EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS Vol. 11, No. 2, 2018, 505-516 ISSN 1307-5543 – www.ejpam.com Published by New York Business Global Near ideals in near semigroups Nurettin Bağırmaz Mardin Artuklu University, Mardin, Turkey Abstract. In this paper, we introduced the notion of near subsemigroups, near ideals, near bi- ideals and homomorphisms of near semigroups on near approximation spaces. Then we give some properties of these near structures. 2010 Mathematics Subject Classifications: 03E99, 20M99 Key Words and Phrases: Near set, near semigroup, near ideal, near bi-ideal, homomorphism 1. Introduction Rough sets were introduced by Z. Pawlak in his paper [16]. Algebraic structures of rough sets have been studied by many authors, for example, Bonikowaski [4], Iwinski [8], and Pomykala and Pomykala [24]. In 1994, Biswas and Nanda [3] introduced the notion of rough group and rough subgroups that their notion depends on the upper approximation and does not depend on the lower approximation. Miao et al. [14] improve definitions of rough group and rough subgroup, and prove their new properties. On the other hand, Kuroki and Wang [11] presented some properties of the lower and upper approximations with respect to the normal subgroups in 1996. In addition, some properties of the lower and the upper approximations with respect to the normal subgroups were studied in [5, 13, 26– 28]. Also, Kuroki [12], introduced the notion of a rough ideal in a semigroup. Davvaz [6], introduced the notion of rough subring with respect to an ideal of a ring. Xiao and Zhang [30], studied the notions of rough prime ideals and rough fuzzy prime ideals in a semigroup. Bağırmaz and Özcan [1], studied the notion of rough semigroup on approximation space. Moreover, Bağırmaz [2], investigated rough prime ideals on approximation spaces. Near sets were introduced by Peters [17] on the basis of a generalization of rough set theory. The algebraic properties of near sets are described in [19]. Recent work has considered near groups [9] and near semigroups [10]. The fundamental idea of near set theory is object description and classification according to perceptual knowledge. It is supposed that perceptual knowledge about objects is always given with respect to probe functions, i.e., real-valued functions which represent features of a physical object [7, 15, 20– 23, 25, 29]. Email address: nurettinbagirmaz@artuklu.edu.tr (N. Bağırmaz) http://www.ejpam.com 505 c© 2018 EJPAM All rights reserved. N. Bağırmaz / Eur. J. Pure Appl. Math, 11 (2) (2018), 505-516 506 The main purpose of this paper is to introduce near ideals and give some properties of such ideals on nearness approximation spaces. Finally, near image and near inverse image of near ideal are discussed. We introduced the notion of near ideal that our notion depends on the upper approximation and does not depend on the lower approximation. So, our definition of near ideal is similar to the definition of rough ideal [1]. 2. Preliminaries In this section, we will give some definitions and properties regarding near sets as in [18]. Table 1: Description Symbols Symbol Interpretation R Set of real numbers, O Set of perceptual objects, X X ⊆ O, set of sample objects, x x ∈ O, sample objects, F a set of functions representing object features, B B ⊆ F , Φ Φ : O → RL, object description, L L is a description length, i i ≤ L, Φ(x) Φ(x) = (φ1(x), φ2(x), φ3(x), ..., φi(x), ..., φL(x)). Objects are known by their descriptions. An object description is defined by means of a tuple of function values Φ(x) associated with an object x ∈ X. The important thing to notice is the choice of functions φi ∈ B used to describe an object of interest. Assume that B ⊆ F (see Table 1) is a given set of functions representing features of sample objects X ⊆ O. Let φi ∈ B, where φi : O → R. In combination, the functions representing object features provide a basis for an object description Φi : O → RL , a vector containing measurements associated with each functional value φi (x) , where the description length |Φ| = L. Object Description : Φ(x) = (φ1(x), φ2(x), φ3(x), ..., φi(x), ..., φL(x)). The intuition underlying a description Φ(x) is a recording of measurements from sen- sors, where each sensor is modelled by a function φi. Table 2: Nearness Approximation Space Symbols N. Bağırmaz / Eur. J. Pure Appl. Math, 11 (2) (2018), 505-516 507 Symbol Interpretation B B ⊆ F , Br r ≤ |B| probe functions in B, ∼Br Indiscernibility relation defined using Br, [x]Br [x]Br = {x′ ∈ O | x ∼Br x ′}, equivalence class, O� ∼Br O� ∼Br= { [x]Br | x ′ ∈ O } , quotient set, ξO,Br Partition ξO,Br = O� ∼Br , r (|B| r ) , i.e., |B| functions φi ∈ B taken r at a time, Nr(B) Nr(B) = {ξO,Br |Br ⊆ B} set of partitions, Nr(B)∗X Nr(B)∗X = ∪ x∈O {[x]Br : [x]Br ⊆ X} , lower approximation, Nr(B)∗X Nr(B)∗X = ∪ x∈O {[x]Br : [x]Br ∩X 6= ∅} , upper approximation, BndNr(B)(X) Nr(B)∗X�Nr(B)∗X = {x|x ∈ Nr(B)∗X and x /∈ Nr(B)∗X. A nearness approximation space (NAS) is denoted by NAS = (O, F,∼Br , Nr, vNr) which is defined with a set of perceived objects O, a set of probe functions F represent- ing object features, an indiscernibility relation ∼Br defined relative to Br ⊆ B ⊆ F , a collection of partitions (families of neighbourhoods) Nr(B), and a neighbourhood overlap function Nr. The relation ∼Br is the usual indiscernibility relation from rough set theory restricted to a subset Br ⊆ B. The subscript r denotes the cardinality of the restricted sub- set Br, where we consider (|B| r ) , i.e., |B| functions i ∈ F taken r at a time to define the re- lation ∼Br . This relation defines a partition of O into non-empty, pairwise disjoint subsets that are equivalence classes denoted by [x]Br , where [x]Br = {x′ ∈ O | x ∼Br x ′}. These classes form a new set called the quotient setO� ∼Br , whereO� ∼Br= { [x]Br | x ′ ∈ O } . In effect, each choice of probe functions Br defines a partition ξO,Br on a set of objects O, namely, ξO,Br = O� ∼Br . Every choice of the set Br leads to a new partition of O. Let F denote a set of features for objects in a set X, where each φi ∈ F that maps X to some value set Vφi (range of φi). The value of φi (x) is a measurement associated with a feature of an object x ∈ X. The overlap function vNr is defined by vNr : P (O)×P (O)→ [0, 1],where P (O) is the powerset of O. The overlap function vNr maps a pair of sets to a number in [0, 1] representing the degree of overlap between sets of objects with features defined by probe functions Br ⊆ B. For each subset Br ⊆ B of probe functions, define the binary relation ∼Br= {(x, x′ ) ∈ O×O : ∀φi ∈ Br, φi(x) = φi(x ′ )}. Since each ∼Br is, in fact, the usual indiscernibility relation [16], for Br ⊆ B and x ∈ O, let [x]B denote the equivalence class containing x, i.e., [x]Br = {x′ ∈ O|∀φ ∈ Br, φ(x ′ ) = φ(x)}. If (x, x ′ ) ∈∼Br (also written x ∼Br x ′ ), then x and x ′ are said to be B indiscernible with respect to all feature probe functions in Br. Then define a collection of partitions Nr(B) (families of neighbor- hoods), where Nr(B) = {ξO,Br |Br ⊆ B}. Families of neighborhoods are constructed for each combination of probe functions in B using (|B| r ) , i.e., |B| probe functions taken r at a time. Proposition 1. [10] Let (O, F,∼Br , Nr, vNr) be a nearness approximation space and X,Y ⊆ O. Then, the approximations have the following properties: N. Bağırmaz / Eur. J. Pure Appl. Math, 11 (2) (2018), 505-516 508 (1) Nr(B)∗X ⊆ X ⊆ Nr(B)∗X, (2) Nr(B)∗ (X ∪ Y ) = Nr(B)∗X ∪Nr(B)∗Y, (3) Nr(B)∗ (X ∩ Y ) = Nr(B)∗X ∩Nr(B)∗X, (4) X ⊆ Y implies Nr(B)∗X ⊆ Nr(B)∗Y, (5) X ⊆ Y implies Nr(B)∗X ⊆ Nr(B)∗Y, (6) Nr(B)∗ (X ∩ Y ) ⊆ Nr(B)∗X ∩Nr(B)∗Y, (7) Nr(B)∗ (X ∪ Y ) ⊇ Nr(B)∗X ∪Nr(B)∗Y. A nonempty subset H of a semigroup S is said to be a subsemigroup of S, if ab ∈ H for all a, b ∈ T , i.e., H2 ⊆ H. A nonempty subset I of a semigroup S is said to be a left (resp. right) ideal of S if SI ⊆ I (resp. IS ⊆ I). A nonempty subset I of S is called an ideal of S if I is both a left and a right ideal of S. A subsemigroup H of S is called a bi-ideal of S if HSH ⊆ H. Let S be a semigroup. An element x ∈ S is a left identity of S, if ∀y ∈ S : xy = y . Similarly, x is a right identity of S, if ∀y ∈ S : yx = y . If x is both a left and a right identity of S, then x is called an identity of S. A semigroup is a monoid, if it has an identity. The identity of a monoid S is usually denoted by 1S , or just by 1, for short. A monoid G is a group, if every x ∈ G has a (group) inverse x−1 ∈ G : xx−1 = 1 = x−1x 3. Near ideals In this section, we introduce the notions of near subsemigroup, near ideal and near bi-ideal on a near approximation space, and study some of its properties. Definition 1. [10] Let (O, F,∼Br , Nr, vNr) be a nearness approximation space and let (·) be a binary operation defined on O. A subset S of the set of perceptual objects O is called a near semigroup on nearness approximation space, provided the following properties are satisfied: (1) For all x, y ∈ S, x · y ∈ Nr(B)∗S, (2) For all x, y, z ∈ S, (x · y) · z = x · (y · z) property holds in Nr(B)∗S. Let (O, F,∼Br , Nr, vNr) be a nearness approximation space and (·) be a binary opera- tion defined on O. Let S be a near semigroup. There is only one an element x ∈ Nr(B)∗S is a left identity of S, if ∀y ∈ S : xy = y . Similarly, x is a right identity of S, if ∀y ∈ S : yx = y . If x is both a left and a right identity of S, then x is called a near identity of S. A near semigroup is a near monoid, if it has a near identity. N. Bağırmaz / Eur. J. Pure Appl. Math, 11 (2) (2018), 505-516 509 Lemma 1. A near semigroup S can have at most one identity. In fact, if S has a left identity x and a right identity y, then x = y. In particular, the identity of a near monoid is unique. Proof. By the definitions, y = xy = x. The identity of a near monoid S is denoted by e. A near monoid G is a near group, if every x ∈ G has a inverse x−1 ∈ G : x−1x = e = x−1x. Definition 2. Let (O, F,∼Br , Nr, vNr) be a nearness approximation space and (·) be a binary operation defined on O. Let S be a near semigroup and H a nonempty subset of S. A nonempty subset H of a near semigroup S is said to be a near subsemigroup of S, if ab ∈ Nr(B)∗H for all a, b ∈ H, i.e., HH ⊆ Nr(B)∗H. Another difference between near semigroup and semigroup is the following: Proposition 2. Let (O, F,∼Br , Nr, vNr) be a nearness approximation space and (·) be a binary operation defined on O. Let H1 and H2 be two near subsemigroups of the near semigroup S. A sufficient condition for intersection of two near subsemigroups of a near semigroup to be a near subsemigroup is Nr(B)∗H1 ∩Nr(B)∗H2 = Nr(B)∗ (H1 ∩H2). Proof. Suppose H1 and H2 are two near subsemigroups of S. It is obvious that H1 ∩ H2 ⊂ S. Consider x, y ∈ H1 ∩ H2. Because H1 and H2 are near subsemigroups, we have xy ∈ Nr(B)∗H1, xy ∈ Nr(B)∗H2, i.e. xy ∈ Nr(B)∗H1 ∩ Nr(B)∗H2. Assuming Nr(B)∗H1 ∩ Nr(B)∗H2 = Nr(B)∗ (H1 ∩H2), we have xy ∈ Nr(B)∗ (H1 ∩H2). Thus H1 ∩H2 is a near subsemigroup of S. Definition 3. Let (O, F,∼Br , Nr, vNr) be a nearness approximation space and (·) be a binary operation defined on O. A nonempty subset I of a near semigroup S is said to be a near left (resp. right) ideal of S if SI ⊆ Nr(B)∗I (resp. IS ⊆ Nr(B)∗I). A nonempty subset I of S is called an near ideal of S if I is both a left and a right near ideal of S. Proposition 3. Let (O, F,∼Br , Nr, vNr) be a nearness approximation space and (·) be a binary operation defined on O and S ⊆ O. Then (1) If H is a subsemigroup of semigroup S, then H is a near subsemigroup of near semi- group S. (2) If I is a left (right, two-sided) ideal of semigroup S, then I is a near left (right, two-sided) ideal of near semigroup S. Proof. (1) Let H be a subsemigroup of semigroup S, that is, HH ⊆ H. By Propo- sition 1 (1), we have that H ⊆ Nr(B)∗H. Thus HH ⊆ Nr(B)∗H. Hence, H is a near subsemigroup of near semigroup S. (2) Let I be a left ideal of semigroup S, that is, SI ⊆ I. Since I ⊆ S, by Proposition 1 (5), we know that Nr(B)∗I ⊆ Nr(B)∗S. Then, by Proposition 1.(1), we have that N. Bağırmaz / Eur. J. Pure Appl. Math, 11 (2) (2018), 505-516 510 I ⊆ Nr(B)∗I. Thus SI ⊆ I ⊆ Nr(B)∗I. This means that I is a near left ideal of near semigroup S. Also, we can easily show that I is a near right ideal of near semigroup S. The other cases can be seen in a similar way. The following example shows that the converse of by Proposition 3 is not true. Example 1. Let O = {1, 2, 3, 4, 5} be a set of perceptual objects with the following multi- plication table 2 and B = {φ1, φ2, φ3} ⊆ F be a set of probe functions with the following multiplication table 3, respectively. · a b c d e f a a a a b b a b a b b c d f c c c c e d f d b a b d f a e a b c d e f f b f b f f a a b c d e f φ1 1 1 2 3 2 4 φ2 1 2 2 3 4 4 φ3 1 1 3 2 4 4 Table 2. Table 3. Since [a]φ1 = {a, b} , [c]φ1 = {c, e} , [d]φ1 = {d} , [f ]φ1 = {f} we have ξφ1 = {[a]φ1 , [c]φ1 , [d]φ1 , [f ]φ1} . Since [a]φ2 = {a} , [b]φ2 = {b, c} , [d]φ2 = {d} , [e]φ2 = {e, f} we have ξφ2 = {[a]φ2 , [b]φ2 , [d]φ2 , [e]φ2} . Since [a]φ3 = {a, b} , [c]φ3 = {c} , [d]φ3 = {d} , [e]φ3 = {e, f} . we have ξφ3 = {[a]φ3 , [c]φ3 , [d]φ3 , [e]φ3} . Therefore, for r = 1, a set partitions of O is N1(B) = {ξφ1 , ξφ2 , ξφ3} . Let S = {b, c, d} be a subset of perceptual O as in table 4. · b c d b b b c c c c e d a b d N. Bağırmaz / Eur. J. Pure Appl. Math, 11 (2) (2018), 505-516 511 Table 4. Then, we have that N1(B)∗S = ∪ x∈O {[x]B1 : [x]B1 ∩ S 6= ∅} = {a, b} ∪ {c, e} ∪ {d} ∪ {b, c} ∪ {d} ∪ {c} = {a, b, c, d, e} . From Definition 1, S ⊆ O is a near semigroup. Because d · b = a /∈ S we have S is not a semigroup. Let H = {b, c} , then N1(B)∗H = ∪ x∈O {[x]B1 : [x]B1 ∩H 6= ∅} = {a, b} ∪ {c, e} ∪ {a, b, c} = {a, b, c} . From Definition 2, H is a near subsemigroup of near semigroup S. Let I = {c, d} , then N1(B)∗I = ∪ x∈O {[x]B1 : [x]B1 ∩ I 6= ∅} = {c, e} ∪ {d} ∪ {b, c} ∪ {c} = {b, c, d, e} . From Definition 3, I is a near left ideal of near semigroup S. The Proposition 3 shows that the notion of a near semigroup (left ideal, right ideal, two-sided ideal) is an extended notion of an ordinary semigroup (left ideal, right ideal, two-sided ideal). Another difference between near left (right, two-sided) ideal and left (right, two-sided) ideal is the following: Proposition 4. Let (O, F,∼Br , Nr, vNr) be a nearness approximation space and (·) be a binary operation defined on O. Let I1 and I2 be two left (right, two-sided) ideals of the near semigroup S. A sufficient condition for intersection of two left (right, two-sided) ideal of a near semigroup to be a near left (right, two-sided) ideal is Nr(B)∗I1 ∩ Nr(B)∗I2 = Nr(B)∗ (I1 ∩ I2) . Proof. Suppose I1 and I2 are two near left ideals of S. It is obvious that I1 ∩ I2 ⊂ S. Consider x ∈ S and y ∈ I1 ∩ I2. Because I1 and I2 are near left ideals, we have xy ∈ Nr(B)∗I1, xy ∈ Nr(B)∗I2, i.e. xy ∈ Nr(B)∗I1 ∩ Nr(B)∗I2. Assuming Nr(B)∗I1 ∩ Nr(B)∗I2 = Nr(B)∗ (I1 ∩ I2), we have xy ∈ Nr(B)∗ (I1 ∩ I2). Thus I1 ∩ I2 is a near left ideal of S. The other cases can be seen in a similar way. N. Bağırmaz / Eur. J. Pure Appl. Math, 11 (2) (2018), 505-516 512 Definition 4. Let (O, F,∼Br , Nr, vNr) be a nearness approximation space and (·) be a binary operation defined on O. A near subsemigroup I of near semigroup S is called a near bi-ideal of S if ISI ⊆ Nr(B)∗I. Proposition 5. Let (O, F,∼Br , Nr, vNr) be a nearness approximation space and (·) be a binary operation defined on O. Let S ⊆ O be a semigroup. If I is a bi- ideal of S, then I is a near bi- ideal of near semigroup S. Proof. Let I be a bi-ideal of S, i.e., ISI ⊆ I. Since I ⊆ S, by Proposition 1 (5), we know that Nr(B)∗I ⊆ Nr(B)∗S. Then, by Proposition 1 (1), we have that I ⊆ Nr(B)∗I. Hence, I is a near bi- ideal of near semigroup S. Lemma 2. Let (O, F,∼Br , Nr, vNr) be a nearness approximation space and (·) be a binary operation defined on O. Let X ⊆ O. Then Nr(B)∗ (Nr(B)∗X) = Nr(B)∗ (X) . Proof. Since Nr(B)∗ (X) ⊆ Nr(B)∗ (X), then, by Proposition 1 (1) Nr(B)∗ (X) ⊆ Nr(B)∗ (Nr(B)∗X) . Conversely, let [x]Br ∈ Nr(B)∗ (Nr(B)∗X) . Then [x]Br ∩Nr(B)∗X 6= ∅. Thus [x]Br ∈ Nr(B)∗X. Hence Nr(B)∗ (Nr(B)∗X) = Nr(B)∗ (X) . Proposition 6. Let (O, F,∼Br , Nr, vNr) be a nearness approximation space and (·) be a binary operation defined on O and S ⊆ U be a near semigroup. If I is a near right ideal of S and J is a near left ideal of S, then Nr(B)∗ (IJ) ⊆ Nr(B)∗I ∩Nr(B)∗J. Proof. Let I be a near right ideal of S and J be a near left ideal of S, then IJ ⊆ IS ⊆ Nr(B)∗I and IJ ⊆ SJ ⊆ Nr(B)∗J. Thus IJ ⊆ Nr(B)∗I ∩Nr(B)∗J. Thus, it follows from Proposition 1.(5) and (6) that Nr(B)∗ (IJ) ⊆ Nr(B)∗ (Nr(B)∗I ∩Nr(B)∗J) ⊆ Nr(B)∗ (Nr(B)∗I) ∩Nr(B)∗ (Nr(B)∗J) . Then, by Lemma 2, we have that Nr(B)∗ (Nr(B)∗I) = Nr(B)∗ (I) and Nr(B)∗ (Nr(B)∗J) = Nr(B)∗ (J) . Hence Nr(B)∗ (IJ) ⊆ Nr(B)∗ (I) ∩Nr(B)∗ (J) . 4. Homomorphisms of near semigroups Let (O1, F1,∼Br1 , Nr1 , vNr1 ), (O2, F2,∼Br2 , Nr2 , vNr2 ) be two nearness approximation spaces, and (·) , (◦) be binary operations over universes O1 and O2, respectively. Definition 5. Let S1 ⊂ O1, S2 ⊂ O2 be near semigroups. If there exists a surjection φ : Nr1(B)∗ (S1)→ Nr2(B)∗ (S2) such that φ(x ·y) = φ(x)◦φ(y) for all x, y ∈ Nr1(B)∗ (S1) then ϕ is called a near homomorphism and S1, S2 are called near homomorphic semigroups. Proposition 7. Let S1 and S2 be near homomorphic semigroups. If (·) satisfies the commutative law, then (◦) also satisfies it. N. Bağırmaz / Eur. J. Pure Appl. Math, 11 (2) (2018), 505-516 513 Proof. Consider S1, S2, and φ such that φ (x · y) = φ (x) ◦ φ(y) for all x, y ∈ G1. For every φ(x), φ(y) ∈ S2, since φ is surjection, there exist x, y ∈ S1 such that x 7→ φ(x), y 7→ φ(y). Thus φ(x · y) = φ(x) ◦ φ(y), and φ(y · x) = φ(y) ◦ φ(x). Now, assuming x · y = y · x, we obtain φ(x) ◦ φ(y) = φ(y) ◦ φ(x). That means that (◦) satisfies the commutative law. Proposition 8. Let S1 ⊂ O1, S2 ⊂ O2 be near homomorphic semigroups and let Nr2(B)∗ (φ (S1)) = Nr2(B)∗ (S2) .Then φ (S1) is a near semigroup. Proof. (1) ∀x′, y′ ∈ φ(S1), consider x, y ∈ S1 such that x 7→ x′, y 7→ y′. We have φ(x ·y) = φ(x)◦φ(y) ∈ Nr2(B)∗ (S2) = Nr2(B)∗ (φ (S1)), that is x′ ◦y′ ∈ Nr2(B)∗ (φ (S1)). (2) S1 is a near semigroup, so ∀x, y, z ∈ S1, x · (y · z) = (x · y) · z. Hence, φ(x · (y · z)) = φ(x) ◦ φ(y · z) = φ(x) ◦ (φ(y) ◦ φ(z)) φ((x · y) · z) = φ(x · y) ◦ φ(z) = (φ(x) ◦ φ(y)) ◦ φ(z), i.e., (φ(x) ◦ φ(y)) ◦ φ(z) = φ(x) ◦ (φ(y) ◦ φ(z)). Consequently, we can conclude that φ (S1) is a near semigroup. Proposition 9. Let S1 ⊂ O1, S2 ⊂ O2 be near homomorphic semigroups and let H, I be a near subsemigroup and a near left (right, two-sided) ideal of S1, respectively. Then; (a) φ (H) is a near subsemigroup of S2 if φ (Nr1(B)∗ (H)) = Nr2(B)∗ (φ (H)), (b) φ (I) is a near left (right, two-sided) ideal of S2 if φ (Nr1(B)∗ (I)) = Nr2(B)∗ (φ (I)). Proof. (a) Consider a onto mapping φ from Nr1(B)∗ (S1) to Nr2(B)∗ (S2) such that ∀x, y ∈ Nr1(B)∗S1, φ(x · y) = φ(x) ◦ φ(y). For all φ(x), φ(y) ∈ φ(H), by the definition of φ, there exists x, y ∈ H such that x 7→ φ(x) and φ(x) ◦ φ(y) = φ(x · y) ∈ φ(Nr1(B)∗H). Since φ (Nr1(B)∗H) = Nr2(B)∗φ (H), we have φ(x) ◦ φ(y) ∈ Nr2(B)∗φ (H). (b) By (a), it is easy to see that φ (I) is a near subsemigroup of S2 if ϕ (Nr1(B)∗I) = Nr2(B)∗φ (I). Since ∀φ(x) ∈ S2, φ(y) ∈ φ(I) there is φ(x) ◦ φ(y) = φ(x · y) and I is a near left ideal of S1, we have x · y ∈ Nr1(B)∗I. Thus φ(x · y) ∈ φ (Nr1(B)∗I) . Since φ (Nr1(B)∗ (I)) = Nr2(B)∗ (φ (I)), we have φ(x) ◦ φ(y) ∈ Nr2(B)∗φ (I) . Hence ϕ (I) is a rough left ideal of S2. Similarly, we can prove the other statement. Proposition 10. Let S1 ⊂ O1, S2 ⊂ O2 be near homomorphic semigroups and let I be a near bi-ideal of S1. Then, φ (I) is a near bi-ideal of S2 if φ (Nr1(B)∗I) = Nr2(B)∗φ (I) . Proof. By Proposition 9 item (a), φ (I) is a near subsemigroup of S2 if φ (Nr1(B)∗ (I)) = Nr2(B)∗ (φ (I)) . Since ∀φ(x) ∈ S2, φ(y) ∈ φ(I) there is φ(y) ◦φ(x) ◦φ(y) = φ(y ·x · y) and I is a near bi-ideal of S1, we have y ·x · y ∈ Nr1(B)∗ (I). Thus φ(y ·x · y) ∈ φ (Nr1(B)∗I) . Since φ (Nr1(B)∗I) = Nr2(B)∗φ (I), we have φ(y) ◦ φ(x) ◦ φ(y) ∈ Nr2(B)∗φ (I) . Hence φ (I) is a near bi-ideal of S2. Proposition 11. Let S1 ⊂ O1, S2 ⊂ O2 be near homomorphic semigroups and let H2, I2 be a near subsemigroup and a near left (right, two-sided) ideal of S2. Then, (a) φ−1 (H2) = H1 is a near subsemigroup of S1 if φ (Nr1(B)∗ (H1)) = Nr2(B)∗ (φ (H1)), REFERENCES 514 (b) φ−1 (I2) = I1 is a near left (right, two-sided) ideal of S1 if φ (Nr1(B)∗ (I1)) = Nr2(B)∗ (φ (I1)). Proof. (a) Since φ−1 (H2) = H1, we have φ (H1) = H2, and so Nr2(B)∗H2 = Nr2(B)∗ (φ (H1)) = φ (Nr1(B)∗H1) . ∀x, y ∈ H1, we have φ(x), φ(y) ∈ H2. Since H2 is a near subsemigroup, we get φ(x) ◦φ(y) ∈ Nr2(B)∗H2. Thus φ(x · y) ∈ φ (Nr1(B)∗H1) . Thus x · y ∈ Nr1(B)∗H1. (b) By (a), it is easy to see that ϕ−1 (I2) = I1 is a near subsemigroup of S1 if φ (Nr1(B)∗I1) = Nr2(B)∗φ (I1) . Since φ−1 (I2) = I1, we have φ (I1) = I2, and so Nr2(B)∗I2 = Nr2(B)∗φ (I1) = φ (Nr1(B)∗I1) . ∀x ∈ S1 and y ∈ I1, we have φ(x) ∈ φ (S1) and φ(y) ∈ I2. Since I2 is a near left ideal of S2, we have φ(x) ◦ φ(y) ∈ Nr2(B)∗I2. Thus φ(x · y) ∈ φ (Nr1(B)∗I1) . Thus x · y ∈ Nr1(B)∗I1. Therefore, ϕ−1 (I2) = I1 is a near left ideal of S1. Similarly, we can prove the other statement. Proposition 12. Let S1 ⊂ O1, S2 ⊂ O2 be near homomorphic semigroups and let I2 be a near bi-ideal of S2. Then, φ−1 (I2) = I1 is a near bi-ideal of S1 if φ (Nr1(B)∗I1) = Nr2(B)∗φ (I1) . Proof. By Proposition 11 item (a), φ−1 (I2) = I1 is a near subsemigroup of S1 if φ (Nr1(B)∗I1) = Nr2(B)∗φ (I1) . Since ϕ−1 (I2) = I1, we have ϕ (I1) = I2, and so Nr2(B)∗I2 = Nr2(B)∗φ (I1) = φ (Nr1(B)∗I1) . ∀x ∈ S1 and y ∈ I1, we have φ(x) ∈ φ (S1) and φ(y) ∈ I2. Since I2 is a near bi-ideal of S2, we have φ(y) ◦ φ(x) ◦ φ(y) ∈ Nr2(B)∗I2. Thus φ(y · x · y) ∈ φ (Nr1(B)∗I1) . Thus y · x · y ∈ Nr1(B)∗I1. Therefore, φ−1 (I2) = I1 is a near bi-ideal of S1. References [1] N. Bağırmaz and A. F. Özcan, Rough semigroups on approximation spaces, Interna- tional Journal of Algebra, Vol. 9, no. 7, 339-350, 2015. [2] N. Bağırmaz, Rough Prime Ideals in Rough Semigroups, International Mathematical Forum, Vol. 11, no. 8, 369 - 377, 2016. [3] R. Biswas and S. Nanda, Rough groups and rough subgroups, Bull. Polish Acad. Sci. Math., 42 251–254, 1994. [4] Z. Bonikowaski, Algebraic structures of rough sets, in: W.P. 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