International Journal of Analysis and Applications Volume 19, Number 3 (2021), 455-464 URL: https://doi.org/10.28924/2291-8639 DOI: 10.28924/2291-8639-19-2021-455 ON ROUGH FUZZY PRIME IDEALS IN LEFT ALMOST SEMIGROUPS AHMED ELMOASRY1,2,∗ 1Department of Mathematics, College of Science Al-Zulfi, Majmaah University, Al-Zulfi, Saudi Arabia 2Mathematics Department Faculty of Science, Aswan University, Aswan, Egypt ∗Corresponding author: a.elmoasry@mu.edu.sa Abstract. In this paper we shall introduce the notion of rough prime ideals and rough fuzzy prime ideals in LA-semigroups. We proved that the lower and the upper approximation of a prime ideal is a prime ideal and we also proved that a fuzzy subset f of an LA-semigroup S is a fuzzy prime ideal of S iff fλ 6= ∅ (fsλ 6= ∅) is a prime ideal of S for every λ ∈ [0, 1]. 1. Introduction The notion of a rough set was originally proposed by Z. Pawlak [26] as a formal tool for modeling and processing incomplete information in information systems. The theory of rough set is an extension of set theory. The equivalence classes are the building blocks for the construction of the lower and upper approximations. The lower approximation of a given set is the union of all equivalence classes which are subsets of the set, and the upper approximation is the union of all equivalence classes which have a nonempty intersection with the set. Some authors have studied the algebraic properties of rough sets. Biswas and Nanda [3], introduced the notion of rough subgroups. Kuroki, in [14], introduced the notion of a rough ideal in a semigroup. Also, Kuroki and Mordeson in [13] studied the structure of rough sets and rough groups. Y. B. Jun applied the rough set theory to BCK-algebras [9]. Received December 9th, 2020; accepted January 7th, 2021; published April 28th, 2021. 2010 Mathematics Subject Classification. 20M10, 20M12. Key words and phrases. LA-semigroups; rough prime ideals; rough fuzzy prime ideals. ©2021 Authors retain the copyrights of their papers, and all open access articles are distributed under the terms of the Creative Commons Attribution License. 455 https://doi.org/10.28924/2291-8639 https://doi.org/10.28924/2291-8639-19-2021-455 Int. J. Anal. Appl. 19 (3) (2021) 456 A fuzzy subset f of a set S is a function from S to a closed interval [0, 1], this concept of a fuzzy set was introduced by Zadeh [31], in 1965. Rosenfeld [27], was the first who studied fuzzy sets in the structure of groups. Kuroki [16], has studied the fuzzy ideals and fuzzy bi-ideals in semigroups. M. Banerjee [2], give the concept of roughness of a fuzzy set. The fuzzy theory provides the underlying structure for the generalization of many fields including logic, differential equations and groups. The fuzzy theory on algebraic structures have been widely explored. This paper concerns the relationship between rough fuzzy sets and left almost semigroups. The left almost semigroup abbreviated as an LA-semigroup, was first introduced by M. A. Kazim and M. Naseerudin [10]. They generalized some useful results of semigroup theory. They introduced braces on the left of the ternary commutative law abc = cba, to get a new pseudo associative law, that is (ab)c = (cb)a, and named it as left invertive law. Later, Q. Mushtaq (in [19], [20], [23] and [25]) and others investigated the structure further and added many useful results to the theory of LA-semigroups. An LA-semigroup is the midway structure between a commutative semigroup and a groupoid. Despite the fact, the structure is non-associative and non- commutative. It nevertheless possesses many interesting properties which we usually find in commutative and associative algebraic structures. Q. Mushtaq and S. M. Yusuf produced useful results [20], on locally associative LA-semigroups in 1979. In this structure they defined powers of an element and congruences using these powers. They constructed quotient LA-semigroups using these congruences. Relations between LA-semigroups, Quasi-groups, commutative monoids and commutative groups were established in [18], [21] and [22]. It is a useful nonassociative structure with wide applications in theory of flocks. In this paper, we have studied ρ-upper and ρ-lower rough prime ideals and also studied ρ-upper and ρ-lower rough fuzzy prime ideals in LA-semigroups. 2. Preliminaries Let S be an LA-semigroup. A subset A of an LA-semigroup S is called an LA-subsemigroup of S if AA ⊆ A. A subset A of an LA-semigroup S is called a left [right] ideal of S if SA ⊆ A [AS ⊆ A], and A is called a two sided ideal of S if it is both a left and a right ideal of S. Let S denote an LA-semigroup unless otherwise specified. Let ρ be a congruence relation on S, that is, ρ is an equivalence relation on S such that (a,b) ∈ ρ implies (ax,bx) ∈ ρ and (xa,xb) ∈ ρ for all x ∈ S. If ρ is a congruence relation on S, then for every x ∈ S, [x]ρ stands for the congruence class of x with respect to ρ. A congruence ρ on S is called complete if [a]ρ[b]ρ = [ab]ρ for all a,b ∈ S. Int. J. Anal. Appl. 19 (3) (2021) 457 Definition 2.1. [1] Let A be a nonempty subset of an LA-semigroup S and ρ be a congruence relation on S. Then the sets Apr ρ (A) = { x ∈ S : [x]ρ ⊆ A } and Aprρ(A) = { x ∈ S : [x]ρ ∩A 6= ∅ } are called ρ-lower and ρ-upper approximations of A respectively. For a nonempty subset A of S, Aprρ(A) = (Aprρ(A),Aprρ(A)) is called a rough set with respect to ρ if Apr ρ (A) 6= Aprρ(A). A subset A of an LA-semigroup S is called a ρ-upper [ρ-lower] rough ideal of S if Aprρ(A) [Aprρ(A)] is an ideal of S. Theorem 2.1. [1] Let ρ be a congruence relation on an LA-semigroup S. If A is a left [right, two-sided] ideal of S. Then (1) Aprρ(A) is a left [right, two-sided] ideal of S. (2) If ρ is complete, then Apr ρ (A) is, if it is nonempty, a left [right, two-sided] ideal of S. 3. Rough Prime Ideals in LA-semigroups An ideal A of an LA-semigroup S is said to be a prime ideal of S, if for x,y ∈ S, xy ∈ A implies x ∈ A or y ∈ A. Let ρ be a congruence relation on an LA-semigroup S. Then a subset A of S is called a ρ-upper rough prime ideal of S if Aprρ(A) is a prime ideal of S. A ρ-lower rough prime ideal of S is defined analogously. A is called a rough prime ideal of S if A is a ρ-upper and a ρ-lower rough prime ideal of S. Theorem 3.1. Let ρ be a complete congruence relation on an LA-semigroup S. If A is a prime ideal of S. Then A is a ρ-upper rough prime ideal of S. Proof. Since A is a prime ideal of S, then by Theorem 2.1(1), Aprρ(A) is an ideal of S. Then for xy ∈ Aprρ(A) for some x,y ∈ S. Then we have [xy]ρ ∩A = [x]ρ[y]ρ ∩A 6= φ. Thus there exist a ∈ [x]ρ and b ∈ [y]ρ such that ab ∈ A. Since A is a prime ideal, we have a ∈ A or b ∈ A. Thus a ∈ [x]ρ ∩A or b ∈ [y]ρ ∩A. This implies [x]ρ ∩A 6= φ or [y]ρ ∩A 6= φ, and so x ∈ Aprρ(A) or y ∈ Aprρ(A). Therefore Aprρ(A) is a prime ideal of S. � Int. J. Anal. Appl. 19 (3) (2021) 458 The following example shows that the upper approximation of a prime ideal is not a prime ideal in general on the same conditions of Theorem 2.1(1). Example 3.1. Let S = {0, 1, 2, 3}, the binary operation ”·” on S be defined as follows: · 0 1 2 3 0 2 2 2 3 1 0 2 2 3 2 2 2 2 3 3 3 3 3 3 Clearly, 0 = 1·(1·0) 6= (1·1)·0 = 2. This shows that S is an LA-semigroup. Now let ρ be a congruence relation on S such that ρ-congruence classes are the subsets {0}, {1}, {2, 3}. Then for A = {3}⊆ S, Aprρ(A) = {2, 3}. It is clear that A is a prime ideal of S. The set Aprρ(A) is not a prime ideal for 0 · 1 = 2 ∈ Aprρ(A) but 0 /∈ Aprρ(A) and 1 /∈ Aprρ(A). Theorem 3.2. Let ρ be a complete congruence relation on an LA-semigroup S and A is a prime ideal of S. Then Apr ρ (A) is, if it is nonempty, a prime ideal of S. Proof. Since A is a prime ideal of S, then by Theorem 2.1(2), we know that Apr ρ (A) is an ideal of S. Let xy ∈ Apr ρ (A) for some x,y ∈ S, then [x]ρ[y]ρ = [xy]ρ ⊆ A. Suppose x /∈ Apr ρ (A) and y /∈ Apr ρ (A). This implies [x]ρ * A and [y]ρ * A, then there exist a ∈ [x]ρ and b ∈ [y]ρ such that a,b /∈ A. Thus ab ∈ [x]ρ[y]ρ = [xy]ρ ⊆ A. Since A is a prime ideal, we have a ∈ A or b ∈ A. It contradicts the supposition. This means that Apr ρ (A) is, if it is nonempty, a prime ideal of S. � We call A a rough prime ideal of S if it is both a ρ-upper and a ρ-lower rough prime ideal of S. From the above, we know that a prime ideal is a rough prime ideal with respect to a complete congruence relation on an LA-semigroup. The following example shows that the converse of Theorems 3.1 and 3.2, does not hold in general. Int. J. Anal. Appl. 19 (3) (2021) 459 Example 3.2. Let S = {0, 1, 2, 3, 4}, the binary operation ”·” on S be defined as follows: · 0 1 2 3 4 0 0 0 0 0 4 1 0 0 3 0 4 2 0 1 2 3 4 3 0 0 1 0 4 4 4 4 4 4 4 Clearly, 3 = 1·(2·2) 6= (1·2)·2 = 1. This shows that S is an LA-semigroup. Now let ρ be a complete congruence relation on S such that ρ-congruence classes are the subsets {0, 1, 2, 3}, {4}. Then for A = {0, 4} ⊆ S, Aprρ(A) = {0, 1, 2, 3, 4}, and Aprρ(A) = {4}. It is clear that Aprρ(A) and Aprρ(A) are prime ideals of S. The ideal A is not a prime ideal for 1 · 3 = 0 ∈ A but 1 /∈ A and 3 /∈ A. 4. Rough Prime Ideals in the Quotient LA-semigroups Let ρ be a congruence relation on an LA-semigroup S and A be a subset of S. The ρ-upper and the ρ-lower approximations can be presented in an equivalent form as shown below Aprρ(A) = {[x]ρ ∈ S/ρ : [x]ρ ∩A 6= ∅} and Apr ρ (A) = {[x]ρ ∈ S/ρ : [x]ρ ⊆ A}. Now we discuss these sets as subsets of a quotient LA-semigroup S/ρ of an LA-semigroup S. Theorem 4.1. [1] Let ρ be a congruence relation on an LA-semigroup S. If A is a left [right, two-sided] ideal of S. Then (1) Aprρ(A) is a left [right, two-sided] ideal of S/ρ. (2) Apr ρ (A) is, if it is nonempty, a left [right, two-sided] ideal of S/ρ. Theorem 4.2. Let ρ be a complete congruence relation on an LA-semigroup S. If A is a ρ-upper rough prime ideal of S, then Aprρ(A) is a prime ideal of S/ρ. Proof. Since A is a ρ-upper rough prime ideal of S, then by Theorem 4.1(1), we know that Aprρ(A) is an ideal of S/ρ. Suppose [x]ρ[y]ρ ∈ Aprρ(A) for some [x]ρ, [y]ρ ∈ S/ρ such that [xy]ρ ∈ Aprρ(A) for some [x]ρ, [y]ρ ∈ S/ρ then [xy]ρ ∩A 6= φ. Thus xy ∈ Aprρ(A). Since A is a ρ-upper rough prime ideal of S, that is Aprρ(A) is a prime ideal, thus we have x ∈ Aprρ(A) or y ∈ Aprρ(A) Int. J. Anal. Appl. 19 (3) (2021) 460 so [x]ρ ∩A 6= φ or [y]ρ ∩A 6= φ. Hence [x]ρ ∈ Aprρ(A) or [y]ρ ∈ Aprρ(A). Therefore Aprρ(A) is a prime ideal of S/ρ. This completes the proof. � Theorem 4.3. Let ρ be a complete congruence relation on an LA-semigroup S. If A is a ρ-lower rough prime ideal of S, then Apr ρ (A) is a prime ideal of S/ρ. Proof. Since A is a ρ-lower rough prime ideal of S, then by Theorem 4.1(2), we know that Apr ρ (A) is an ideal of S/ρ. Suppose [x]ρ[y]ρ ∈ Apr ρ (A) for some [x]ρ, [y]ρ ∈ S/ρ such that [xy]ρ ∈ Apr ρ (A) for some [x]ρ, [y]ρ ∈ S/ρ then [xy]ρ ⊆ A. Thus xy ∈ Aprρ(A). Since A is a ρ-lower rough prime ideal of S, that is Aprρ(A) is a prime ideal, we have x ∈ Apr ρ (A) or y ∈ Apr ρ (A) so [x]ρ ⊆ A or [y]ρ ⊆ A. Hence [x]ρ ∈ Apr ρ (A) or [y]ρ ∈ Apr ρ (A). Therefore Apr ρ (A) is a prime ideal of S/ρ. This completes the proof. � 5. Rough Fuzzy Prime Ideals in LA-semigroups A function f from S to the unit interval [0, 1] is called a fuzzy subset of S. A fuzzy subset f of an LA-semigroup S is called a fuzzy subsemigroup of S if f(xy) ≥ f(x) ∧f(y) for all x,y ∈ S. A fuzzy subset f of an LA-semigroup S is called a fuzzy ideal of S if f(xy) ≥ f(x) ∨f(y) for any x,y ∈ S. Let f be a fuzzy subset of S and λ ∈ [0, 1]. Then the sets fλ = {x ∈ S : f(x) ≥ λ} and fsλ = {x ∈ S : f(x) > λ} are called, respectively, λ-levelset and λ-strong levelset of the fuzzy set f. Theorem 5.1. Let f be a fuzzy subset of an LA-semigroup S. Then (1) f is a fuzzy ideal of S iff fλ 6= ∅ is an ideal of S for every λ ∈ [0, 1]. (2) f is a fuzzy ideal of S iff fsλ 6= ∅ is an ideal of S for every λ ∈ [0, 1]. Int. J. Anal. Appl. 19 (3) (2021) 461 Proof. (1) Assume f is a fuzzy ideal of S. Then f(xy) ≥ f(x) ∨f(y) for any x,y ∈ S. Assume fλ 6= ∅. Let x ∈ fλ, y ∈ S. Thus f(x) ≥ λ. Since f is a fuzzy ideal of S, f(xy) ≥ f(x) ∨ f(y) ≥ f(x) ≥ λ. Therefore xy ∈ fλ. Similarly, yx ∈ fλ. Hence fλ is an ideal of S. Conversely, assume for all λ ∈ [0, 1], if fλ 6= ∅, then fλ is an ideal of S. Let x,y ∈ S. Case 1 : f(x) ≥ f(y). Let λ = f(x). Then x ∈ fλ. By assumption, we have fλ is an ideal of S. So xy ∈ fλ. Then f(xy) ≥ λ = f(x) = f(x) ∨f(y). Case 2 : f(x) < f(y). Let λ = f(y). Then y ∈ fλ. By assumption, we have fλ is an ideal of S. So xy ∈ fλ. Then f(xy) ≥ λ = f(y) = f(x) ∨f(y). Therefore f is a fuzzy ideal of S. (2) Assume f is a fuzzy ideal of S. Then f(xy) ≥ f(x) ∨ f(y) for any x,y ∈ S. Assume fsλ 6= ∅. Let x ∈ fsλ , y ∈ S. Thus f(x) > λ. Since f is a fuzzy ideal of S, f(xy) ≥ f(x) ∨ f(y) ≥ f(x) > λ. Therefore xy ∈ fsλ . Similarly, yx ∈ f s λ . Hence f s λ is an ideal of S. Conversely, assume for all λ ∈ [0, 1], if fsλ 6= ∅, then f s λ is an ideal of S. Let x,y ∈ S. Case 1 : f(x) ≥ f(y). Thus x ∈ fsλ for all λ < f(x). By assumption, we have f s λ is an ideal of S for all λ < f(x). So xy ∈ fsλ for all λ < f(x). Then f(xy) > λ for all λ < f(x). Then f(xy) ≥ f(x) = f(x)∨f(y). Case 2 : f(x) < f(y). Thus y ∈ fsλ for all λ < f(y). By assumption, we have f s λ is an ideal of S for all λ < f(y). So xy ∈ fsλ for all λ < f(y). Then f(xy) > λ for all λ < f(y). Then f(xy) ≥ f(y) = f(x) ∨f(y). Therefore f is a fuzzy ideal of S. � Let f be a fuzzy subset of S. Let Aprρ(f)(x) and Aprρ(f)(x) be fuzzy subsets of S defined by Aprρ(f)(x) = ∨ a∈[x]ρ f(a) and Apr ρ (f)(x) = ∧ a∈[x]ρ f(a) are called, respectively, the ρ-upper and ρ-lower approximations of the fuzzy set f. Aprρ(f) = (Aprρ(f),Aprρ(f)) is called a rough fuzzy set with respect to ρ if Aprρ(f) 6= Aprρ(f). Theorem 5.2. Let ρ be a complete congruence relation on an LA-semigroup S. Let f be a fuzzy subset of S. If f is a fuzzy ideal of S. Then (1) Aprρ(f) is a fuzzy ideal of S. (2) Apr ρ (f) is, if it is nonempty, a fuzzy ideal of S. Proof. (1) Assume f is a fuzzy ideal of S. Let x,y ∈ S. Then f(xy) ≥ f(x) ∨f(y). We have Aprρ(f)(xy) = ∨ s∈[xy]ρ f(s) = ∨ s∈[x]ρ[y]ρ f(s) = ∨ p∈[x]ρ, q∈[y]ρ f(pq) ≥   ∨ p∈[x]ρ f(p)  ∨   ∨ q∈[y]ρ f(q)   = Aprρ(f)(x) ∨Aprρ(f)(y). Int. J. Anal. Appl. 19 (3) (2021) 462 Then Aprρ(f)(xy) ≥ Aprρ(f)(x) ∨Aprρ(f)(y). Therefore we obtain that Aprρ(f) is a fuzzy ideal of S. (2) Assume f is a fuzzy ideal of S. Let x,y ∈ S. Then f(xy) ≥ f(x) ∨f(y). We have Apr ρ (f)(xy) = ∧ s∈[xy]ρ f(s) = ∧ s∈[x]ρ[y]ρ f(s) = ∧ p∈[x]ρ, q∈[y]ρ f(pq) ≥   ∧ p∈[x]ρ f(p)  ∨   ∧ q∈[y]ρ f(q)   = Apr ρ (f)(x) ∨Apr ρ (f)(y). Then Apr ρ (f)(xy) ≥ Apr ρ (f)(x) ∨ Apr ρ (f)(y). Therefore we obtain that Apr ρ (f) is, if it is nonempty, a fuzzy ideal of S. This completes the proof. � A fuzzy ideal f of an LA-semigroup S is called a fuzzy prime ideal of S if f(xy) = f(x) or f(xy) = f(y) for all x,y ∈ S. Theorem 5.3. Let f be a fuzzy subset of an LA-semigroup S. Then f is a fuzzy prime ideal of S iff fλ 6= ∅ is a prime ideal of S for every λ ∈ [0, 1]. Proof. Assume f is a fuzzy ideal of S. Then f is a fuzzy ideal of S. Assume fλ 6= ∅. By Theorem 5.1, fλ is a ideal of S. Let x,y ∈ S such that xy ∈ fλ. Since f is a fuzzy prime ideal of S, f(xy) = f(x) or f(xy) = f(y). This implies x ∈ fλ or y ∈ fλ. Therefore fλ is a prime ideal of S. Conversely, assume for all λ ∈ [0, 1], if fλ 6= ∅, then fλ is a prime ideal of S. Let x,y ∈ S. By Theorem 5.1, f is a fuzzy ideal of S. This implies f(xy) ≥ f(x) and f(xy) ≥ f(y). Let λ = f(xy). Thus xy ∈ fλ. Since fλ is a prime ideal of S, x ∈ fλ or y ∈ fλ. This implies that f(x) ≥ λ = f(xy) or f(y) ≥ λ = f(xy). Hence f(xy) = f(x) or f(xy) = f(y). Hence f is a fuzzy prime ideal of S. � Theorem 5.4. Let f be a fuzzy subset of an LA-semigroup S. Then f is a fuzzy prime ideal of S iff fsλ 6= ∅ is a prime ideal of S for every λ ∈ [0, 1]. Proof. Assume f is a fuzzy prime ideal of S. Then f is a fuzzy ideal of S. Assume fsλ 6= ∅. By Theorem 5.1, fsλ is a ideal of S. Let x,y ∈ S such that xy ∈ f s λ. Then f(xy) > λ. Since f is a fuzzy prime ideal of S, f(xy) = f(x) or f(xy) = f(y). This implies that f(x) > λ or f(y) > λ. Hence x ∈ fsλ or y ∈ f s λ. Therefore fsλ is a prime ideal of S. Conversely, assume for all λ ∈ [0, 1], if fsλ 6= ∅, then f s λ is a prime ideal of S. Let x,y ∈ S. By Theorem 5.1, f is a fuzzy ideal of S. This implies f(xy) ≥ f(x) and f(xy) ≥ f(y). We have xy ∈ fsλ for all λ < f(xy). Since fsλ is a prime ideal of S for all λ < f(xy), x ∈ f s λ or y ∈ f s λ for all λ < f(xy). This implies that f(x) > λ or f(y) > λ for all λ < f(xy). Then f(x) ≥ f(xy) or f(y) ≥ f(xy). Hence f(xy) = f(x) or f(xy) = f(y). Hence f is a fuzzy prime ideal of S. � Int. J. Anal. Appl. 19 (3) (2021) 463 Let ρ be a congruence relation on an LA-semigroup S. A fuzzy subset f of S is called a ρ-upper [a ρ-lower] rough fuzzy prime ideal of S if Aprρ(f) [Aprρ(f)] is a fuzzy prime ideal of S. We call f a rough fuzzy prime ideal of S if it is both a ρ-upper and a ρ-lower rough fuzzy prime ideal of S. Lemma 5.1. Let ρ be a congruence relation on an LA-semigroup S. If f is a fuzzy subset of S and λ ∈ [0, 1]. Then (i) (Apr ρ (f))λ = Aprρ(fλ) and (ii) (Aprρ(f)) s λ = Aprρ(f s λ) Proof. (i) Let x ∈ (Apr ρ (f))λ ⇐⇒ Aprρ(f)(x) ≥ λ ⇐⇒ ∧ y∈[x]ρ f(y) ≥ λ ⇐⇒ for all y ∈ [x]ρ, f(y) ≥ λ ⇐⇒ [x]ρ ⊆ fλ ⇐⇒ x ∈ Aprρ(fλ). (ii) Let x ∈ (Aprρ(f))sλ ⇐⇒ Aprρ(f)(x) > λ ⇐⇒ ∨ y∈[x]ρ f(y) > λ ⇐⇒ there exist y ∈ [x]ρ, f(y) > λ ⇐⇒ [x]ρ ∩fsλ 6= ∅⇐⇒ x ∈ Aprρ(f s λ). � Theorem 5.5. Let f be a fuzzy prime ideal of an LA-semigroup S and ρ be a complete congruence relation on S. Then f is a rough fuzzy prime ideal of S. Proof. Let f be a fuzzy prime ideal of an LA-semigroup S and ρ a complete congruence on S. By Theorem 5.3, for all λ ∈ [0, 1], if fλ 6= ∅, then fλ is a prime ideal of S. By Theorem 3.2, for all λ ∈ [0, 1], if Apr ρ (fλ) 6= ∅, then Aprρ(fλ) is a prime ideal of S. From this and Lemma 5.1(i), for all λ ∈ [0, 1], if (Apr ρ (f))λ 6= ∅, (Aprρ(f))λ is a prime ideal of S. By Theorem 5.3, Aprρ(f) is a fuzzy prime ideal of S. Hence f is a ρ-lower rough fuzzy prime ideal of S. Similarly, f is a ρ-upper rough fuzzy prime ideal of S. 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