Interval-Valued Semiprime Fuzzy Ideals of Semigroups

Weintroducethenotionof(i-v)semiprime(irreducible)fuzzyidealsofsemigroupsandinvestigateitsdifferentalgebraicproperties. Westudytheinterrelationamong(i-v)primefuzzyideals,(i-v)semiprimefuzzyideals,and(i-v)irreduciblefuzzyidealsand characterizeregularsemigroupsbyusingthese(i-v)fuzzyideals.


Introduction
Zadeh [1] first introduced the concept of fuzzy sets in 1965.After that it has become an important research tool in mathematics as well as in other fields.It has many applications in many areas like artificial intelligence, coding theory, computer science, control engineering, logic, information sciences, operations research, robotics, and others.Likewise, an idea of connecting the fuzzy sets and algebraic structures came first in Rosenfeld's mind.He first introduced the notion of fuzzy subgroup [2] in 1971 and studied many results related to groups.After that fuzzification of any algebraic structures has become a new area of research for the researchers.Some of fuzzy algebraic structures are mentioned in [3][4][5][6][7][8][9].
During the progress of the research on fuzzy sets, several types of extensions of fuzzy subsets were introduced.Interval-valued (in short, (i-v)) fuzzy subset is one of such extensions.In 1975, the concept of interval-valued fuzzy subset was introduced by Zadeh [10].In this concept, the degree of membership of each element is a closed subinterval in [0,1].Using such concept, it is possible to describe an object in a more precise way.There are many applications of (i-v) fuzzy subsets in different areas: Davvaz [11] on near rings, Hedayati [12] on semirings, Gorzałczany [13] on approximate reasoning, Turksen [14] on multivalued logic, Mendel [15] on intelligent control, Roy and Biswas [16] on medical diagnosis, and so on.
Similar to fuzzy set theory, (i-v) fuzzy set theory gradually developed on different algebraic structures.Biswas [17] defined the (i-v) fuzzy subgroups of Rosenfeld's nature and investigated some elementary properties.Narayanan and Manikantan [18] introduced the notions of (i-v) fuzzy subsemigroup and various (i-v) fuzzy ideals in semigroups.In [19], Kar et al. introduced the concept of (i-v) prime (completely prime) fuzzy ideal of semigroups and studied their properties.Khan et al. [20] introduced the concept of a quotient semigroup by an interval-valued fuzzy congruence relation on a semigroup.In [21], Thillaigovindan and Chinnadurai introduced the notion of (i-v) fuzzy interior (quasi, bi) ideals of semigroup and studied their properties.However, the concept of (i-v) semiprime (irreducible) fuzzy ideals of semigroups has not been considered so far in the best of our knowledge.
In this paper our main goal is to study the semiprime (completely semiprime) ideal of a semigroup by using (i-v) fuzzy concept and discuss their properties.Also, we prove by an example that every (i-v) semiprime fuzzy ideal may not be (i-v) prime fuzzy ideal, although the converse is true.Finally, we define (i-v) irreducible fuzzy ideal of a semigroup and discuss different relations among (i-v) prime fuzzy ideal, (i-v) semiprime fuzzy ideal, and (i-v) irreducible fuzzy ideal.

Preliminaries
In this section we give some basic definitions and results of fuzzy algebra which will be used in this paper.
In this paper, we assume that any two interval numbers in [0, 1] are comparable; that is, for any two interval numbers ã, b ∈ [0, 1], we have either ã ≤ b or ã > b.

(i-v) Semiprime Fuzzy Ideal of Semigroups
In this section we define (i-v) semiprime fuzzy ideals as a generalization of semiprime ideals of a semigroup and discuss its different algebraic properties.Proof.Let  be a semiprime ideal of .Then it is easy to check that χ is a nonconstant (i-v) fuzzy ideal of .Consider μ1 to be an (i-v) fuzzy ideal of  such that μ1 ∘ μ1 ⊆ χ .Let us assume that μ1 ̸ ⊆ χ .Then there exists  ∈  such that μ1 ()  χ ().Since any two interval numbers in [0, 1] are comparable, μ1 () > χ ().This implies χ () = 0 ⇒  ∉ .Since  is a semiprime ideal of , by Proposition 2 it follows that  ∉  for some  ∈ ; that is, χ () = 0. Again, which contradict the fact that μ1 ∘ μ1 ⊆ χ .Therefore, μ1 ⊆ χ and, hence, it follows that χ is an (i-v) semiprime fuzzy ideal of .
Conversely, let χ be an (i-v) semiprime fuzzy ideal of .Then χ is a nonconstant (i-v) fuzzy ideal of  and hence  is a proper ideal of .Let  be an ideal of  such that  ⊆ .Then χ is an (i-v) fuzzy ideal of  and χ ∘ χ = χ ⊆ χ .Therefore, by our hypothesis, χ ⊆ χ ; that is,  ⊆ .Thus,  is a semiprime ideal of .Proposition 5. A nonconstant (i-v) fuzzy ideal μ of a semigroup  is an (i-v) semiprime fuzzy ideal of  if and only if a level ideal Ũ( μ, ã) is a semiprime ideal of  for every ã ∈ Im μ.Lemma 6.Let  be a semiprime ideal of a semigroup  and μ an (i-v) fuzzy subset of  defined by where [, ] ∈ [0, 1] \ { 1}.Then μ is an (i-v) semiprime fuzzy ideal of .
Note.Lemma 6 is an example of an (i-v) semiprime fuzzy ideal of a semigroup.Now, in the following we give an example of an (i-v) semiprime fuzzy ideal which is not an (iv) prime fuzzy ideal, although every (i-v) prime fuzzy ideal is an (i-v) semiprime fuzzy ideal.
Example 7. Let  = Z + 0 , set of nonnegative integers.Then,  forms a semigroup with respect to usual multiplication.Define an (i-v) fuzzy subset μ of  by where [19,Theorem 3.8]).
In the following theorem we try to extend Proposition 2 and characterize an (i-v) semiprime fuzzy ideal.Theorem 8.If μ is a nonconstant (i-v) fuzzy ideal of a semigroup , then the following conditions are equivalent.
Theorem 13.Let  1 and  2 be two semigroups and  :  1 →  2 an epimorphism.Then there is a one-to-one correspondence between the -invariant (i-v) semiprime fuzzy ideals of  1 and (i-v) semiprime fuzzy ideals of  2 .
In the following we try to give the definition of (i-v) fuzzy -system and characterize (i-v) semiprime fuzzy ideal using it.Definition 14.A nonempty subset  of a semigroup  is called a -system of  if for every  ∈  there exists  ∈  such that  ∈ .Definition 15.A nonempty (i-v) fuzzy subset μ of a semigroup  is called an (i-v) fuzzy -system of  if for any  ∈  and ã ∈ [0, 1] \ { 1} μ() > ã implies μ() > ã for some  ∈ .

Theorem 17. A nonconstant (i-v) fuzzy ideal μ of a semigroup 𝑆 is an (i-v) semiprime fuzzy ideal of 𝑆 if and only if μ𝑐 is an (i-v) fuzzy p-system of 𝑆.
Proof.Let μ be an (i-v) semiprime fuzzy ideal of .Since μ is nonconstant, there exists  * ∈  such that μ( * ) ̸ = 1 and hence μ ( that is, μ () > ã.Consequently, it follows that μ  is an (i-v) fuzzy -system of .
The concept of completely semiprime ideal (see Definition 18) is defined in [22], in which it is known as semiprime ideal.Now, we try to generalize this concept using (i-v) fuzzy points and define (i-v) completely semiprime fuzzy ideal.Also, we investigate its various properties.Definition 18.A proper ideal  of a semigroup  is said to be completely semiprime if for any  ∈   2 ∈  implies  ∈ .Definition 19.A nonconstant (i-v) fuzzy ideal μ of a semigroup  is called an (i-v) completely semiprime fuzzy ideal of  if for any (i-v) fuzzy point  ã of   ã ∘  ã ∈ μ implies  ã ∈ μ.
Proposition 23.Let  :  1 →  2 be a semigroup epimorphism.Then the following statements are true.
(ii) From Proposition 10, it follows that ( θ) is an Hence, it follows that ( θ) is an (i-v) completely semiprime fuzzy ideal of  2 .Proposition 24.Every (i-v) completely semiprime fuzzy ideal of a semigroup  is an (i-v) semiprime fuzzy ideal of .
Note.But the converse of Proposition 24 is not always true; that is, every (i-v) semiprime fuzzy ideal of a semigroup  may not be an (i-v) completely semiprime fuzzy ideal of .
Corollary 27.In a commutative semigroup , a nonconstant (i-v) fuzzy ideal μ of  is an (i-v) semiprime fuzzy ideal of  if and only if μ() = μ( 2 ) for every  ∈ .

Theorem 28. A commutative semigroup 𝑆 is regular if and only if every nonconstant (i-v) fuzzy ideal of 𝑆 is an (i-v) semiprime fuzzy ideal of 𝑆.
Proof.Let μ be a nonconstant (i-v) fuzzy ideal of a regular semigroup  and μ1 ∘ μ1 ⊆ μ for any (i-v) fuzzy ideal μ1 of .Then μ1 = μ1 ∩ μ1 = μ1 ∘ μ1 (since  is regular) ⊆ μ.This implies that μ is an (i-v) semiprime fuzzy ideal of .
Definition 30 (see [19] Theorem 31 (see [19]).Let μ be an (i-v) prime fuzzy ideal of a semigroup .Then μ is an (i-v) completely prime fuzzy ideal of  if and only if for any two (i-v) fuzzy points  ã and  b of Theorem 32.Let μ be an (i-v) prime fuzzy ideal of a semigroup .Then μ is an (i-v) completely prime fuzzy ideal of  if and only if μ is an (i-v) completely semiprime fuzzy ideal of .
Proof.Let μ be an (i-v) completely prime fuzzy ideal of .Then it is clear that μ is an (i-v) completely semiprime fuzzy ideal of .
Conversely, let μ be an (i-v) completely semiprime fuzzy ideal of  and consider two (i-v) fuzzy points  ã and  Definition 33 (see [22]).A semigroup  is called intra-regular if for each element  ∈  there exist elements ,  ∈  such that  =  2 .
Theorem 34.In a semigroup , the following statements are equivalent.
(ii)  is a semilattice of simple semigroups.
Advances in Fuzzy Systems 7 (iii) Every ideal of  is completely semiprime.
(iv) Every (i-v) fuzzy ideal of  is an (i-v) completely semiprime fuzzy ideal of .
Theorem 35.The following conditions are equivalent in a semigroup .
(ii) Every (i-v) fuzzy interior ideal of  is an (i-v) completely semiprime fuzzy ideal of .
Proof.Since proof is simple, we omit the proof.

(i-v) Irreducible Fuzzy Ideal of Semigroups
In this section we have defined (i-v) irreducible fuzzy ideal of a semigroup  and we study its several properties.
Proof of Propositions 41-45 is straightforward and so we omit the proof.
In the following theorem we try to find the homomorphic image and preimage of an (i-v) irreducible fuzzy ideal of a semigroup.
Theorem 54.Let  be an ideal of a semigroup  and  ∈  such that  ∉ .Then, there exists an irreducible ideal  of  such that  ⊆  and  ∉ .
Proof.Let  be a set of all (i-v) fuzzy ideals δ of  such that μ ⊆ δ and δ() = μ() > 0. Then  ̸ =  and, under inclusion,  is a poset.Now, if we consider an -chain C = {γ : γ ∈ } of , then it is easy to show that the (i-v) fuzzy ideal ⋃ γ of  is an upper bound of C. Therefore, from Zorn's Lemma, we can say that  has a maximal element, say, θ, which is an (i-v) fuzzy ideal of  containing μ such that θ() = μ().Let θ1 and θ2 be two (i-v) fuzzy ideals of  such that θ = θ1 ∩ θ2 .Then either θ ⊆ θ1 or θ ⊆ θ2 .Therefore, by maximality condition, it implies that either θ1 = θ or θ2 = θ.Hence, θ is an (i-v) irreducible fuzzy ideal of .
Theorem 56.The following conditions are equivalent in a semigroup .
(i)  is regular.
(iv) Every (i-v) fuzzy ideal of  is an intersection of (i-v) prime fuzzy ideals of  containing it.
Theorem 57.Every (i-v) fuzzy ideal of a semigroup  is an (iv) prime fuzzy ideal of  if and only if  is regular and all (i-v) fuzzy ideals of  form an -chain.
Proof.Let every (i-v) fuzzy ideal of  be an (i-v) prime fuzzy ideal of .Then each of these ideals is also an (i-v) semiprime fuzzy ideal of  and hence, by Theorem 56,  is regular.Again, for any two (i-v) fuzzy ideals μ1 and μ2 of , μ1 ∘ μ2 ⊆ μ1 ∩ μ2 and μ1 ∩ μ2 is an (i-v) fuzzy ideal of .Therefore, by our assumption, μ1 ∩ μ2 is an (i-v) prime fuzzy ideal of .Thus, it follows that either μ1 ⊆ μ1 ∩ μ2 or μ2 ⊆ μ1 ∩ μ2 ; that is, either μ1 ⊆ μ2 or μ2 ⊆ μ1 .Thus, the set of all (i-v) fuzzy ideals of  forms an -chain.

Conclusion
Interval-valued fuzzy ideals are new tools to study fuzzy algebra.Interval-valued semiprime fuzzy ideals may be used to further study of fuzzy semigroups and fuzzy semirings and certainly give some important aspects of fuzzy algebra as a whole.

Proposition 41 .
Let μ be an (i-v) irreducible fuzzy ideal of a semigroup  and  ∈ .If μ() = [ − (),  + ()] for some fuzzy ideals  − and  + of , then  − and  + are both irreducible fuzzy ideals of .Proposition 42.A proper ideal  of a semigroup  is an irreducible ideal of  if and only if the characteristic function χ is an (i-v) irreducible fuzzy ideal of .Proposition 43.An (i-v) fuzzy ideal μ of a semigroup  is an (i-v) irreducible fuzzy ideal of  if and only if a level ideal Ũ( μ, ã) is an irreducible ideal of  for every ã ∈ Im μ.Proposition 44.If μ1 and μ2 are two (i-v) irreducible fuzzy ideals of a semigroup , then μ1 ∩ μ2 is an (i-v) irreducible fuzzy ideal of , provided μ1 ∩ μ2 is nonempty.Proposition 45.Let  and  be two nonempty sets and  :  →  a function.If μ1 , μ2 and θ1 , θ2 are the (i-v) fuzzy ideals of  and , respectively, then the following statements are true.

(
Definition 1.A proper ideal  of a semigroup  is said to be semiprime if for any ideal  of   ⊆  implies  ⊆ .In a semigroup , an ideal  of  is a semiprime ideal of  if and only if  ⊆  implies  ∈ . Definition 36.A proper ideal  of a semigroup  is called an irreducible ideal of  if for any two ideals  and  of  ∩ =  implies either  =  or  = .Definition 38.A proper ideal  of a semigroup  is called a strongly irreducible ideal of  if for any two ideals  and  of   ∩  ⊂  implies either  ⊂  or  ⊂ .Consider a semigroup (, * ) where  = {, , , } and a binary operation " * " on  is defined by *     Then  1 = {},  2 = {, },  3 = {, , },  4 = {, , }, and  are the ideals of  in which,  3 ,  4 are the irreducible ideals of , but not the strongly irreducible ideals of .Define an (i-v) fuzzy subset μ of  such that Proposition 40.Every (i-v) strongly irreducible fuzzy ideal of a semigroup  is an (i-v) irreducible fuzzy ideal of .But the converse is not true in general.Proof.For the converse part, we give a counter example.