N,k2∈IKβ with k1≤ k2. Then k1∈IKβ as IKβ is an interior ideal of N. Suppose k1. Then k1 for some tβ∈ [ −1, 0). So k2∈IKtβ but k1∉IKtβ, a contradiction. Thus IK(k1)≥IK(k2).
Let k1∈N,k2∈FKγ with k1≤ k2. Then k1∈FKγ as FKγ is an interior ideal of N. Suppose FK(k1)>FK(k2). Then FK(k1)>tγ≥FK(k2) for some tγ ∈ [ −1, 0). So k2∈FKtγ but k1∉FKtγ, a contradiction. Thus FK(k1)≤FK(k2).
Hence by Theorem 3.12, NK is neutrosophic ϰ-interior ideal.
Let (N,.,≤) be an ordered semigroup. Following [14], we define ɛ-neutrosophic ϰ-subsemigroup, ɛ-neutrosophic ϰ-ideal and ɛ-neutrosophic ϰ-interior ideal in N as follows:
Definition 3.3. Let (N,.,≤) be an ordered semigroup. A neutrosophic ϰ-structure NP in N is called an ɛ-neutrosophic ϰ-subsemigroup in N if the given assertion is valid:
(∀k1,k2∈N)(TP(k1k2)≤∨{TP(k1),TP(k2),εT}IP(k1k2)≥∧{IP(k1),IP(k2),εI}FP(k1k2)≤∨{FP(k1),FP(k2),εF}) where ɛT, ɛI, ɛF∈ [ −1, 0] such that −3≤ ɛT +ɛI +ɛF≤ 0.
Definition 3.4. Let (N,.,≤) be an ordered semigroup. A neutrosophic ϰ-structure NP in N is called an ɛ-neutrosophic ϰ-ideal of N if the given assertions are valid: (i) (∀k1,k2∈N)(TP(k1k2)≤∨{TP(k2),εT}IP(k1k2)≥∧{IP(k2),εI}FP(k1k2)≤∨{FP(k2),εF}). (ii) (∀k1,k2∈N)(TP(k1k2)≤∨{TP(k1),εT}IP(k1k2)≥∧{IP(k1),εI}FP(k1k2)≤∨{FP(k1),εF}). (iii) (∀k1,k2∈N),k1≤k2⇒(TP(k1)≤TP(k2)∨εTIP(k1)≥IP(k2)∧εIFP(k1)≤FP(k2)∨εF) where ɛT, ɛI, ɛF∈ [ −1, 0] such that −3≤ ɛT +ɛI +ɛF≤ 0.
NP is called an ɛ-neutrosophic ϰ-left ideal of N if it satisfies the assertions (i) and (iii).
NP is called an ɛ-neutrosophic ϰ-right ideal of N if it satisfies the assertions (ii) and (iii).
Definition 3.5. Let (N,.,≤) be an ordered semigroup. A neutrosophic ϰ-subsemigroup NP in N is called a ɛ-neutrosophic ϰ-interior ideal in N if the following assertions are valid:
where ɛT, ɛI, ɛF∈ [ −1, 0] such that −3≤ ɛT +ɛI +ɛF≤ 0.
Theorem 3.14. Let (N,.,≤) be an ordered semigroup. If NK and NP are an ɛ-neutrosophic ϰ-subsemigroup and a δ-neutrosophic ϰ-subsemigroup, respectively in N for any ɛT, ɛI, ɛF, δT, δI, δF ∈ [ −1, 0] with −3≤ ɛT +ɛI +ɛF≤ 0 and −3≤ δT +δI +δF≤ 0, then NK∩NP is a ν-neutrosophic ϰ-subsemigroup of N for ν: = ɛ∧ δ, that is, (νT, νI, νF) = (ɛT∨ δT, ɛI∧ δI, ɛF∨ δF).
Proof. The proof is similar to Theorem 4.14 of [14].
Theorem 3.15. Let (N,.,≤) be an ordered semigroup. If NK and NP are an ɛ-neutrosophic ϰ-interior ideal and a δ-neutrosophic ϰ-interior ideal, respectively in N for any ɛT, ɛI, ɛF, δT, δI, δF ∈ [ −1, 0] with −3≤ ɛT +ɛI +ɛF≤ 0 and −3≤ δT +δI +δF≤ 0, then NK∩NP is a ν-neutrosophic ϰ-interior ideal in N for ν: = ɛ∧ δ, that is, (νT, νI, νF) = (ɛT∨ δT, ɛI∧ δI, ɛF∨ δF).
Proof. For any k1,k2,k3∈N, we have
TK∩P(k1k2k3)=∨{TK(k1k2k3),TP(k1k2k3)}≤∨{∨{TK(k2),εT},∨{TP(k2),δT}}≤∨{∨{TK(k2),νT},∨{TP(k2),νT}}=∨{TK(k2),TP(k2),νT}=∨{TK∩P(k2),νT},IK∩P(k1k2k3)=∧{IK(k1k2k3),IP(k1k2k3)}≥∧{∧{IK(k2),εI},∧{IP(k2),δI}}≥∧{∧{IK(k2),νI},∧{IP(k2),νI}}=∧{IK(k2),IP(k2),νI}=∧{IK∩P(k2),νI},FK∩P(k1k2k3)=∨{FK(k1k2k3),FP(k1k2k3)}≤∨{∨{FK(k2),εF},∨{FP(k2),δF}}≤∨{∨{FK(k2),νF},∨{FP(k2),νF}}=∨{FK(k2),FP(k2),νF}=∨{FK∩P(k2),νF}.
For any k1,k2∈N with k1≤ k2, we have
TK∩P(k1)=TK(k1)∨TP(k1)≤{TK(k2)∨εT}∨{TP(k2)∨δT}=TK∩P(k2)∨νT,IK∩P(k1)=IK(k1)∧IP(k1)≥{IK(k2)∧εT}∧{IP(k2)∧δT}=IK∩P(k2)∧νT,FK∩P(k1)=FK(k1)∨FP(k1)≤{FK(k2)∨εT}∨{FP(k2)∨δT}=FK∩P(k2)∨νT.
Therefore NK∩NP is a ν-neutrosophic ϰ-interior ideal in N.
Theorem 3.16. Let (N,.,≤) be an ordered semigroup. Let NK be an ɛ-neutrosophic ϰ-interior ideal in N. If
κ:=(κT,κI,κF)=(⋁k1∈NTK(k1),⋀k1∈NIK(k1),⋁k1∈NFK(k1)), then the set
Ω:={k∈N|TK(k)≤κT∨εT,IK(k)≥κI∧εI,FK(k)≤κF∨εF} is an interior ideal of N.
Proof. Let k1,k2∈N. If k1, k2∈ Ω, then
TK(k1)≤κT∨εT=⋁k1∈N{TK(k1)}∨εT,TK(k2)≤κT∨εT=⋁k2∈N{TK(k2)}∨εT,IK(k1)≥κI∧εI=⋀k1∈N{IK(k1)}∧εI,IK(k2)≥κI∧εI=⋀k2∈N{IK(k2)}∧εI,FK(k1)≤κF∨εF=⋁k1∈N{FK(k1)}∨εF,FK(k2)≤κF∨εF=⋁k2∈N{FK(k2)}∨εF.
Now for any k1,k2,k3∈N,
TK(k1k2k3)≤⋁{TK(k2),εT}≤⋁{κT∨εT,εT}=κT∨εT,IK(k1k2k3)≥⋀{IK(k2),εI}≥⋀{κI∧εI,εI}=κI∧εI,FK(k1k2k3)≤⋁{FK(k2),εF}≤⋁{κF∨εF,εF}=κF∨εF.
So k1k2k3∈ Ω.
Let k1,k2∈N with k1≤ k2 and k2∈ Ω. Then TK(k2)≤κT∨εT,IK(k2)≥κI∧εI,FK(k2)≤κF∨εF. Since NK is an ɛ-neutrosophic ϰ-interior ideal in N, we have TK(k1)≤TK(k2),IK(k1)≥IK(k2) and FK(k1)≤FK(k2). So TK(k1)≤κT∨εT,IK(k1)≥κI∧εI,FK(k1)≤κF∨εF which imply k1∈ Ω and hence Ω is an interior ideal of N.
Following [26], let (N,.,≤) and (M,∗,⪯) be ordered semigroups and a mapping f:N→M. f is known as isotone if k1,k2∈N,k1≤k2 implies f(k1) ≼ f(k2). f is called inverse isotone if k1,k2∈N,f(k1)⪯f(k2) implies k1 ≤ k2 [each inverse isotone mapping is (1–1)]. f is said to be a homomorphism if (i) f is isotone and (ii) f(k1.k2) = f(k1) * f(k2) for all k1,k2∈N. f is known as an isomorphism if it is homomorphism, onto and inverse isotone.
For a map f:N→M of ordered semigroups and a neutrosophic ϰ-structure NK:=M(TK,IK,FK) over M and ɛ = (ɛT, ɛI, ɛF) with −3≤ ɛT +ɛI +ɛF≤ 0, define a neutrosophic ϰ-structure NKε:=N(TKε,IKε,FKε) over N by:
TKε:N→[−1,0],k↦⋁{TK(f(k)),εT},IKε:N→[−1,0],k↦⋀{IK(f(k)),εI},FKε:N→[−1,0],k↦⋁{FK(f(k)),εF}.
Theorem 3.17. Let (N,.,≤) and (M,∗,⪯) be ordered semigroups and f:N→M be homomorphism. If a neutrosophic ϰ-structure NK:=M(TK,IK,FK) over M is an ɛ-neutrosophic ϰ-subsemigroup of M, then NKε:=N(TKε,IKε,FKε) is an ɛ-neutrosophic ϰ-subsemigroup in N.
Proof. The proof is similar to Theorem 4.16 of [14].
Theorem 3.18. Let (N,.,≤) and (M,∗,⪯) be ordered semigroups and f:N→M be homomorphism. If a neutrosophic ϰ-structure NK:=M(TK,IK,FK) over M is an ɛ-neutrosophic ϰ-left ideal (resp., right ideal, ideal) of M, then NKε:=N(TKε,IKε,FKε) is an ɛ-neutrosophic ϰ-left ideal (resp., right ideal, ideal) of N.
Proof. Let NK:=M(TK,IK,FK) is an ɛ-neutrosophic ϰ-left ideal in M. For any k1,k2∈N, we have
TKε(k1k2)=⋁{TK(f(k1k2)),εT}=⋁{TK(f(k1)∗f(k2)),εT}⪯⋁{TK(f(k2)),εT}=TKε(k2),IKε(k1k2)=⋀{IK(f(k1k2)),εI}=⋀{IK(f(k1)∗f(k2)),εI}⪰⋀{IK(f(k2)),εI}=IKε(k2),FKε(k1k2)=⋁{FK(f(k1k2)),εF}=⋁{FK(f(k1)∗f(k2)),εF}⪯⋁{FK(f(k2)),εF}=FKε(k2).
Let k1≤ k2. Then
TKε(k1)=TK(f(k1))∨εT⪯TK(f(k2))∨εT=TKε(k2),IKε(k1)=IK(f(k1))∧εI⪰IK(f(k2))∧εI=IKε(k2),FKε(k1)=FK(f(k1))∨εF⪯FK(f(k2))∨εF=FKε(k2).
Therefore NKε is an ɛ-neutrosophic ϰ-left ideal in N.
Theorem 3.19. Let (N,.,≤) and (M,∗,⪯) be ordered semigroups and f:N→M be homomorphism. If a neutrosophic ϰ-structure NK:=M(TK,IK,FK) over M is an ɛ-neutrosophic ϰ-interior ideal of M, then NKε:=N(TKε,IKε,FKε) is an ɛ-neutrosophic ϰ-interior ideal in N.
Proof. For any k1,k2,k3∈N, we have
TKε(k1k2k3)=⋁{TK(f(k1k2k3)),εT}=⋁{TK(f(k1)∗f(k2)∗f(k3)),εT}⪯⋁{TK(f(k2)),εT}=TKε(k2),IKε(k1k2k3)=⋀{IK(f(k1k2k3)),εI}=⋀{IK(f(k1)∗f(k2)∗f(k3)),εI}⪰⋀{IK(f(k2)),εI}=IKε(k2),FKε(k1k2k3)=⋁{FK(f(k1k2k3)),εF}=⋁{FK(f(k1)∗f(k2)∗f(k3)),εF}⪯⋁{FK(f(k2)),εF}=FKε(k2).
Let k1≤ k2. Then
TKε(k1)=TK(f(k1))∨εT⪯TK(f(k2))∨εT=TKε(k2),IKε(k1)=IK(f(k1))∧εI⪰IK(f(k2))∧εI=IKε(k2),FKε(k1)=FK(f(k1))∨εF⪯FK(f(k2))∨εF=FKε(k2).
Hence by Theorem 3.17, NKε is an ɛ-neutrosophic ϰ-interior ideal in N.
Let (N,.,≤) and (M,∗,⪯) be ordered semigroups. Consider a map f:N→M. If MP:=M(TP,IP,FP) is a neutrosophic ϰ-structures over M, then the preimage of MP under f is defined to be a neutrosophic ϰ-structures
f−1(MP)=N(f−1(TP),f−1(IP),f−1(FP)) over N, where f−1(TP)(k)=TP(f(k)),f−1(IP)(k)=IP(f(k)) and f−1(FP)(k)=FP(f(k)) for all k∈N.
Theorem 3.20. Let (N,.,≤) and (M,∗,⪯) be ordered semigroups and f:N→M be homomorphism. If MK:=M(TP,IP,FP) is a neutrosophic ϰ-subsemigroup of M, then the preimage of MK under f is a neutrosophic ϰ-subsemigroup of N.
Proof. The proof is similar to Theorem 4.17 of [14].
Theorem 3.21. Let (N,.,≤) and (M,∗,⪯) be ordered semigroups and f:N→M be homomorphism. If MP:=M(TP,IP,FP) is a neutrosophic ϰ-right ideal (resp., left ideal, ideal) of M, then the preimage of MP under f is a neutrosophic ϰ-right ideal (resp., left ideal, ideal) of N.
Proof. Let f−1(MP)=N(f−1(TP),f−1(IP),f−1(FP)) be the preimage of MP under the map f. For any k1,k2∈N, we can have
f−1(TP)(k1k2)=TP(f(k1k2))=TP(f(k1)∗f(k2))⪯TP(f(k1))=f−1(TP)(k1),f−1(IP)(k1k2)=IP(f(k1k2))=IP(f(k1)∗f(k2))⪰IP(f(k1))=f−1(IP)(k1),f−1(FP)(k1k2)=FP(f(k1k2))=FP(f(k1)∗f(k2))⪯FP(f(k1))=f−1(FP)(k1).
Let k1,k2∈N with k1≤ k2. Then
f−1(TP)(k1)=TP(f(k1))⪯TP(f(k2))=f−1(TP)(k2),f−1(IP)(k1)=IP(f(k1))⪰IP(f(k2))=f−1(IP)(k2),f−1(FP)(k1)=FP(f(k1))⪯FP(f(k2))=f−1(FP)(k2).
Therefore f−1(MP) is a neutrosophic ϰ-right ideal in N.
Theorem 3.22. Let (N,.,≤) and (M,∗,⪯) be ordered semigroups. Consider the homomorphism f:N→M. If MP:=M(TP,IP,FP) is a neutrosophic ϰ-interior ideal of M, then the preimage of MP under f is a neutrosophic ϰ-interior ideal in N.
Proof. Let f−1(MP)=N(f−1(TP),f−1(IP),f−1(FP)) be the preimage of MP under f. For any k1,k2,k3∈N, we have
f−1(TP)(k1k2k3)=TP(f(k1k2k3))=TP(f(k1)∗f(k2)∗f(k3))⪯TP(f(k2))=f−1(TP)(k2),f−1(IP)(k1k2k3)=IP(f(k1k2k3))=IP(f(k1)∗f(k2)∗f(k3))⪰IP(f(k2))=f−1(IP)(k2),f−1(FP)(k1k2k3)=FP(f(k1k2k3))=FP(f(k1)∗f(k2)∗f(k3))⪯FP(f(k2))=f−1(FP)(k2).
Let k1,k2∈N with k1≤ k2. Then
f−1(TP)(k1)=TP(f(k1))⪯TP(f(k2))=f−1(TP)(k2),f−1(IP)(k1)=IP(f(k1))⪰IP(f(k2))=f−1(IP)(k2),f−1(FP)(k1)=FP(f(k1))⪯FP(f(k2))=f−1(FP)(k2).
Therefore, by Theorem 3.20, f−1(MP) is a neutrosophic ϰ-interior ideal in N.
Let (N,.,≤) and (M,∗,⪯) be ordered semigroups. Consider the onto function f:N→M. If NK:=N(TK,IK,FK) is a neutrosophic ϰ-structures over N, then the image of NK under f is defined to be a neutrosophic ϰ-structures
f(NK)=M(f(TK),f(IK),f(FK)) over M, where for all k2∈M,
f(TK)(k2)=⋀k1∈f−1(k2)TK(k1),f(IK)(k2)=⋁k1∈f−1(k2)IK(k1),f(FK)(k2)=⋀k1∈f−1(k2)FK(k1).
Theorem 3.23. Let (N,.,≤) and (M,∗,⪯) be ordered semigroups. Consider an onto homomorphism f:N→M. Let NK:=N(TK,IK,FK) be a neutrosophic ϰ-structure over N such that
(∀Q⊆N)(∃x0∈Q)(TK(x0)=⋀z∈QTK(z)IK(x0)=⋁z∈QIK(z)FK(x0)=⋀z∈QFK(z)).
If NK is a neutrosophic ϰ-subsemigroup of N, then the image of NK is a neutrosophic ϰ-subsemigroup of M under f.
Proof. The proof is similar to Theorem 4.18 of [14].
Theorem 3.24. Let (N,.,≤) and (M,∗,⪯) be ordered semigroups. Consider an onto homomorphism f:N→M. Let NK:=N(TK,IK,FK) be a neutrosophic ϰ-structure over N such that
(∀Q⊆N)(∃x0∈Q)(TK(x0)=⋀z∈QTK(z)IK(x0)=⋁z∈QIK(z)FK(x0)=⋀z∈QFK(z)).
If NK is a neutrosophic ϰ-left ideal in N, then the image of NK is a neutrosophic ϰ-left ideal in M under f.
Proof. Let f(NK)=M(f(TK),f(IK),f(FK)) be the image in NK under f and k1,k2∈M. Then f−1(k1) ≠ φ and f−1(k2) ≠ φ. So there exist s ∈ f−1(k1) and t ∈ f−1(k2) such that
TK(s)=⋀u∈f−1(k1)TK(u),TK(t)=⋀v∈f−1(k2)TK(v),IK(s)=⋁u∈f−1(k1)IK(u),IK(t)=⋁v∈f−1(k2)IK(v),FK(s)=⋀u∈f−1(k1)FK(u),FK(t)=⋀v∈f−1(k2)FK(v).
Let k1, k2∈ M with k1≤ k2. Then
f(TK)(k1)=⋀x∈f−1(k1)TK(x)=TK(s)≤TK(t)=⋀v∈f−1(k2)TK(v)=f(TK)(k2),f(IK)(k1)=⋁x∈f−1(k1)IK(x)=IK(s)≥IK(t)=⋁v∈f−1(k2)IK(v)=f(IK)(k2),f(FK)(k1)=⋀x∈f−1(k1)FK(x)=FK(s)≤FK(t)=⋀v∈f−1(k2)FK(v)=f(FK)(k2).
Therefore f(NK) is a neutrosophic ϰ-left ideal in M.
Theorem 3.25. Let (N,.,≤) and (M,∗,⪯) be ordered semigroups. Consider an onto homomorphism f:N→M. Let NK:=N(TK,IK,FK) be a neutrosophic ϰ-structure over N such that
(∀Q⊆N)(∃x0∈Q)(TK(x0)=⋀z∈QTK(z)IK(x0)=⋁z∈QIK(z)FK(x0)=⋀z∈QFK(z)).
If NK is a neutrosophic ϰ-interior ideal in N, then the image of NK under f is a neutrosophic ϰ-interior ideal in M.
Proof. Let f(NK)=M(f(TK),f(IK),f(FK)) be the image of NK under f. Let k1,k2,k3∈M. Then f−1(k1) ≠ φ, f−1(k2) ≠ φ and f−1(k3) ≠ φ. Then there exist s ∈ f−1(k1), t ∈ f−1(k2) and u ∈ f−1(k3) such that
TK(s)=⋀z∈f−1(k1)TK(z),TK(t)=⋀w∈f−1(k2)TK(w),TK(u)=⋀v∈f−1(k3)TK(v),IK(s)=⋁z∈f−1(k1)IK(z),IK(t)=⋁w∈f−1(k2)IK(w),IK(u)=⋁v∈f−1(k3)IK(v),FK(s)=⋀z∈f−1(k1)FK(z),FK(t)=⋀w∈f−1(k2)FK(w),FK(u)=⋀v∈f−1(k3)FK(v).
Let k1,k2∈M with k1≤ k2. Then
f(TK)(k1)=⋀x∈f−1(k1)TK(x)=TK(s)≤TK(t)=⋀v∈f−1(k2)TK(v)=f(TK)(k2),f(IK)(k1)=⋁x∈f−1(k1)IK(x)=IK(s)≥IK(t)=⋁v∈f−1(k2)IK(v)=f(IK)(k2),f(FK)(k1)=⋀x∈f−1(k1)FK(x)=FK(s)≤FK(t)=⋀v∈f−1(k2)FK(v)=f(FK)(k2).
Hence, by Theorem 3.23, f(NK) is a neutrosophic ϰ-interior ideal in M.
Conclusion
In ordered semigroups, the concepts of neutrosophic ϰ-ideals, neutrosophic ϰ-bi-ideals, and neutrosophic ϰ-interior ideals were introduced and their properties were investigated. We defined ordered semigroups by employing various neutrosophic ϰ-ideals, neutrosophic ϰ-bi-ideals, neutrosophic ϰ-interior ideals and so on. In our future work, we intend to define different types of notions in neutrosophic ϰ-structures over-ordered semigroups, such as neutrosophic ϰ-prime, neutrosophic ϰ-quasi-prime, and investigate the structural properties of ordered semigroups using the concepts and results in ordered semigroups. Hopefully, our research work will continue in this direction and will create a platform for other algebraic structures.
Funding Statement: The authors are grateful to the anonymous referees for careful checking of the details and for helpful comments that improved this paper. This work was supported by the Taif University Researchers Supporting Project (TURSP-2020/246), Taif University, Taif, Saudi Arabia.
Conflicts of Interest: The authors declare that they have no conflicts of interest to report regarding the present study.
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