${protect elax protect �egingroup immediate write @unused def MessageBreak let protect edef You're in trouble here. Try typing to proceed.MessageBreak If that doesn't work, type X to quit. errhelp let def MessageBreak def errmessage LaTeX Error: mathcal allowed only in math mode. See the LaTeX manual or LaTeX Companion for explanation. Type H for immediate help endgroup elax N}$-soft $p$-ideals of $BCI$-algebras EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS Vol. 12, No. 1, 2019, 79-87 ISSN 1307-5543 – www.ejpam.com Published by New York Business Global N -soft p-ideals of BCI-algebras G. Muhiuddin1,∗, Shuaa Aldhafeeri2 1 Department of Mathematics, University of Tabuk, Tabuk 71491, Saudi Arabia 2 Department of Mathematics, College of Basic Education, Public Authority for Applied Education and Training, Kuwait Abstract. In this paper, using the notions of soft sets and N -structures, the notion of N -soft p-ideals in BCI-algebras is introduced, and related properties are investigated. Furthermore, relations between N -soft ideals and N -soft p-ideals are discussed. Finally, conditions for an N -soft ideal to be an N -soft p-ideal are established. 2010 Mathematics Subject Classifications: 06D72, 06F35, 03G25 Key Words and Phrases: p-ideal, N -ideal, pN -ideal, N -soft ideal, N -soft p-ideal 1. Introduction Uncertainties can’t be handled using traditional mathematical tools but may be dealt with using a wide range of existing theories such as the probability theory, the theory of (intuitionistic) fuzzy sets, the theory of vague sets, the theory of interval mathematics, and the theory of rough sets. However, all of these theories have their own limitations which are pointed out in [15]. Maji et al. [13] and Molodtsov [15] suggested that one reason for these difficulties may be due to the inadequacy of the parametrization tool of the theory. To overcome these difficulties, Molodtsov [15] introduced the concept of soft set as a new mathematical tool for dealing with uncertainties that is free from the difficulties that have troubled the usual theoretical approaches. He pointed out several directions for the applications of soft sets. Later on, Maji et al. [13] described the ap- plication of soft set theory to a decision making problem. Maji et al. [12] also studied several operations on the theory of soft sets. Chen et al. [4] presented a new definition of soft set parametrization reduction, and compared this definition to the related concept of attributes reduction in rough set theory. The algebraic structure of set theories dealing with uncertainties has been studied by some authors. The most appropriate theory for dealing with uncertainties is the theory of fuzzy sets developed by Zadeh [21]. Roy et al. [20] presented some results on an application of fuzzy soft sets in decision making problem. ∗Corresponding author. DOI: https://doi.org/10.29020/nybg.ejpam.v12i1.3343 Email addresses: chishtygm@gmail.com (G. Muhiuddin), saldhafeeri@yahoo.com (S. Aldhafeeri) http://www.ejpam.com 79 c© 2019 EJPAM All rights reserved. G. Muhiuddin, S. Aldhafeeri / Eur. J. Pure Appl. Math, 12 (1) (2019), 79-87 80 Aygünoǧlu et al. [2] introduced the notion of fuzzy soft group and studied its properties. Ali et al. [3] discussed new operations in soft set theory. Jun [6] applied the notion of soft set to BCK/BCI-algebras, and Jun et al. [8] considered applications of soft set theory in the ideals of d-algebras. Also, Muhiuddin et al. studied the soft set theory on various aspects (see for e.g., [1], [16], [17]), [18], [19]). A (crisp) set A in a universe X can be defined in the form of its characteristic function µA : X → {0, 1} yielding the value 1 for elements belonging to the set A and the value 0 for elements excluded from the set A. So far most of the generalization of the crisp set have been conducted on the unit interval [0, 1] and they are consistent with the asymme- try observation. In other words, the generalization of the crisp set to fuzzy sets relied on spreading positive information that fit the crisp point {1} into the interval [0, 1]. Because no negative meaning of information is suggested, so one should be interested to deal with negative information and to supply a mathematical tool for the same. Considering this fact, Jun et al. [9] introduced a new function which is called negative-valued function, and constructed N -structures. They applied N -structures to BCK/BCI-algebras, and discussed N -subalgebras and N -ideals in BCK/BCI-algebras. Jun et al. [10] considered closed ideals in BCH-algebras based on N -structures. Jun et al. [11] introduced the no- tion of N -soft sets which are a soft set based on N -structures, and then they applied it to both a decision making problem and a BCK/BCI-algebra. Jun et al [7] introduced the notion of (closed) N -ideal over a BCI-algebra based on soft sets and N -structures, and investigated related properties. They established relations between N -BCI-algebras and N -ideals. They also provided characterizations of a (closed) N -ideal over a BCI-algebra, and considered conditions for an N -ideal to be an N -BCI-algebra. In this paper, we apply the soft sets and N -structures to p-ideals in BCI-algebras. We introduce the notion ofN -soft p-ideals in BCI-algebras, and investigate related properties. We provide relations between N -soft ideals and N -soft p-ideals, and establish conditions for an N -soft ideal to be an N -soft p-ideal. 2. Preliminaries A BCK/BCI-algebra is an important class of logical algebras introduced by K. Iséki and was extensively investigated by several researchers. An algebra (X; ∗, 0) of type (2, 0) is called a BCI-algebra if it satisfies the following conditions: (I) (∀x, y, z ∈ X) (((x ∗ y) ∗ (x ∗ z)) ∗ (z ∗ y) = 0), (II) (∀x, y ∈ X) ((x ∗ (x ∗ y)) ∗ y = 0), (III) (∀x ∈ X) (x ∗ x = 0), G. Muhiuddin, S. Aldhafeeri / Eur. J. Pure Appl. Math, 12 (1) (2019), 79-87 81 (IV) (∀x, y ∈ X) (x ∗ y = 0, y ∗ x = 0 ⇒ x = y). Define a binary relation ≤ on X by letting x ∗ y = 0 if and only if x ≤ y. Then (X,≤) is a partially ordered set. A BCI-algebra X satisfying 0 ≤ x for all x ∈ X, is called BCK-algebra. Theorem 1. Let X be a BCI-algebra. Then following hold (a1) (∀x ∈ X) ((x ∗ 0 = x)), (a2) (∀x, y, z ∈ X) ((x ∗ y) ∗ z = (x ∗ z) ∗ y), (a3) (∀x ∈ X) (0 ∗ (0 ∗ (0 ∗ x)) = 0 ∗ x), (a4) (∀x, y, z ∈ X) (0 ∗ (0 ∗ ((x ∗ z) ∗ (y ∗ z))) = (0 ∗ y) ∗ (0 ∗ x)), (a5) (∀x, y ∈ X) (0 ∗ (0 ∗ (x ∗ y)) = (0 ∗ y) ∗ (0 ∗ x)) A non-empty subset S of a BCK/BCI-algebra X is called a subalgebra of X if x∗y ∈ S for all x, y ∈ S. A subset A of a BCK/BCI-algebra X is called an ideal of X if it satisfies: (c1) 0 ∈ A, (c2) (∀x, y ∈ X) (x ∗ y ∈ A, y ∈ A ⇒ x ∈ A). We refer the reader to the books [5, 14] for further information regarding BCK/BCI- algebras. For any family {ai | i ∈ Λ} of real numbers, we define∨ {ai | i ∈ Λ} := { max{ai | i ∈ Λ} if Λ is finite, sup{ai | i ∈ Λ} otherwise. ∧ {ai | i ∈ Λ} :=  min{ai | i ∈ Λ} if Λ is finite, inf{ai | i ∈ Λ} otherwise. Denote by F (X, [−1, 0]) the collection of functions from a set X to [−1, 0]. We say that an element of F (X, [−1, 0]) is a negative-valued function from X to [−1, 0] (briefly, N -function on X). By an N -structure we mean an ordered pair (X, η) of X and an N -function η on X. Definition 1 ([9]). By a subalgebra of a BCK/BCI-algebra X based on N -function η (briefly, N -subalgebra of X), we mean an N -structure (X, η) in which η satisfies the following assertion: (∀x, y ∈ X) ( η(x ∗ y) ≤ ∨ {η(x), η(y)} ) . (1) G. Muhiuddin, S. Aldhafeeri / Eur. J. Pure Appl. Math, 12 (1) (2019), 79-87 82 Definition 2 ([9]). By an ideal of a BCK/BCI-algebra X based on N -function η (briefly, N -ideal of X), we mean an N -structure (X, η) in which η satisfies the following assertion: (∀x, y ∈ X) ( η(0) ≤ η(x) ≤ ∨ {η(x ∗ y), η(y)} ) . (2) 3. p-ideals based on N -soft sets In what follows let E denote a set of attributes unless otherwise specified. We will use the terminology “soft machine” which means that it produces a BCI-algebra, that is, consider a soft machine “](−,−)” for which ](x, y) = z means that if we input a couple (x, y) of informations to ](−,−) then we get a new information z. Definition 3 ([11]). Let X be an initial universe set. By an N -soft set over X we mean a pair (η,A) where A ⊂ E and η is a mapping from A to F (X, [−1, 0]), i.e., for each a ∈ A, η(a) := ηa is an N -function on X. Denote by N (X,E) the collection of all N -soft sets over X with attributes from E and we call it an N -soft class. Definition 4 ([11]). Let (η,A) be an N -soft set over a BCK/BCI-algebra X where A is a subset of E. If there exists an attribute u ∈ A for which the N -structure (X, ηu) is an N -subalgebra of X, then we say that (η,A) is an N -soft BCK/BCI-algebra related to the attribute u (briefly, Nu-soft BCK/BCI-algebra). If (η,A) is an Nu-soft BCK/BCI- algebra for all u ∈ A, we say that (η,A) is an N -soft BCK/BCI-algebra. Definition 5 ([7]). Let (η,A) be an N -soft set over a BCK/BCI-algebra X where A is a subset of E. If there exists an attribute u ∈ A for which the N -structure (X, ηu) is an N -ideal of X, then we say that (η,A) is an N -soft ideal of X related to the attribute u (briefly, Nu-soft ideal). If (η,A) is an Nu-soft ideal of X for all u ∈ A, we say that (η,A) is an N -soft ideal over X. Definition 6. By a p-ideal of a BCI-algebra X based on N -function ψ (briefly, pN -ideal of X), we mean an N -structure (X,ψ) in which ψ satisfies the following assertions: (i) (∀x ∈ X) (ψ(0) ≤ ψ(x)) , (ii) (∀x, y, z ∈ X) (ψ(x) ≤ ∨ {ψ ((x ∗ z) ∗ (y ∗ z)) , ψ(y)}) . Definition 7. Let (η,A) be an N -soft set over a BCI-algebra X where A is a subset of E. If there exists an attribute u ∈ A for which the N -structure (X, ηu) is a pN -ideal of X, then we say that (η,A) is an N -soft p-ideal of X related to the attribute u (briefly, Nu-soft p-ideal). If (η,A) is an Nu-soft p-ideal of X for all u ∈ A, we say that (η,A) is an N -soft p-ideal over X. Example 1. Let U be a initial universe set consists of ‘white’, ‘reddish’, ‘green’ and ‘yellow’. The soft machine “](−,−)” is equipped as follows: G. Muhiuddin, S. Aldhafeeri / Eur. J. Pure Appl. Math, 12 (1) (2019), 79-87 83 Table 1: Tabular representation of (η,A) (η,A) white reddish green yellow beautiful −0.8 −0.7 −0.3 −0.3 fine −0.6 −0.5 −0.4 −0.4 smart −0.7 −0.5 −0.1 −0.1 ](x, y) = y if x = white and y ∈ U, ](x, y) =  reddish if (x, y) = (reddish,white), white if (x, y) = (reddish, reddish), yellow if (x, y) = (reddish, green), green if (x, y) = (reddish, yellow), ](x, y) =  green if (x, y) = (green,white), yellow if (x, y) = (green, redish), white if (x, y) = (green, green), reddish if (x, y) = (green, yellow), ](x, y) =  yellow if (x, y) = (yellow,white), green if (x, y) = (yellow, redish), reddish if (x, y) = (yellow, green), white if (x, y) = (yellow, yellow). Then the soft machine “](−,−)” makes U into a BCI-algebra. Consider a set of at- tributes: A := {beautiful, fine, smart}, and let (η,A) be an N -soft sets over U with the tabular representations which is given by Table 1. Then the N -structures (U, ηbeautiful) , (U, ηfine) and (U, ηsmart) are pN -ideals of U. Hence (η,A) is an N -soft p-ideal over U. Proposition 1. For any attribute u ∈ A, every Nu-soft p-ideal (η,A) over a BCI-algebra X satisfies the following inequality: (∀x ∈ X) (ηu(x) ≤ ηu (0 ∗ (0 ∗ x))) . (3) Proof. Using Definition 6(ii), we have ηu(x) ≤ ∨ {ηu((x ∗ z) ∗ (y ∗ z)), ηu(y)} (4) for all x, y, z ∈ X. If we substitute x for z, and 0 for y in (4), then ηu(x) ≤ ∨ {ηu((x ∗ x) ∗ (0 ∗ x)), ηu(0)} = ∨ {ηu(0 ∗ (0 ∗ x)), ηu(0)} = ηu(0 ∗ (0 ∗ x)) G. Muhiuddin, S. Aldhafeeri / Eur. J. Pure Appl. Math, 12 (1) (2019), 79-87 84 by using (III) and Definition 6(i). This completes the proof. Corollary 1. Every N -soft p-ideal (η,A) over a BCI-algebra X satisfies the following inequality: (∀x ∈ X) (η(x) ≤ η(0 ∗ (0 ∗ x))) . Theorem 2. For any attribute u ∈ A, every Nu-soft p-ideal over a BCI-algebra X is an Nu-soft ideal over X. Proof. Let (η,A) be an Nu-soft p-ideal over a BCI-algebra X. Since x ∗ 0 = x for all x ∈ X, it follows from Definition 6(ii) and (a1) that ηu(x) ≤ ∨ {ηu((x ∗ 0) ∗ (y ∗ 0)), ηu(y)} = ∨ {ηu(x ∗ y), ηu(y)} for all x, y ∈ X. Therefore (η,A) is an Nu-soft ideal over X. Corollary 2. Every N -soft p-ideal over a BCI-algebra X is an N -soft ideal over X. The converse of Theorem 2 is not true as seen in the following example. Example 2. Let U be an initial universe set consists of ‘white’, ‘blackish’, ‘reddish’, ‘green’ and ‘yellow’. The soft machine “](−,−)” is equipped as follows: ](x, y) = x if x ∈ U and y = blackish, ](x, y) = { blackish if (x, y) ∈ {(blackish, reddish), (reddish, reddish)}, x if (x, y) ∈ {(green, reddish), (yellow, reddish), (white, reddish)}, ](x, y) =  white if (x, y) ∈ {(blackish, green), (reddish, green)}, blackish if (x, y) = (green, green), green if (x, y) = (yellow, green), yellow if (x, y) = (white, green), ](x, y) =  yellow if (x, y) ∈ {(blackish, yellow), (reddish, yellow)}, white if (x, y) = (green, yellow), blackish if (x, y) = (yellow, yellow), green if (x, y) = (white, yellow), ](x, y) =  green if (x, y) ∈ {(blackish,white), (reddish,white)}, yellow if (x, y) = (green,white), white if (x, y) = (yellow,white), blackish if (x, y) = (white,white). Then the soft machine “](−,−)” makes U into a BCI-algebra. Consider a set of at- tributes: A := {beautiful, fine,moderate}, and let (η,A) be an N -soft sets over U with the tabular representation which is given by Table 2. Then (η,A) is an Nfine-soft ideal over U, but it is not an Nfine-soft p-ideal over U since ηfine(reddish) = −0.5 > −0.7 = ηfine(blackish) = ∨ {ηfine(](](reddish, green),](blackish, green))), ηfine(blackish)} . G. Muhiuddin, S. Aldhafeeri / Eur. J. Pure Appl. Math, 12 (1) (2019), 79-87 85 Table 2: Tabular representation of (η,A) (η,A) blackish reddish green yellow white beautiful −0.9 −0.2 −0.4 −0.6 −0.1 fine −0.7 −0.5 −0.2 −0.3 −0.4 moderate −0.8 −0.4 −0.3 −0.2 −0.5 Proposition 2. For any attribute u ∈ A, every Nu-soft p-ideal (η,A) over a BCI-algebra X satisfies the following inequality: (∀x, y, z ∈ X) (ηu(x ∗ y) ≥ ηu ((x ∗ z) ∗ (y ∗ z))) . (5) Proof. Let u ∈ A. If (η,A) is an Nu-soft p-ideal over a BCI-algebra X, then it is an Nu-soft ideal over X by Theorem 2. Note that ((x ∗ z) ∗ (y ∗ z)) ∗ (x ∗ y) = 0 for all x, y, z ∈ X. Thus we have ηu ((x ∗ z) ∗ (y ∗ z)) ≤ ∨ {ηu (((x ∗ z) ∗ (y ∗ z)) ∗ (x ∗ y)) , ηu(x ∗ y)} = ∨ {ηu(0), ηu(x ∗ y)} = ηu(x ∗ y) for all x, y, z ∈ X. We provide conditions for an N -soft ideal to be an N -soft p-ideal. Theorem 3. For any attribute u ∈ A, let (η,A) be an Nu-soft ideal over a BCI-algebra X that satisfies: (∀x, y, z ∈ X) (ηu (x ∗ y) ≤ ηu ((x ∗ z) ∗ (y ∗ z))) . (6) Then (η,A) is an Nu-soft p-ideal over X. Proof. If an Nu-soft ideal (η,A) over a BCI-algebra X satisfies (6), then ηu(x) ≤ ∨ {ηu(x ∗ y), ηu(y)} ≤ ∨ {ηu ((x ∗ z) ∗ (y ∗ z)) , ηu(y)} for all x, y, z ∈ X. Therefore (η,A) is an Nu-soft p-ideal over X. Lemma 1 ([7]). For any attribute u ∈ A, every Nu-soft ideal (η,A) over a BCI-algebra X satisfies the following inequality: (∀x ∈ X) (ηu(0 ∗ (0 ∗ x)) ≤ ηu(x)) . (7) Theorem 4. For any attribute u ∈ A, let (η,A) be an Nu-soft ideal over a BCI-algebra X that satisfies: (∀x ∈ X) (ηu(x) ≤ ηu (0 ∗ (0 ∗ x))) . (8) Then (η,A) is an Nu-soft p-ideal over X. REFERENCES 86 Proof. By using Lemma 1, (a4), (a5) and (8), we have ηu ((x ∗ z) ∗ (y ∗ z)) ≥ ηu (0 ∗ (0 ∗ ((x ∗ z) ∗ (y ∗ z)))) = ηu ((0 ∗ y) ∗ (0 ∗ x)) = ηu (0 ∗ (0 ∗ (x ∗ y)))) ≥ ηu(x ∗ y) for all x, y, z ∈ X. It follows from Theorem 3 that (η,A) is an Nu-soft p-ideal over X. Acknowledgements The authors would like to express their sincere thanks to the learned referee(s) for valuable comments and several useful suggestions that improved the overall presentation of this paper. The first author was partially supported by the research grant S-0064-1439, Deanship of Scientific Research (DRS), University of Tabuk, Tabuk-71491, Saudi Arabia. References [1] A. Al-roqi, G. Muhiuddin and S. Aldhafeeri, Normal Unisoft Filters in R0-algebras, Cogent Mathematics, Vol. 1, No.4, 1-9 (2017). [2] A. Aygünoǧlu and H. Aygün, Introduction to fuzzy soft groups, Comput. Math. Appl. 58 (2009) 1279–1286. [3] M. I. Ali, F. Feng, X, Liu, W. K. Min and M. Shabir, On some new operations in soft set theory, Comput. Math. Appl. 57 (2009) 1547–1553. [4] D. Chen, E. C. C. Tsang, D. S. Yeung and X. Wang, The parametrization reduction of soft sets and its applications, Comput. Math. Appl. 49 (2005) 757–763. [5] Y. S. Huang, BCI-algebra, Science Press, Beijing, 2006. [6] Y. B. Jun, Soft BCK/BCI-algerbas, Comput. Math. Appl. 56 (2008) 1408–1413. [7] Y. B. Jun, K. J. Lee and M. S. Kang, Ideal theory in BCK/BCI-algebras based on soft sets and N -structures, Discrete Dyn. Nat. Soc. Volume 2012, Article ID 910450, 13 pages [8] Y. B. Jun, K. J. Lee and C. H. Park, Soft set theory applied to ideals in d-algebras, Comput. Math. Appl. 57 (2009) 367–378. [9] Y. B. Jun, K. J. Lee and S. Z. Song, N -ideals of BCK/BCI-algebras, J. Chungcheong Math. Soc. 22 (2009) 417–437. [10] Y. B. Jun, M. A. Öztürk and E. H. Roh, N -structures applied to closed ideals in BCH-algebras, Int. J. Math. Math. Sci. Volume 2010, Article ID 943565, 9 pages. REFERENCES 87 [11] Y. B. Jun, S. Z. Song and K. J. Lee, The combination of soft sets and N -structures with applications, Journal of Applied Mathematics Volume 2013, Article ID 420312, 10 pages. [12] P. K. Maji, R. Biswas and A. R. Roy, Soft set theory, Comput. Math. Appl. 45 (2003) 555–562. [13] P. K. Maji, A. R. Roy and R. Biswas, An application of soft sets in a decision making problem, Comput. Math. Appl. 44 (2002) 1077–1083. [14] J. Meng and Y. B. Jun, BCK-algebras, Kyungmoon Sa Co. Seoul, 1994. [15] D. Molodtsov, Soft set theory - First results, Comput. Math. Appl. 37 (1999) 19–31. [16] G. Muhiuddin, Abdullah M. Al-roqi and Shuaa Aldhafeeri, Filter theory in MTL- algebras based on Uni-soft property, Bulletin of the Iranian Mathematical Society, Vol. 43, No.7 (2017) 2293–2306. [17] G. Muhiuddin and Abdullah M. Al-roqi, Unisoft Filters in R0-algebras, Journal of Computational Analysis and Applications, 19, No. 1, (2015) 133–143. [18] G. Muhiuddin, Feng Feng and Young Bae Jun, Subalgebras of BCK/BCI-Algebras Based on Cubic Soft Sets, The Scientific World Journal, Volume 2014, Article ID 458638, (2014) 9 pages. [19] G. Muhiuddin and Abdullah M. Al-roqi, Cubic soft sets with applications in BCK/BCI-algebras, Annals of Fuzzy Mathematics and Informatics, Volume 8, No. 2, (2014) 291–304. [20] A. R. Roy and P. K. Maji, A fuzzy soft set theoretic approach to decision making problems, J. Comput. Appl. Math. 203 (2007) 412–418. [21] L. A. Zadeh, Fuzzy sets, Inform. Control 8 (1965) 338–353.