New generalization of Fuzzy Interior Gamma Ideals of Ordered
Gamma Semigroups
Ibrahim Gambo
1, Nor Haniza Sarmin
2∗, Hidayat Ullah Khan
3& Faiz Muhammad Khan
41,2
Department of Mathematical Sciences, Faculty of Science,
Universiti Teknologi, Malaysia, 81310 UTM Johor Bahru, Malaysia
3
Department of Mathematics
University of Malakand Khyber Pukhtunkhwa, Pakistan
4
Department of Mathematics and Statistics
University of Swat Khyber Pakhtunkhwa, Khyber Pakhtunkhwa, Pakistana
∗
nhs@utm.my
Abstract
Ordered Gamma semigroups as a generalization of ordered semigroup, it is a significance structure in algebra with the associative property and hence has a wide broad of applications in various fields of, coding theory, error correction, computer science, automata theory, and artificial intelligence among others. In this paper, the new form of fuzzy interior ideals in ordered Gamma semigroups which is a new form of generalization of interior ideals is studied and investigated. The characterization and some properties of this new form of generalization in ordered Gamma semigroups is given in this paper.
Keywords : Fuzzy interior Gamma Ideal; Ordered Gamma Semigroups; Regular Ordered Semigroups; Gamma Ideals.
2020 Mathematical Subject Classification Codes : 20M75, 03E72
1
Introduction
Several mathematical problems involving unprecise and uncertainty cannot be solve through classi-cal mathematiclassi-cal methods. The use of fuzzy set theory [1] in such problems has been accomplishing landmark achievements in contemporary mathematics. Zadeh’s seminal paper [1] has received much attention and opens a new direction for researchers to tackle problems of uncertainties with a more appropriated mathematical tool. The new findings of fuzzy set theory and other related theories of uncertainties (soft sets, rough sets) are much relevant due to the diverse applications in automata theory, coding theory, decision making, computer sciences, artificial Intelligence and control engi-neering [2-8] Rosenfeld [9] was the first to apply Zadeh’s pioengi-neering idea of fuzzy sets to algebraic structures and introduced fuzzy subgroups. The inception of fuzzy subgroups provides a platform for other researchers to use this icebreaking idea in other algebraic structures along with several applications. Among other algebraic structures, ordered semigroups are having a lot of applications in error correcting codes, control engineering, performance of super computer and information sci-ences. Mordeson et al. [10] initiated a novel concept i.e., fuzzy subsemigroups along with fuzzy
ideals in semigroups while Kehayopulu and Tsingelis [11-13] used fuzzy sets in ordered semigroups to developed fuzzy ideal theory. Khan et al. [14] investigated fuzzy Γ−ideals and fuzzy generalized bi Γ−ideals in ordered Γ−semigroups with characterization of different classes based on the new notion. In this paper, we apply the concept of generalized quasi-coincident with relation in ordered Γ−semigroup and introduced the new notion of fuzzy interior Γ−ideals in ordered Γ−semigroup. Based on this concept of quasi-coincident with relation, various classes of ordered Γ−semigroups like regular ordered Γ−semigroups with characterization based on the idea of fuzzy interior Γ−ideals.
2
Definitions and Preliminaries
Based on the fact that ordered Γ−semigroups are the basic algebraic structures in some of the advanced fields of computer sciences, error correcting codes, automata theory, robotics, control en-gineering and formal languages. Therefore, we develop new algebraic structures based on interior Γ−ideals to tackle the more complicated problems that can be applied in the aforementioned fields of research. In the following, we present some fundamental definitions and previous results that would be used in this paper.
Definition 1. Ordered fuzzy point
Given S an ordered semigroup, and let a ∈ S with t ∈ (0, 1]. An ordered fuzzy point at of S is defined by the rule that
at(x) =
t, if x ∈ (a] 0, if x /∈ (a].
It is accepted that atis a mapping from S in to [0, 1] , then ordered fuzzy point of S is a fuzzy subset λ of S, denoted as at⊆ λ by at∈ λ in sequel.
Definition 2. Ordered Γ−semigroup
If G and Γ are non-empty sets, then a structure (G, Γ, ≤) is called an ordered Γ-semigroup if: 1. (aαb)βc = aα(bβc) for all a, b, c ∈ G and α, β ∈ Γ,
2. If a ≤ b → aαx ≤ bαx and xαa ≤ xαb for all a, b, x ∈ G and α ∈ Γ. Definition 3. Fuzzy Subsemigroup
A non-empty fuzzy subset λ of a Γ−semigroup S is called a fuzzy subsemigroup of S if λ (aαb) ≥ min {λ (a) , λ (b)} ∀a, b ∈ S and α ∈ Γ.
3
Fuzzy Interior Γ−Ideal of the Form (∈, ∈ ∨q
k) in Ordered
Γ−Semigroups
This section, provides a new generalization of interior fuzzy interior Γ−ideals using the idea of fuzzy point and quasi coincident with relation as another new form of generalization. The fuzzy interior Γ−ideal of the form (∈, ∈ ∨qk) in ordered Γ−semigroup G is introduced, where k ∈ [0, 1), unless or otherwise stated.
Definition 4. A fuzzy subset λ of G is called a fuzzy interior Γ−ideal of G of the form (∈, ∈ ∨qk), if it satisfies the following conditions:
1. If a ≤ b → (bt∈ λ → at∈ ∨qkλ) . ∀a, b ∈ G with t ∈ (0, 1] .
2. If at∈ λ, bt∈ λ → (aαb)t∈ ∨qkλ (resp. (bαa)t∈ ∨qkλ) , ∀a, b ∈ G, α, β ∈ Γ and t ∈ (0, 1] . 3. If xt∈ λ, → (aαxβc)t∈ ∨qkλ, ∀x, a, c ∈ G, α, β ∈ Γ and t ∈ (0, 1] .
A fuzzy subset λ is said to be of the form (q, ∈ ∨qk) , if given a, b ∈ G with a ≤ b such that btqλ implies λ (b) + t > 1 implies that a ∈ A, for any A ⊆ G and t ∈ (0, 1] .
The characterization of the new form of interior Γ−ideal of the form (∈, ∈ ∨qk), and (q, ∈ ∨qk) are given below.
Theorem 1. Given a fuzzy subset λ of an ordered Γ−semigroup G and let I be an interior Γ−ideal of G and λ is defined as:
λ (a) = 1−k
2 , if a ∈ I, 0, if a /∈ I. Then,
1. λ is a fuzzy interior Γ−ideal of G of the form (∈, ∈ ∨qk). 2. λ is a fuzzy interior Γ−ideal of G of the form (q, ∈ ∨qk).
Proof:
1. Suppose a, b ∈ G, and α, β ∈ Γ with t, r ∈ (0, 1] , such that at ∈ λ and br ∈ λ then a, b ∈ I. Hence, λ (aαb) ≥ 1−k2 , then, the following cases are established.
Case I: If min {t, r} >1−k
2 , then λ (aαb) + min {t, r} + k > 1−k
2 +
1−k
2 + k = 1. Thus, it shows that (aαb)min{t,r}∈ qkλ.
Case II: If min {t, r} ≤ 1−k2 . Then, λ (aαb) ≥ min {t, r} and hence, (aαb)min{t,r} ∈ qkλ. Let a, x, c ∈ G, and α, β ∈ Γ with t ∈ (0, 1] , such that at∈ λ. Thus, a ∈ I. Therefore, aαxβc ∈ I. Hence, λ (aαxβc) ≥ 1−k2 . If t > 1−k2 . Since, λ (aαxβc) + t + k > 1−k2 +1−k2 + 1−k2 + k = 1 and so, (aαxβc)t∈ qkλ. Similarly, If t ≤ 1−k2 , then λ (aαxβc) ≥ t and so λ (aαxβc) ∈ λ thus, (aαxβc)t∈ ∨qkλ. Therefore, by Theorem ?? the remaining part of the theorem follows. 2. Suppose a, b ∈ G, and α, β ∈ Γ with t, r ∈ (0, 1] , such that atqλ and brqλ. Thus, a, b ∈ I, so
λ (a) + t > 1. Similarly, λ (a) + r > 1. Since I is an interior Γ−ideal by the hypothesis. Then λ (aαb) ≥ 1−k2 . Also, if min {t, r} > 1−k2 , thus, λ (aαb) + min {t, r} + k > 1−k2 +1−k2 + k = 1, and so (aαb)min{t,r}qkλ. If min {t, r} ≤ 1−k2 hence, λ (aαb) ≥ min {t, r} and therefore, (aαb)min{t,r}λ which lead to (aαb)min{t,r}∈ ∨qkλ. In the same way, let a, x, c ∈ G, and α, β ∈ Γ with t, r ∈ (0, 1] , such that atqλ. Then a ∈ I, λ (a) + t > 1. But I is an interior Γ−ideal of G, then aαxβc ∈ I. Thus, (aαxβc) ≥ 1−k2 .
If t > 1−k 2 , then
λ (aαxβc) + t + k > 1−k2 +1−k2 + k = 1, which implies that (aαxβc)tqkt, thus (aαxβc)t∈ λ. Hence, (aαxβc)t∨ qkλ.
If k = 0, with usual binary operation putting in the above theorem, then a corollary below is developed.
Corollary 1. [?] Given a fuzzy subset λ of an ordered Γ−semigroup G and let I be an interior Γ−ideal of G and λ is defined as:
λ (a) = 1
2, if a ∈ I, 0, if a /∈ I. Then,
1. λ is a fuzzy interior Γ−ideal of G of the form (∈, ∈ ∨q). 2. λ is a fuzzy interior Γ−ideal of G of the form (q, ∈ ∨q).
The necessary and sufficient conditions for a fuzzy subset λ to be fuzzy interior Γ−ideal of the form (q, ∈ ∨qk) are provided in the following.
Theorem 2. A fuzzy subset λ of an ordered Γ−semigroup G, is a fuzzy interior Γ−ideal of G if and only if all of the following are satisfied:
1. (∀a, b ∈ G) a ≤ b → λ (a) ≥ minλ (b) ,1−k 2 , 2. (∀a, b ∈ G) λ (aαb) ≥ minλ (a) , λ (b) ,1−k
2 , 3. (∀a, c, x ∈ G) λ (aαxβc) ≥ minλ (x) ,1−k
2 .
Proof: Suppose λ is a fuzzy interior Γ−ideal of (∈, ∈ ∨qk). Suppose that there exist a, b ∈ G, with a ≤ b, then λ (a) ≥ minλ (b) ,1−k
2 which follows from Theorem ??. Now, suppose on the contrary there exist a, b ∈ G, α ∈ Γ with a ≤ b such that λ (aαb) ≤ minλ (a) , λ (b) ,1−k2 at t ∈ (0, 1] , by choosing t from the (0, 1] , such that λ (aαb) < t ≤ minλ (a) , λ (b) ,1−k2 then at∈ λ, bt ∈ λ, while λ (aαb) < t and λ (aαb) + t + k < 1−k2 + 1−k2 + k = 1, So λ (aαb)tqkλ. Hence, λ (aαb)t∈ ∨qkλ which is a contradiction. Thus, λ (aαb) ≥ minλ (a) , λ (b) ,1−k2 for all a, b ∈ G, α ∈ Γ. Suppose there exist a, x, c ∈ G, such that λ (aαxβc) < minλ (x) ,1−k
2 . Now, for some t ∈ (0, 1] , with λ (aαxβc) < t ≤ minλ (x) ,1−k
2 , then it shows that xt ∈ λ but λ (aαxβc) < t and λ (aαxβc) + t + k < 1−k2 +1−k2 + k = 1, so λ (aαxβc)tqkλ. Thus, λ (aαxβc)t∈ ∨qkλ which is λ (aαxβc) ≥ minλ (x) ,1−k2 , for all a, x, c ∈ G.
Conversely, suppose bt∈ λ for some t, r ∈ (0, 1] . Using Theorem ??, at∈ ∨qkλ. Let at ∈ λ,br∈ λ, then λ (a) ≥ t and λ (b) ≥ r. Hence, λ (aαb) ≥ minλ (a) , λ (b) ,1−k2 ≥ mint, r,1−k
2 , now if min {t, r} > 1−k2 , then λ (aαb) ≥ 1−k2 and λ (aαb) + min {t, r} + k > 1−k2 + 1−k2 + k = 1. There-fore, λ (aαb)min{t,r}qkλ. Similarly, if min {t, r} ≤ 1−k2 , then λ (aαb) ≥ min {t, r} which implies that (aαb)min{t,r} ∈ λ. Thus, (aαb)min{t,r} ∈ ∨qkλ. Suppose at∈ λ, then λ (a) ≥ t, hence, λ (aαxβc) ≥ minλ (a) ,1−k 2 ≥ min λ (a) , 1−k 2 ≥ min t, 1−k 2 . Now, if t > 1−k 2 , then λ (aαxβc) ≥ 1−k 2 and λ (aαxβc)+t+k > 1−k2 +1−k2 +k = 1, and so, λ (aαxβc) qkλ. Similarly, if t ≤ 1−k2 , then λ (aαxβc) ≥ t, thus, λ (aαxβc)tλ. Hence, (aαxβc)t∈ ∨qkλ which proves that λ is a fuzzy interior Γ−ideal of G of the form (∈, ∈ ∨qk).
Taking k = 0 in the above theorem and considering the Γ operation to be a normal binary operation, the following corollary is obtained which lead to the existing literature result.
Corollary 2. [?] Let λ be a fuzzy subset of an ordered semigroup G. Then λ is a fuzzy interior ideal of G if and only if:
1. (∀a, b ∈ G) a ≤ b → λ (a) ≥ minλ (b) ,1 2 , 2. (∀a, b ∈ G) λ (ab) ≥ minλ (a) , λ (b) ,12 , 3. (∀a, c, x ∈ G) λ (axc) ≥ minλ (x) ,1
2 .
Using characteristic function and level subset a link between interior Γ−ideals of the form (∈, ∈ ∨qk).
Theorem 3. A fuzzy subset λ of an ordered Γ−semigroup is a fuzzy interior Γ−ideal of G of the form (∈, ∈ ∨qk) if and only if the level subset U (λ, t) 6= ∅ is an interior Γ−ideal of G for all t ∈ 0,1−k2 . Proof: Assume that λ is a fuzzy interior Γ−ideal of G of the form (∈, ∈ ∨qk). Let a, b ∈ U (λ, t) for some t ∈ 0,1−k
2 . Then, λ (a) ≥ t and λ (b) ≥ t by the hypothesis. λ (aαb) ≥ min λ (a) , λ (b) , 1−k 2 ≥ mint, t,1−k 2 = t. Since t ∈ 0, 1−k
2 . Therefore, aαb ∈ U (λ, t) . Thus, if a, x, b ∈ G and α, β ∈ Γ, with x ∈ U (λ, t) for some t ∈ 0,1−k2 , then, λ (x) ≥ t which shows by the hypothesis, λ is an interior Γ−ideal which shows that
λ (aαxβc) ≥ min λ (x) ,1 − k 2 ≥ min t,1 − k 2 = t since t ∈ 0,1−k2 .
Therefore, aαxβc ∈ U (λ, t) . Now suppose a, b ∈ G, and a ≤ b with b ∈ U (λ, t) for some t ∈ 0,1−k2 which shows that a ∈ U (λ, t) by Theorem ??.
Conversely, suppose that U (λ; t) (6= ∅) is an interior Γ−ideal of G for all t ∈ 0,1−k2 . If there exist a, b ∈ G,α ∈ Γ, with λ (aαb) < minλ (a) , λ (b) ,1−k2 then, there exists a t ∈ 0,1−k2 , such that λ (aαb) < t ≤ minλ (a) , λ (b) ,1−k2 . Hence, a, b ∈ U (λ; t) but aαb /∈ U (λ; t) which is a contradiction. Thus, λ (aαb) ≥ minλ (a) , λ (b) ,1−k2 , ∀a, b ∈ G,α ∈ Γ, and k ∈ [0, 1) . Now, if there exist a, x, c ∈ G, and α, β ∈ Γ, with λ (aαxβc) < minλ (x) ,1−k
2 , then there exist t ∈ 0, 1−k
2 , such that λ (aαxβc) < t < minλ (x) ,1−k
2 . Hence, a ∈ U (λ; t) but aαxβc /∈ U (λ; t) which is a contradiction. Thus, λ (aαxβc) ≥ minλ (a) ,1−k
2 ∀a, b, x ∈ G,α, β ∈ Γ, and k ∈ [0, 1) (By Theorem ??). λ (x) ≥ minλ (b) ,1−k
2 ∀a, b ∈ G. Hence, λ is a fuzzy interior Γ−ideal of G.
Example 1. Consider the ordered Γ−semigroup G = {a, b, c, d} and Γ = {α, β} with ordered relation “ ≤ ” defined as:
≤:= {(a, a) , (b, b) , (c, c) , (d, d) , (a, b)} and a binary operation is defined in the following multiplication table
Table 1: Multiplication table for Example 1
α a b c d a a a a a b a a b a c a a b b d a a a a β a b c d a a a a a b a a a a c a a b a d a a b b
Then, {a} ,{a, b} ,{a, c} ,{a, d} ,{a, b, c} , {a, c, d} and {a, b, c, d} , are interior Γ−ideals of G. Now, define a fuzzy subset λ of G as follows:
λ : G → [0, 1] |x → λ (x) = 0.7, if x = a, 0.6, if x = b, 0.3, if x = c, 0.2, if x = d. Then, U (λ; t) = G, if 0 < t ≤ 0.2, {a, b, c} , if 0.2 < t ≤ 0.3, {a, b} , if 0.3 < t ≤ 0.6, {a, c, d} , if 0.6 < t ≤ 0.7, ∅, if 0.7 < t ≤ 1.
Hence, by Theorem 3, the above λ is a fuzzy interior Γ−ideal of G of the form (∈, ∈ ∨qk) for all t ∈ 0,1−k2 and k = 0.6.
The link between interior Γ−ideal and the fuzzy interior Γ−ideal of G of the form (∈, ∈ ∨qk) is provided in the following proposition.
Proposition 1. Let λ be a nonzero fuzzy interior Γ−ideal of an ordered Γ−semigroup of G. Then, the set λ0= {a ∈ G | λ (a) > 0} is an interior Γ−ideal of G.
Proof: Suppose λ is a fuzzy interior Γ−ideal of G. Let a, b ∈ G, with a ≤ b and b ∈ λ0, then, λ (b) > 0. But λ is a fuzzy Γ−ideal of G, then, from Proposition ??, it follows that, λ (a) ≥ minλ (b) ,1−k
2 > 0. since λ (b) > 0. Thus, λ (a) > 0 and so a ∈ λ0. Let a, b ∈ λ0. Similarly, a ∈ λ0 it shows that if a, c ∈ λ0 α, β ∈ Γ. λ (aαxβc) ≥ minλ (a) ,1−k2 > 0, Since λ (a) > 0. Therefore, aαxβc ∈ λ0, which implies that λ0is an interior Γ−ideal of G.
Interior Γ−ideal and fuzzy interior Γ−ideal of the form (∈, ∈ ∨qk) is linked in the proposition provided below using the characteristic function.
Proposition 2. Given a non-empty subset A of an ordered Γ−semigroup G, is an interior Γ−ideal if and only if the characteristic function λA of A is a fuzzy interior Γ−ideals of G of the form (∈, ∈ ∨qk).
Proof: Let A be an interior Γ−ideal of G. Suppose a, b ∈ G, with a ≤ b such that bt ∈ λA. Hence, λA(b) ≥ t and t ∈ (0, 1] which shows that λA(b) = 1. Thus, b ∈ A. But A is an interior Γ−ideal of G and a ≤ b, hence, a ∈ A which lead to χA(a) = 1 ≥ t, thus, at∈ λAwhich implies that at∈ ∨qkχA. Let a, b ∈ G, α ∈ Γ, with bt∈ λA. Then b ∈ A which follows that aαb ∈ A, which shows that (aαb)t∈ ∨qkχA. Similarly, let a, x, c ∈ A, and α, β ∈ Γ, then it follows that (aαxβc) ∈ A. Since A is interior Γ−ideal, thus (aαxβc) ∈ ∨qkχA. Therefore, λA is a fuzzy interior Γ−ideal of G of the form (∈, ∈ ∨qk). Conversely, Suppose that χAis a fuzzy interior Γ−ideal of G of the form (∈, ∈ ∨qk). Let a, b ∈ G, with b ∈ A, then χA(b) = 1. Hence, χA(b) > t ∈ (0, 1] , thus bt∈ χA. But χAis a fuzzy interior Γ−ideal, then at∈ ∨qkχA. So, if at ∈ χA, then χA(a) ≥ t ∈ (0, 1] . Therefore, χA(a) = 1 which shows that a ∈ A. Similarly, if atqkχAthen χA(a) + t + k > 1 for k ∈ [0, 1) . Thus, χA(a) 6= 0 which shows that a ∈ A. Now, let a, b ∈ G, α ∈ Γ such that b ∈ A. Then bt ∈ χA. But χA is an interior Γ−ideal of G of the form (∈, ∈ ∨qk) then (aαb)t ∈ ∨qkχA which shows that aαb ∈ A. In a similar way, let a, b ∈ G,α, β ∈ Γ with a ∈ A then at∈ χAand (aαxβc)t∈ ∨qkχAwhich implies that aαxβc ∈ A. As a result of that A is an interior Γ−ideal of G.
Proposition 3. A non empty subset λ of G is an interior Γ−ideal if and only if the characteristic function χλ of G is the fuzzy interior Γ−ideal of the form (∈, ∈ ∨qk) of G.
Proof: The proof follows from Proposition 2.
The fact that every Γ−ideal is an interior Γ−ideal, then the following proposition shows that for every Γ−ideal of the form (∈, ∈ ∨qk) is also an interior Γ−ideal of the form (∈, ∈ ∨qk).
Proposition 4. Every fuzzy Γ−ideal of G of the form (∈, ∈ ∨qk) is a fuzzy interior Γ−ideal of G of the form (∈, ∈ ∨qk).
Proof: Suppose λ is a fuzzy Γ−ideal of G of the form (∈, ∈ ∨qk). Let a, b ∈ G, with a ≤ b but since λ is a fuzzy Γ−ideal of the form (∈, ∈ ∨qk), therefore, λ (a) ≥ minλ (b) ,1−k2 and if a, b ∈ G, α ∈ Γ, then λ (aαb) ≥ minλ (a) ,1−k
2 and λ (aαb) ≥ min λ (b) , 1−k
2 for λ a fuzzy left Γ−ideal of G and fuzzy right Γ−ideal of G respectively by condition (1) and (2) of Theorem ?? which implies that λ (aαb) ≥ minλ (a) , λ (b) ,1−k
2 for all a, b ∈ G, α ∈ Γ. Now, let a, x, c ∈ G, α, β ∈ Γ. Then,
λ (aαxβc) = λ (aα (xβc)) ≥ minλ (xβc) ,1−k
2 since λ a fuzzy left Γ−ideal. Also, λ (aαxβc) ≥ minλ (x) ,1−k
2 since λ a fuzzy right Γ−ideal. Therefore, using Theorem 2 above it shows that λ a fuzzy interior Γ−ideal of G of the form (∈, ∈ ∨qk).
The converse of the above stated proposition is not always true as can be seen in the following example.
Example 2. Consider the ordered Γ−semigroup G = {a, b, c, d} and Γ = {α, β} with ordered relation defined “ ≤ ” as:
≤:= {(a, a) , (b, b) , (c, c) , (d, d) , (a, b)} with the binary operations define in the following multiplication table
Table 2: Multiplication table for Example 2
α a b c d a a a a a b a a a a c a a a b d a a b c β a b c d a a a a a b a a a b c a a b c d a a a a
Define λ : G → [0, 1] , be a fuzzy subset as shown below.
λ : G → [0, 1] |x → λ (x) = 0.7, if x = a, 0.3, if x = b, 0.6, if x = c, 0, if x = d. If x = a
λ (aαxβc) = λ (aβc) = λ (a) = 0.7 > 0.4 ≥ min {0.7, 0.4} . Now, if aαb = a, then λ (aαb) = λ (a) = 0.7 > 0.4 ≥ minλ (a) , λ (b) ,1−k2
λ (aαb) = 0.7 > 0.3. If aαb = b, then λ (aαb) = λ (b) = 0.3 > 0 ≥ min λ (a) , λ (b) ,1 − k 2 0.3 > 0 = min {0.7, 0.3, 0.4} . If aαb = c, then λ (aαb) = λ (c) = 0.6 > 0 = min λ (a) , λ (b) ,1 − k 2
Hence, λ is a fuzzy interior Γ−ideal of G of the form (∈, ∈ ∨qk) for all k = 0.2 but not a fuzzy Γ−ideal of G of the form (∈, ∈ ∨qk).
The following result shows the coinciding of fuzzy Γ−ideal and interior Γ−ideal all of the form (∈, ∈ ∨qk).
Proposition 5. In a regular ordered Γ−semigroup G, every fuzzy interior Γ−ideal is a fuzzy Γ−ideal of G all of the form (∈, ∈ ∨qk).
Proof: Suppose λ is a fuzzy interior Γ−ideal of G and let a, b ∈ G, α, β ∈ Γ. Then, there exist x ∈ G such that a ≤ aαxβa. Therefore,
λ (aαb) ≥ min λ ((aαxβa) αb) ,1 − k 2 = min λ ((aαx) βaαb) ,1 − k 2 ≥ min λ (a) ,1 − k 2 ,1 − k 2 = min λ (a) ,1 − k 2 . Since λ is a fuzzy interior Γ−ideal of G.
In a similar way, it also can be shown that λ (aαb) ≥ minλ (b) ,1−k2 for every a, b ∈ G,α, β ∈ Γ. Therefore, λ is a fuzzy interior Γ−ideal of G of the form (∈, ∈ ∨qk).
Corollary 3. In ordered Γ−semigroup, a fuzzy interior Γ−ideal and the fuzzy Γ−ideal of G of the form (∈, ∈ ∨qk) coincide.
Proof: The proof follows from the Proposition 5.
In what follows, the proposition give condition for fuzzy interior Γ−ideal of G to be fuzzy Γ−ideal of the form (∈, ∈ ∨qk) in ordered Γ−semigroup.
Proposition 6. In semisimple ordered Γ−semigroup G, every fuzzy interior Γ−ideal I of G, then, I is also a fuzzy Γ−ideal of G of the form (∈, ∈ ∨qk).
Proof: Let G be a semisimple ordered Γ−semigroup G, and λ is a fuzzy interior Γ−ideal of G. Let a, b ∈ G,α, β ∈ Γ then there exist x, y, z ∈ G, such that a ≤ xαaβyαaβz. Hence,
λ (aαb) ≥ min λ (xαaβyαaβz) αb,1 − k 2 = min λ (xαaβy) αaβ (zαb) ,1 − k 2 ≥ min λ (a) ,1 − k 2 ,1 − k 2
Since λ is a fuzzy interior Γ−ideal of G.
λ (aαb) = min λ (a) ,1 − k 2 .
In a similar way, it can also be shown that λ (aαb) ≥ minλ (b) ,1−k2 . For every a, b ∈ G,α ∈ Γ. Therefore, λ is a fuzzy Γ−ideal of G of the form (∈, ∈ ∨qk).
Theorem 4. In a semisimple ordered Γ−semigroup, the fuzzy interior Γ−ideal with fuzzy Γ−ideal of the form (∈, ∈ ∨qk) coincide.
Proof: The proof of this proposition follows from Proposition 6.
Acknowledgment
The first author would gracefully like to acknowledge Universiti Teknologi Malaysia (UTM)for Post Doctoral fellowship and The Second Author would like to acknowledge Ministry of Higher Edu-cation Malaysia (MoHE) for the financial support through Fundamental Research Grant Scheme (FRGS1/2020/STG06/UTM/01/2) and (UTMFR) Vote Number 20H70.
Conclusion
Fuzzy interior Γ−ideal of ordered Γ−semigroup and some characterizations play a remarkable role in the study of algebraic structures of ordered Γ−semigroups. In this Paper, the results on fuzzy interior Γ−ideals of the form (∈, ∈ ∨qk) of an ordered Γ−semigroup gave more characterizations where simple ordered Γ−semigroup, Γ−regular, intra Γ−regular or semiprime coincide. Some more characterization of fuzzy interior Γ−ideal of the form (∈, ∈ ∨qk), with several properties of an ordered of fuzzy interior Γ−ideals of the form (∈, ∈ ∨qk) have been studied and connections between interior Γ−ideals and the introduced fuzzy interior Γ−ideals of the form (∈, ∈ ∨qk) using characteristic function are established.
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