Approximation of soft ideals by soft relations in semigroups

Abstract

A soft set handles indeterminate data. By using an equivalence relation, Pawlak introduced the rough set concept for dealing uncertainty to approximate a set. Many authors generalized the concept and used binary relations to approximate a set. A soft set is approximated in this paper by soft binary relations in the context of the aftersets and foresets. Along these lines, we get two sets of soft sets, called the lower approximation and upper approximation with respect to the aftersets and foresets. We applied these concepts on semigroups and approximations of soft subsemigroups, soft left (right) ideals, soft interior ideals and soft bi-ideals of semigroups are studied. Moreover, for the illustration of the concept, some examples are considered.

Keywords

Soft relations Soft set approximation Soft substructures

1 Introduction

Molodtsov [27] displayed soft set theory to overcome the troubles related with different theories implied for to deal with uncertainty. Soft sets have many operations which are exceptionally helpful to manage different kinds of circumstances. In soft set theory, many authors [1, 2, 25, 27, 28] described several operations. Aktas and Çagman [1] started soft groups and demonstrated that fuzzy groups are an uncommon instance of soft groups. Jun connected soft sets to BCK/BCI-algebras [16]. Jun and Park announced uses of soft sets in BCK/BCI-algebras [17]. Soft semirings along with related ideas are characterized keeping in mind a connection to set up an association between soft sets and semirings [3]. There is a quick development of interest for soft set theory and its applications now-a-days.

Another theory, which handles uncertainty in a non-customary way is the rough sets theory, displayed by Pawlak [30]. Since its initiation, it inspired specialists and researchers. Numerous expansions of rough sets in literature have been proposed, for example, covering based rough sets, variable precision rough sets and rough sets in the sense of binary relations [51]. Yet at the same time, the first rough set theory can possibly propose solutions of many issues. Rough set theory decreases the information without losing helpful data and handles basic decision making problems eccentrically.

The rough set theory is an augmentation of the set theory for the investigation of intelligent systems described by inadequate and deficient data [29, 30, 34]. The idea of rough sets is persuaded by useful needs especially in characterization and concept formation with deficient data [35]. It is not the same as and corresponding to different generalizations, such as multisets and fuzzy sets [12, 35]. In this new emerging theory, there has been a rich interest. The successful applications of rough set models have shown their benefits in many problems [25, 33, 54, 55, 56].

Broad research has been completed to compare different theories of uncertainty and the theory of rough sets [15]. The consequences of these examinations improve our comprehension of the theory of rough sets. In this way, some more broad rough set models were presented; see [5–9, 11, 14, 31, 32, 36–39, 44, 45, 50, 51]. In algebraic structures, many scholars have studied roughness. The study of roughness in algebra is studied by Iwinski [15]. Roughness in groups and subgroups is seen by Biswas and Nanda [4]. Roughness in semigroups is studied by Kuroki [22]. Davvaz initiated the rough ideals of rings [10]. Shabir and Irshad proposed an approximation in ordered semigroups along with its properties depending on pseudoorder [48]. The properties of rough sets in hemirings is studied by Ali et al. [3]. Generalized roughness or T-roughness in fuzzy algebraic and algebraic structures has been discussed by Liu in [23], Qurashi and Shabir in [41–43], Mahmood et al. in [26] and Pomykala in [40]. In [57], Zhu defined generalized rough set depending on binary relation. Recently, Shabir et al. showed reduction of parameters by soft relations [49] and Kanwal and Shabir showed an approach to a fuzzy set and corresponding decision making, in [19]. Moreover, semigroup is considered to approximate ideals under different environments as in [18, 20]. And, Riaz et al. showed an appraoch in [46, 47] about fuzzy and soft sets.

A soft binary relation is a parameterized family on a universe of binary relations. It is a generalization of ordinary binary relations on a soft set. In rough set theory, rough approximations just address single binary relation. In any case, rough approximations in light of soft binary relations can deal with different binary relations.

The sequel of the present paper is described as follows. In Section 2, some basic thoughts related to soft relations are given. Section 3, is given to the investigation of approximation by soft relations. We approximate a soft set by using the aftersets and foresets. In this way, we get two sets of soft sets, called the lower approximation and the upper approximation with respect to aftersets and foresets. Section 4, presents that we applied these concepts on semigroups and approximations of soft subsemigroups, soft left (right) ideals, soft interior ideals and soft bi-ideals of semigroups are presented along with examples. Further, the conclusion is displayed to explain this work in Section 5.

2 Basic definitions and preliminaries

This section contains a few ideas and results which will be helpful in the following.

δ is an equivalence relation (ER) on U and U is a non-empty finite set, then if (U, δ) is an approximation space [27]. X might be appeared as union of the equivalence classes of U or may not if X is a subset of U. X is definable if X can be appeared as union of some equivalence classes of U. In the other case, it isn’t definable. On the off chance that X isn’t definable, then it very well may be approximated by two definable subsets called the lower and the upper approximations of X as $\begin{matrix} \underline{δ} (X) & = & \cup {[t]_{δ} such that [t]_{δ} \subseteq X} \\ \bar{δ} (X) & = & \cup {[t]_{δ} such that [t]_{δ} \cap X is non - empty} . \end{matrix}$

The pair $(\underline{δ} (X), \bar{δ} (X))$ is a rough set and the set $\bar{δ} (X) - \underline{δ} (X)$ is called the boundary region. Moreover, X is definable if $\underline{δ} (X) = \bar{δ} (X) .$

Proposition 1. [27] If X and Y are subsets of U and δ is an ER on the set U, then the following hold:

(i) $\underline{δ} (X) \subseteq X \subseteq \bar{δ} (X)$

(ii) X ⊆ Y implies $\underline{δ} (X) \subseteq \underline{δ} (Y)$

(iii) X ⊆ Y implies $\bar{δ} (X) \subseteq \bar{δ} (Y)$

(iv) $\underline{δ} (X \cap Y) = \underline{δ} (X) \cap \underline{δ} (Y)$

(v) $\bar{δ} (X \cap Y) \subseteq \bar{δ} (X) \cap \bar{δ} (Y)$

(vi) $\underline{δ} (X \cup Y) \supseteq \underline{δ} (X) \cup \underline{δ} (Y)$

(vii) $\bar{δ} (X \cup Y) = \bar{δ} (X) \cup \bar{δ} (Y)$ .

Now, we give some related definitions of semigroup thoery.

A semigroup is a set S ≠ φ having an associative binary operation "·". We shall denote the product of two elements a, b ∈ S by ab instead of a · b. The product XY in the semigroup S for two subsets X and Y can be defined as $XY = {xy : x \in X, y \in Y} .$

X is said to be a subsemigroup of S if xy ∈ X (for all x, y ∈ X) , for any non-empty subset X of S. A non-empty subset X of a semigroup S is a left (right) ideal of a semigroup S satisfying the condition X ⊇ SX (X ⊇ XS). A 2-sided ideal is a non-empty subset of S which is both a left and a righr ideal of S. A non-empty subset X of a semigroup S is called an interior ideal of S if it satisfies X ⊇ SXS. A subsemigroup X of a semigroup S is called a bi-ideal of S if it satisfies X ⊇ XSX. Throughout this paper, we shall denote a subsemigroup, left ideal, right ideal, bi-ideal and interior ideal by SS, LI, RI, BI and II, respectively.

Definition 1. [28] A pair (G, A) is called a soft set over U, where A is a subset of E (the set of parameters) and G is defined by G : A → P (U).

Definition 2. [2] (1) For two soft sets (L, B) and (G, A) over a common universe U, we say that (G, A) is a soft subset of (L, B) if (i) B ⊇ A and (ii) L (e) ⊇ G (e) for all e ∈ A.

(2) Two soft sets (L, B) and (G, A) over a common universe U are said to be soft equal if (G, A) is a soft subset of (L, B) and (L, B) is a soft subset of (G, A).

(3) The product of two soft sets (G, A) and (L, A) over the common universe U is the soft set (GL, A) such that GL (e) = G (e) L (e) for all e ∈ A .

(4) The intersection of two soft sets (G, A) and (L, A) over the common universe U is the soft set (J, A) such that J (e) = L (e) ∩ G (e) for all e ∈ A .

(5) The union of two soft sets (G, A) and (L, A) over the common universe U is the soft set (J, A) such that J (e) = L (e) ∪ G (e) for all e ∈ A .

Definition 3. Let (G, A) be a soft set over S and S be a semigroup. Then

(1) (G, A) is called a soft SS over S if G (e) is a SS of S for all e ∈ A with G (e) ≠ φ.

(2) A soft set (G, A) over a semigroup S is called a soft ideal (resp. soft LI, soft RI) over S, if G (e) is an ideal (resp. LI, RI) of S for all e ∈ A with G (e) ≠ φ.

(3) A soft set (G, A) over a semigroup S is defined to be a soft BI over S, if G (e) is a BI of S for all e ∈ A with G (e) ≠ φ.

(4) A soft set (G, A) over a semigroup S is said to be a soft II over S, if G (e) is an II of S for all e ∈ A with G (e) ≠ φ.

3 Approximation by soft relations

In this section, we used soft relations from S₁ to S₂ (S₁ and S₂ are semigroups) to approximate a soft set in two ways. We used the afterset and foreset to approximate a soft set over S₂ and a soft set over S₁, respectively. In the way, we get two sets of soft sets corresponding to each soft set, called the lower approximation and the upper approximation with respect to the aftersets and with respect to the foresets, respectively.

[13] If (δ, A) is a soft set over S₁ × S₂, that is δ : A → P (S₁ × S₂), then (δ, A) is said to be a soft binary relation (SBR) from S₁ to S₂ (S₁ and S₂ are semigroups), where A is a subset of E (parametersset) . An SBR δ from S₁ to S₂ (S₁ and S₂ are semigroups) is called soft compatible if (a, b) , (c, d) ∈ δ (e) ⇒ (ac, bd) ∈ δ (e) for all a, c ∈ S₁ and b, d ∈ S₂. A soft compatible relation (δ, A) from S₁ to S₂ (S₁ and S₂ are semigroups) is called soft complete relation with respect to the aftersets if (ab) δ (e) = aδ (e) . bδ (e) for all a, b ∈ S₁ and e ∈ A and is called soft complete relation with respect to the foresets if δ (e) (ab) = δ (e) a . δ (e) b for all a, b ∈ S₂ and e ∈ A. In general, neither soft complete relation with respect to the aftersets implies soft complete relation with respect to the foresets nor soft complete relation with respect to the foresets implies soft complete relation with respect to the aftersets.

Definition 4. [21] Let (δ, A) be an SBR from S₁ to S₂ (S₁ and S₂ are semigroups), that is δ : A → P (S₁ × S₂), where A is a subset of E (parametersset). For a soft set (G, A) over S₂, the lower approximation $({\underline{δ}}^{G}, A)$ and the upper approximation $({\bar{δ}}^{G}, A)$ of (G, A) with respect to the aftersets are basically the two soft sets over S₁, defined as follows: $\begin{matrix} {\underline{δ}}^{G} (e) & = & {s_{1} \in S_{1} : φ \neq s_{1} δ (e) \subseteq G (e)} and \\ {\bar{δ}}^{G} (e) & = & {s_{1} \in S_{1} : s_{1} δ (e) \cap G (e) \neq φ} \\ for all e \in A . \end{matrix}$ And the lower approximation $(L \underline{δ}, A)$ and the upper approximation $(L \bar{δ}, A)$ of a soft set (L, A) over S₁ with respect to the foresets are actually the two soft sets over S₂, as defined below: $\begin{array}{l} L \underline{δ} (e) = {s_{2} \in S_{2} : φ \neq δ (e) s_{2} \subseteq L (e)} \\ L \underline{δ} (e) = {s_{2} \in S_{2} : δ (e) s_{2} \cap L (e) \neq φ} \end{array}$ for all e ∈ A, where s₁δ (e) ={ s₂ ∈ S₂ : (s₁, s₂) ∈ δ (e) } and is called the after set of s₁ and δ (e) s₂ ={ s₁ ∈ S₁ : (s₁, s₂) ∈ δ (e) } and is called the foreset of s₂. Moreover, ${\underline{δ}}^{G} : A \to P (S_{1}),$ ${\bar{δ}}^{G} : A \to P (S_{1})$ , $L \underline{δ} : A \to P (S_{2})$ and $L \bar{δ} : A \to P (S_{2})$ for each soft set (G, A) over S₂ and (L, A) over S₁, respectively.

Theorem 1. Let (δ, A) and (β, A) be two SBRs from a non-empty set S₁ to a non-empty set S₂ and let (G₁, A) and (G₂, A) be two soft sets over S₂. Then the following assertions hold:

(1) (G₁, A) ⊆ (G₂, A) implies $({\underline{δ}}^{G_{1}}, A) \subseteq ({\underline{δ}}^{G_{2}}, A);$

(2) (G₁, A) ⊆ (G₂, A) implies $({\bar{δ}}^{G_{1}}, A) \subseteq ({\bar{δ}}^{G_{2}}, A);$

(3) $({\underline{δ}}^{G_{1}}, A) \cap ({\underline{δ}}^{G_{2}}, A) = ({\underline{δ}}^{G_{1} \cap G_{2}}, A);$

(4) $({\bar{δ}}^{G_{1}}, A) \cap ({\bar{δ}}^{G_{2}}, A) \supseteq ({\bar{δ}}^{G_{1} \cap G_{2}}, A);$

(5) $({\underline{δ}}^{G_{1}}, A) \cup ({\underline{δ}}^{G_{2}}, A) \subseteq ({\underline{δ}}^{G_{1} \cup G_{2}}, A);$

(6) $({\bar{δ}}^{G_{1}}, A) \cup ({\bar{δ}}^{G_{2}}, A) = ({\bar{δ}}^{G_{1} \cup G_{2}}, A);$

(7) (δ, A) ⊆ (β, A) implies $({\underline{δ}}^{G_{1}}, A) \supseteq ({\underline{β}}^{G_{1}}, A);$

(8) (δ, A) ⊆ (β, A) implies $({\bar{δ}}^{G_{1}}, A) \subseteq ({\bar{β}}^{G_{1}}, A)$ .

Theorem 2. Let (δ, A) and (β, A) be two SBRs from a non-empty set S₁ to a non-empty set S₂ and let (L₁, A) and (L₂, A) be two soft sets over S₁. Then the assertions following hold:

(1) (L₁, A)⊆ (L₂, A) ⇒ $(L_{1} \underline{δ}, A) \subseteq$ $(L_{2} \underline{δ}, A);$

(2) (L₁, A)⊆ (L₂, A) ⇒ $(L_{1} \bar{δ}, A) \subseteq$ $(L_{2} \bar{δ}, A);$

(3) $(L_{1} \underline{δ}, A) \cap (L_{2} \underline{δ}, A) = (L_{1} \cap L_{2} \underline{δ}, A);$

(4) $(L_{1} \bar{δ}, A) \cap (L_{2} \bar{δ}, A) \supseteq ({\bar{δ}}^{L_{1} \cap L_{2}}, A);$

(5) $(L_{1} \underline{δ}, A) \cup (L_{2} \underline{δ}, A) \subseteq ({\underline{δ}}^{L_{1} \cup L_{2}}, A);$

(6) $(L_{1} \bar{δ}, A) \cup (L_{2} \bar{δ}, A) = ({\bar{δ}}^{L_{1} \cup L_{2}}, A);$

(7) (δ, A) ⊆ (β, A) implies $(L_{1} \underline{δ}, A) \supseteq (L_{1} \underline{β}, A);$

(8) (δ, A) ⊆ (β, A) implies $(L_{1} \bar{δ}, A) \subseteq (L_{1} \bar{β}, A)$ .

Theorem 3. Let (δ, A) and (β, A) be two SBRs from a non-empty set S₁ to a non-empty set S₂. If (G, A) is a soft set over S₂, then

(1) $({(\bar{δ \cap β})}^{G}, A) \subseteq ({\bar{δ}}^{G}, A) \cap ({\bar{β}}^{G}, A)$ .

(2) $({(\underline{δ \cap β})}^{G}, A) \supseteq ({\underline{δ}}^{G}, A) \cup ({\underline{β}}^{G}, A)$ .

Proof. The results follows from parts (7) and (8) of Theorem 3. □

The converse of above results is not valid as shown by example 1.

Example 1. Let U₁ = {a, b, c, d, e} and U₂ ={ 1, 2, 3, 4, 5 } and A = {e₁, e₂}. Define δ : A → P (U₁ × U₂) and β : A → P (U₁ × U₂) by $δ (e_{1}) = {\begin{matrix} (a, 1), (b, 1), (b, 2), (b, 5), (c, 3), (c, 5), \\ (d, 1), (d, 3), (d, 4), (d, 5), (e, 1), (e, 5) \end{matrix}}$ $δ (e_{2}) = {(a, 1), (b, 2), (c, 3), (d, 4), (e, 5)},$ $β (e_{1}) = {(a, 1), (b, 1), (b, 2), (c, 3), (d, 4), (e, 2), (e, 5)} and$ $β (e_{2}) = {(a, 1), (b, 2), (b, 3), (c, 3), (d, 4), (e, 5)} .$ Therefore, $(δ \cap β) (e_{1}) = {(a, 1), (b, 1), (b, 2), (c, 3), (d, 4), (e, 5)}$ and $(δ \cap β) (e_{2}) = {(a, 1), (b, 2), (c, 3), (d, 4), (e, 5)} .$ Now, $\begin{matrix} a δ (e_{1}) & = & {1}, b δ (e_{1}) = {1, 2, 5}, c δ (e_{1}) = {3, 5}, \\ d δ (e_{1}) & = & {1, 3, 4, 5} and e δ (e_{1}) = {1, 5} \end{matrix}$ and $\begin{matrix} a β (e_{1}) & = & {1}, b β (e_{1}) = {1, 2}, c β (e_{1}) = {3}, \\ d β (e_{1}) & = & {4} and e β (e_{1}) = {2, 5} . \end{matrix}$ Also, $\begin{matrix} a (δ \cap β) (e_{1}) & = & {1}, b (δ \cap β) (e_{1}) = {1, 2}, \\ c (δ \cap β) (e_{1}) = {3}, \\ d (δ \cap β) (e_{1}) & = & {4} and e (δ \cap β) (e_{1}) = {5} . \end{matrix}$ $\begin{matrix} Define (G_{1}, A), a soft set over U_{2} by G_{1} (e_{1}) \\ = {1, 2} and G_{1} (e_{2}) = {2, 4, 5} . \end{matrix}$

Then $\begin{matrix} {\bar{δ}}^{G_{1}} (e_{1}) = {a, b, d, e}, {\bar{β}}^{G_{1}} (e_{1}) = {a, b, e} and \\ {(\bar{δ \cap β})}^{G_{1}} (e_{1}) = {a, b} . \end{matrix}$ This shows that ${\bar{δ}}^{G_{1}} (e_{1}) \cap {\bar{β}}^{G_{1}} (e_{1}) = {a, b, e} \neq {a, b} = {(\bar{δ \cap β})}^{G_{1}} (e_{1}) .$ Now, $\begin{matrix} Define (G_{2}, A), a soft set over U_{2} by G_{2} (e_{1}) = {5} \\ and G_{2} (e_{2}) = {1, 3, 4} . \end{matrix}$ Then ${\underline{δ}}^{G_{2}} (e_{1}) = φ,$ ${\underline{β}}^{G_{2}} (e_{1}) = φ$ and ${(\underline{δ \cap β})}^{G_{2}}$ (e₁) = {e}. This shows that ${\underline{δ}}^{G_{2}} (e_{1}) \cup {\underline{β}}^{G_{2}} (e_{1}) = φ \neq {e} = {(\underline{δ \cap β})}^{G_{2}} (e_{1}) .$

Theorem 4. Let (δ, A) and (β, A) be two soft binary relations from a non-empty set S₁ to a non-empty set S₂. If (L, A) is a soft set over S₁, then

(1) $(L (\bar{δ \cap β}), A) \subseteq (L \bar{δ}, A) \cap (L \bar{β}, A)$ .

(2) $(L (\underline{δ \cap β}), A) \supseteq (L \underline{δ}, A) \cup (\underline{L β}, A)$ .

Proof. The results follows from parts (7) and (8) of Theorem 3(invert). □

The converse of above results is not valid as shown by example 2.

Example 2. Let U₁ = {a, b, c, d, e} and U₂ ={ 1, 2, 3, 4, 5 } and A = {e₁, e₂}. Define δ : A → P (U₁ × U₂) and β : A → P (U₁ × U₂) by $\begin{matrix} δ (e_{1}) = \\ {\begin{matrix} (a, 1), (b, 1), (b, 2), (b, 5), \\ (c, 3), (c, 5), (d, 1), (d, 3), (d, 4), (d, 5), (e, 1), (e, 5)} \end{matrix}}, \end{matrix}$ $δ (e_{2}) = {(a, 1), (b, 2), (c, 3), (d, 4), (e, 5)},$ $β (e_{1}) = {(a, 1), (a, 5), (b, 1), (b, 2), (c, 3), (d, 4), (e, 5)} and$ $β (e_{2}) = {(a, 1), (b, 2), (b, 3), (c, 3), (d, 4), (e, 5)} .$ Therefore, $(δ \cap β) (e_{1}) = {(a, 1), (b, 1), (b, 2), (c, 3), (d, 4), (e, 5)}$ and $(δ \cap β) (e_{2}) = {(a, 1), (b, 2), (c, 3), (d, 4), (e, 5)} .$ Now, $\begin{matrix} δ (e_{1}) 1 & = & {a, b, d, e}, δ (e_{1}) 2 = {b}, δ (e_{1}) 3 = {c, d}, \\ δ (e_{1}) 4 & = & {d} and δ (e_{1}) 5 = {b, c, d, e}, \end{matrix}$ and $\begin{matrix} β (e_{1}) 1 & = & {a, b}, β (e_{1}) 2 = {b}, 3 β (e_{1}) = {c}, \\ β (e_{1}) 4 & = & {d} and β (e_{1}) 5 = {a, e} . \end{matrix}$ Also, $\begin{matrix} (δ \cap β) (e_{1}) 1 & = & {a, b}, (δ \cap β) (e_{1}) 2 = {b}, \\ (δ \cap β) (e_{1}) 3 = {c}, \\ (δ \cap β) (e_{1}) 4 & = & {d} and (δ \cap β) (e_{1}) 5 = {e} . \end{matrix}$ $\begin{matrix} Define (L_{1}, A), a soft set over U_{1} by L_{1} (e_{1}) \\ = {a, b} and L_{1} (e_{2}) = {c, d} . \end{matrix}$

Then $L_{1} \bar{δ} (e_{1}) = {1, 2, 5},$ $L_{1} \bar{β} (e_{1}) = {1, 2, 5}$ and $L_{1} (\bar{δ \cap β}) (e_{1}) = {1, 2}$ . This shows that $L_{1} \bar{δ} (e_{1}) \cap^{L_{1}} \bar{β} (e_{1}) = {1, 2, 5} \neq {1, 2} =^{L_{1}} (\bar{δ \cap β}) (e_{1}) .$ Now, $\begin{matrix} Define (L_{2}, A), a soft set over U_{1} by L_{2} (e_{1}) = {e} \\ and L_{2} (e_{2}) = {a, b} . \end{matrix}$ Then $L_{2} \underline{δ} (e_{1}) = φ,$ $L_{2} \underline{β} (e_{1}) = φ$ and $L_{2} (\underline{δ \cap β})$ (e₁) = {5}. This shows that $L_{2} \underline{δ} (e_{1}) \cup^{L_{2}} \underline{β} (e_{1}) = φ \neq {5} = L_{2} (\underline{δ \cap β}) (e_{1}) .$

Theorem 5. Let (δ, A) be a soft compatible relation with respect to the aftersets from S₁ to S₂ (S₁ and S₂ are semigroups). For any two soft sets (G₁, A) and (G₂, A) over S₂, ${\bar{δ}}^{G_{1}} (e) . {\bar{δ}}^{G_{2}} (e) \subseteq {\bar{δ}}^{G_{1} G_{2}} (e)$ for all e ∈ A .

Proof. Let $u \in {\bar{δ}}^{G_{1}} (e) . {\bar{δ}}^{G_{2}} (e)$ . Then u = g₁g₂ for some $g_{1} \in {\bar{δ}}^{G_{1}} (e)$ and $g_{2} \in {\bar{δ}}^{G_{2}} (e)$ . This implies that g₁δ (e) ∩ G₁ (e) and g₂δ (e) ∩ G₂ (e) are non-empty, so their exist elements a, b ∈ S₂ such that a ∈ g₁δ (e) ∩ G₁ (e) and b ∈ g₂δ (e) ∩ G₂ (e). Thus a ∈ g₁δ (e), b ∈ g₂δ (e), a ∈ G₁ (e) and b ∈ G₂ (e). Now, (g₁, a) ∈ δ (e) and (g₂, b) ∈ δ (e) implies that (g₁g₂, ab) ∈ δ (e), that is ab ∈ g₁g₂δ (e). Also, ab ∈ G₁ (e) G₂ (e), therefore, ab ∈ G₁ (e) G₂ (e) ∩ g₁g₂δ (e). Hence, $u = g_{1} g_{2} \in {\bar{δ}}^{G_{1} G_{2}} (e)$ . □

The proof of Theorem 6 is similar to the proof of Theorem 5.

Theorem 6. Let (δ, A) be a soft compatible relation with respect to the foresets from S₁ to S₂ (S₁ and S₂ are semigroups). For any two soft sets (L₁, A) and (L₂, A) over S₁, $L_{1} \bar{δ} (e) .^{L_{2}} \bar{δ} (e) \subseteq$ $L_{1} L_{2} \bar{δ} (e)$ for all e ∈ A .

Theorem 7. Let (δ,A) be a soft complete relation with respect to the aftersets from S₁ to S₂ (S₁ and S₂ are semigroups). For any two soft sets (G₁,A) and (G₂,A) over S₂, ${\underline{δ}}^{G_{1}} (e) . {\underline{δ}}^{G_{2}} (e) \subseteq {\underline{δ}}^{G_{1} G_{2}} (e)$ for all e ∈ A.

Proof. First we consider that ${\underline{δ}}^{G_{1}} (e)$ and ${\underline{δ}}^{G_{2}} (e)$ non-empty and $u \in {\underline{δ}}^{G_{1}} (e) . {\underline{δ}}^{G_{2}} (e)$ . Then, u = g₁ g₂ for some $g_{1} \in {\underline{δ}}^{G_{1}} (e)$ and $g_{2} \in {\underline{δ}}^{G_{2}} (e)$ . This implies that G₁(e) ⊇ g₁ δ (e) ≠ ϕ and G₂(e) ⊇ g₂ δ (e) ≠ ϕ. As g₁ g₂ δ (e) = g₁ δ (e). g₂ δ (e) ⊆ G₁ (e) G₂ (e), we have $u = g_{1} g_{2} \in {\underline{δ}}^{G_{1} G_{2}}$ . Hence ${\underline{δ}}^{G_{1} G_{2}} (e) \supseteq {\underline{δ}}^{G_{1}} (e) . {\underline{δ}}^{G_{2}} (e)$ . Now, if one of ${\underline{δ}}^{G_{1}} (e)$ and ${\underline{δ}}^{G_{2}} (e)$ is empty then $φ = {\underline{δ}}^{G_{1}} (e) . {\underline{δ}}^{G_{2}} (e) \subseteq {\underline{δ}}^{G_{1} G_{2}} (e) .$ □

The proof of Theorem 8 is similar to the proof of Theorem 7 .

Theorem 8. Let (δ, A) be a soft complete relation with respect to the foresets from S₁ to S₂ (S₁ and S₂ are semigroups). For any two soft sets (L₁, A) and (L₂, A) over S₁, $L_{1} \underline{δ} (e) . L_{2} \underline{δ} (e) \subseteq$ $L_{1} L_{2} \underline{δ} (e)$ for all e ∈ A.

Example 3. For two semigroups S₁ = {1, 2, 3} and S₂ ={ a, b, c } with the multiplication tables as follows:

•	1	2	3
1	1	2	3
2	1	2	3
3	1	2	3

•	a	b	c
a	a	a	c
b	a	b	c
c	a	c	c

and A = {e₁, e₂}. Define δ : A → P (S₁ × S₂) by $δ (e_{1}) = {(1, a), (2, a), (2, b), (2, c), (3, c)}$ and $δ (e_{2}) = {(1, a), (1, b), (1, c), (2, a), (2, b), (3, c)} .$

Then (δ, A) is a soft compatible relation from the semigroup S₁ to the semigroup S₂ with respect to the aftersets. $\begin{matrix} 1 δ (e_{1}) & = & {a}, 2 δ (e_{1}) = {a, b, c} and 3 δ (e_{1}) = {c} \\ 1 δ (e_{2}) & = & {a, b, c}, 2 δ (e_{2}) = {a, b}, 3 δ (e_{2}) = {c} . \end{matrix}$ $\begin{matrix} Define (G_{1}, A) and (G_{2}, A), the two soft sets over S_{2} by \\ G_{1} (e_{1}) & = & {a}, G_{1} (e_{2}) = {a, b} and G_{2} (e_{1}) = {b, c}, G_{2} (e_{2}) = {a, c} . \end{matrix}$

Then ${\bar{δ}}^{G_{1}} (e_{1}) = {1, 2}$ and ${\bar{δ}}^{G_{2}} (e_{1}) = {2, 3}$ . Now, G₁ (e₁) G₂ (e₁) = {a, c} and ${\bar{δ}}^{G_{1} G_{2}} (e_{1}) = {1, 2, 3} \neq {2, 3} = {1, 2} {2, 3} = {\bar{δ}}^{G_{1}} (e_{1}) {\bar{δ}}^{G_{2}} (e_{1}) .$

Example 4. Consider the semigroups and soft relations of Example 3, $\begin{matrix} δ (e_{1}) a & = & {1, 2}, δ (e_{1}) b = {2} and δ (e_{1}) c = {2, 3} \\ δ (e_{2}) a & = & {1, 2}, δ (e_{2}) b = {1, 2}, δ (e_{2}) c = {1, 3} . \end{matrix}$ $\begin{matrix} Define (L_{1}, A) and (L_{2}, A), the two soft sets over S_{1} by \\ L_{1} (e_{1}) & = & {1, 2}, L_{1} (e_{2}) = {3} and L_{2} (e_{1}) = {2, 3}, L_{2} (e_{2}) = {1} . \end{matrix}$

Then, $L_{1} \bar{δ} (e_{2}) = {c}$ and $L_{2} \bar{δ} (e_{2}) = {a, b, c}$ . Now, L₁ (e₂) L₂ (e₂) = {1} and $L_{1} L_{2} \bar{δ} (e_{2}) = {a, b, c} ⊈ {a, c} = {c} {a, b, c} =^{L_{1}} \bar{δ} (e_{2}) .^{L_{2}} \bar{δ} (e_{2}) .$

Example 5. Let S₁ = {a, b, c, d} and S₂ ={ 1, 2, 3, 4 } be two semigroups with the multiplication tables as follows:

•	a	b	c	d
a	a	a	a	d
b	a	b	a	d
c	a	a	c	d
d	d	d	d	d

•	1	2	3	4
1	1	2	3	4
2	2	2	2	2
3	3	3	3	3
4	4	3	2	1

and A = {e₁, e₂}. Define δ : A → P (S₁ × S₂) by $δ (e_{1}) = {(a, 2), (a, 3), (b, 2), (b, 3), (c, 2), (c, 3), (d, 2), (d, 3)}$ and $δ (e_{2}) = {(a, 2), (b, 2), (c, 2), (d, 2)} .$ Then (δ, A) is a soft complete relation from the semigroup S₁ to the semigroup S₂ with respect to the aftersets. $a δ (e_{1}) = {2, 3}, b δ (e_{1}) = {2, 3}, c δ (e_{1}) = {2, 3} and d δ (e_{1}) = {2, 3} .$ Also, $a δ (e_{2}) = {2}, b δ (e_{2}) = {2}, c δ (e_{2}) = {2} and d δ (e_{2}) = {2} .$ $\begin{matrix} Define (G_{1}, A) and (G_{2}, A), the two soft sets over S_{2} by \\ G_{1} (e_{1}) & = & {4}, G_{1} (e_{2}) = {2, 3} and G_{2} (e_{1}) = {1, 2, 3}, G_{2} (e_{2}) = {2, 4} . \end{matrix}$

Then ${\underline{δ}}^{G_{1} (e)} (e_{1}) = φ$ and ${\underline{δ}}^{G_{2}} (e_{1}) = {a, b, c, d}$ . Now, G₁ (e₁) G₂ (e₁) = {2, 3, 4} and ${\underline{δ}}^{G_{1} G_{2}} (e_{1}) = {a, b, c, d} ⊈ φ = φ {a, b, c, d} = {\underline{δ}}^{G_{1}} (e_{1}) {\underline{δ}}^{G_{2}} (e_{1}) .$

Example 6. Consider the semigroup of Example 3

and A = {e₁, e₂}. Define δ : A → P (S₁ × S₂) by $\begin{matrix} δ (e_{1}) & = & {(a, 1), (a, 2), (a, 3), (a, 4), (d, 1), \\ (d, 2), (d, 3), (d, 4)} and \\ δ (e_{2}) & = & {(d, 1), (d, 2), (d, 3), (d, 4)} . \end{matrix}$ Then (δ, A) is a soft complete relation from the semigroup S₁ to the semigroup S₂ with respect to the foresets. $\begin{matrix} δ (e_{1}) 1 & = & {a, d}, δ (e_{1}) 2 = {a, d}, δ (e_{1}) 3 = {a, d} and δ (e_{1}) 4 = {a, d} . \\ Also, δ (e_{2}) 1 & = & {d}, δ (e_{2}) 2 = {d}, δ (e_{2}) 3 = {d} and δ (e_{2}) 4 = {d} . \end{matrix}$ $\begin{matrix} Define (L_{1}, A) and (L_{2}, A), the two soft sets over S_{1} by \\ L_{1} (e_{1}) & = & {a, c, d}, L_{1} (e_{2}) = {a} and L_{2} (e_{1}) = {b}, L_{2} (e_{2}) = {a, b} . \end{matrix}$

Then $L_{1} \underline{δ} (e_{1}) = {1, 2, 3, 4}$ and $L_{2} \underline{δ} (e_{1}) = φ$ . Now, L₁ (e₁) L₂ (e₁) = {a, d} and $L_{1} L_{2} \underline{δ} (e_{1}) = {1, 2, 3, 4} ⊈ φ = {1, 2, 3, 4} φ =^{L_{1}} \underline{δ} (e_{1}) .^{L_{2}} \underline{δ} (e_{1}) .$

4 Approximation of soft ideals in semigroups

In the following, we used soft compatible relation (δ, A) and δ (e) ≠ φ for all e ∈ A to approximate a SS (LI, RI, BI, II) of a semigroup with respect to the aftersets (resp. with respect to the foresets). We prove that upper approximation of a soft SS (LI, RI, BI, II) of a semigroup is a soft SS (LI, RI, BI, II) of the semigroup and give examples that the converse is not true. We also prove that lower approximation of a soft SS (LI, RI, BI, II) of a semigroup by a soft complete relation (with respect to the aftersets and with respect to the foresets) is a soft SS (LI, RI, BI, II) of the semigroup and give examples that the converse is not true. Throughout the remaining paper, (δ, A) is a soft relation from S₁ to S₂ (S₁ and S₂ are semigroups) and uδ (e) ≠ φ for all u ∈ S₁ and e ∈ A and δ (e) w ≠ φ for all w ∈ S₂ and e ∈ A .

Define a soft set (δ, A) over (S, A) by S (e) = S for all e ∈ A . If (δ, A) is a soft relation from S₁ to S₂ and (S₁, A) , (S₂, A) are soft sets over S₁ and S₂ respectively, as defined above, then $\begin{matrix} {\bar{δ}}^{S_{2}} (e) & = & {u \in S_{1} : u δ (e) \cap S_{2} \neq φ} = S_{1} \\ {\underline{δ}}^{S_{2}} (e) & = & {u \in S_{1} : u δ (e) \subseteq S_{2}} = S_{1} \\ {\bar{δ}}^{S_{1}} (e) & = & {w \in S_{2} : δ (e) w \cap S_{1} \neq φ} = S_{2} \\ {\underline{δ}}^{S_{1}} (e) & = & {w \in S_{2} : δ (e) w \subseteq S_{1}} = S_{2} . \end{matrix}$

Definition 5. Let (δ, A) be an SBR from S₁ to S₂ (S₁ and S₂ are semigroups). If the upper approximation $({\bar{δ}}^{G}, A)$ is a soft SS of S₁ for any soft set (G, A) over S₂, then (G, A) is said to be generalized upper soft SS of S₁ with respect to the aftersets. The soft set (G, A) is called generalized upper soft L (R, 2-sided)I of S₁ with respect to the aftersets if $({\bar{δ}}^{G}, A)$ is a soft L (R, 2-sided)I of S₁.

Definition 6. Let (δ, A) be an SBR from S₁ to S₂ (S₁ and S₂ are semigroups). If the upper approximation $(L \bar{δ}, A)$ is a soft SS of S₂ for any soft set (L, A) over S₁, then (L, A) is said to be generalized upper soft SS of S₂ with respect to the foresets. The soft set (L, A) is called generalized upper soft L (R, 2-sided)I of S₂ with respect to the foresets if $(L \bar{δ}, A)$ is a soft L (R, 2-sided)I of S₂.

Theorem 9. Let (δ, A) be a soft compatible relation from S₁ to S₂ (S₁ and S₂ are semigroups). Then

(1) If (G, A) is a soft SS of S₂, then (G, A) is a generalized upper soft SS of S₁ with respect to the aftersets.

(2) If (L, A) is a soft SS of S₁, then (L, A) is a generalized upper soft SS of S₂ with respect to the foresets.

(3) If (G, A) is a soft L (R, 2-sided)I of S₂, then (G, A) is a generalized upper soft L (R, 2-sided)I of S₁ with respect to the aftersets.

(4) If (L, A) is a soft L (R, 2-sided)I of S₁, then (L, A) is a generalized upper soft L (R, 2-sided)I of S₂ with respect to the foresets.

Proof. (1) Suppose (G, A) is a soft SS of S₂. If $φ \neq {\bar{δ}}^{G} (e)$ for e ∈ A, then by Theorem 5, ${\bar{δ}}^{G} (e) . {\bar{δ}}^{G} (e) \subseteq {\bar{δ}}^{GG} (e) \subseteq {\bar{δ}}^{G} (e)$ , that is ${\bar{δ}}^{G} (e)$ is a SS of S₁ for e ∈ A and so (G, A) is a generalized upper soft SS of S₁ with respect to the aftersets.

(2) The proof is similar to the proof of part (1).

(3) Suppose (G, A) is a soft LI of S₂. As we know that ${\bar{δ}}^{S_{2}} (e)$ = S₁ for all e ∈ A. We have from Theorem 3, $S_{1} {\bar{δ}}^{G} (e) = {\bar{δ}}^{S_{2}} (e) . {\bar{δ}}^{G} (e) \subseteq {\bar{δ}}^{S_{2} G} (e)$

$\subseteq {\bar{δ}}^{G} (e)$ . Hence ${\bar{δ}}^{G} (e)$ is a LI of S₁ and so (G, A) is a generalized upper soft LI of S₁ with respect to the aftersets.

(4) The proof of this part is a routine and similar verification to part (3) and hence omitted.

The other cases can be demonstrated comparatively. □

The example below demonstrates that the converse of above parts of theorem is not valid generally.

Example 7. Let S₁ = {a, b, c, d, e} and S₂ ={ 1, 2, 3, 4, 5 } be two semigroups with the multiplication tables as follows:

•	a	b	c	d	e
a	b	b	d	d	d
b	b	b	d	d	d
c	d	d	c	d	c
d	d	d	d	d	d
e	d	d	c	d	c

•	1	2	3	4	5
1	1	5	3	4	5
2	1	2	3	4	5
3	1	5	3	4	5
4	1	5	3	4	5
5	1	5	3	4	5

Let A = {e₁, e₂}. Define δ : A → P (S₁ × S₂) by

$δ (e_{1}) = {\begin{matrix} (a, 1), (b, 1), (b, 2), (b, 5), (c, 3), \\ (c, 5), (d, 1), (d, 3), (d, 4), (d, 5), (e, 5) \end{matrix}} and$

$δ (e_{2}) = {\begin{matrix} (a, 1), (b, 1), (b, 2), (b, 3), (b, 5), (c, 3), \\ (c, 5), (d, 1), (d, 3), (d, 4), (d, 5), (e, 5) \end{matrix}} .$

Then (δ, A) is a soft compatible relation from the semigroups S₁ to the semigroup S₂ . Now,

$\begin{matrix} a δ (e_{1}) & = & {1}, b δ (e_{1}) = {1, 2, 5}, c δ (e_{1}) = {3, 5}, \\ d δ (e_{1}) & = & {1, 3, 4, 5}, e δ (e_{1}) = {5} and \end{matrix}$

$\begin{matrix} a δ (e_{2}) & = & {1}, b δ (e_{2}) = {1, 2, 3, 5}, c δ (e_{2}) = {3, 5}, \\ d δ (e_{2}) & = & {1, 3, 4, 5}, e δ (e_{2}) = {5} . \end{matrix}$

Also, $\begin{matrix} δ (e_{1}) 1 & = & {a, b, d}, δ (e_{1}) 2 = {b}, δ (e_{1}) 3 = {c, d}, \\ δ (e_{1}) 4 & = & {d}, δ (e_{1}) 5 = {b, c, d, e} and \end{matrix}$ $\begin{matrix} δ (e_{2}) 1 & = & {a, b, d}, δ (e_{2}) 2 = {b}, δ (e_{2}) 3 = {b, c, d}, \\ δ (e_{2}) 4 & = & {d}, δ (e_{2}) 5 = {b, c, d, e} . \end{matrix}$

Define (G, A) , a soft set over S₂ by G (e₁) ={ 1, 2, 3 } and G (e₂) ={ 2, 3, 4 } and Define (L, A) , a soft set over S₁ by L (e₁) ={ a, b, c } and L (e₂) = { a, b, e } .

(1) (G, A) is not a soft SS of S₂ but ${\bar{δ}}^{G} (e_{1}) = {a, b, c, d}$ and ${\bar{δ}}^{G} (e_{2}) = {a, b, c, d}$ are SSs of S₁. Hence, (G, A) is a generalized upper soft SS of S₁ with respect to the aftersets.

(2) (L, A) is not a soft SS of S₁ but $L \bar{δ} (e_{1}) = {1, 2, 3, 5}$ and $L \bar{δ} (e_{2}) = {1, 2, 3, 5}$ are SSs of S₂. Hence, (L, A) is a generalized upper soft SS of S₂ with respect to the foresets.

(3) (G, A) is not a soft LI of S₂ but ${\bar{δ}}^{G} (e_{1}) = {a, b, c, d}$ and ${\bar{δ}}^{G} (e_{2}) = {b, c, d}$ are LIs of S₁. Hence, (G, A) is a generalized upper soft LI of S₁ with respect to the aftersets.

(4) (L, A) is not a soft LI of S₁ but $L \bar{δ} (e_{1}) = {1, 2, 3, 5}$ and $L \bar{δ} (e_{2}) = {1, 2, 3, 5}$ are LIs of S₂. Hence, (L, A) is a generalized upper soft LI of S₂ with respect to the foresets.

Definition 7. Let (δ, A) be a soft compatible relation from S₁ to S₂ (S₁ and S₂ are semigroups). A soft set (G, A) over S₂ is said to be generalized lower soft SS of S₁ with respect to the aftersets if $({\underline{δ}}^{G}, A)$ is a soft SS of S₁. The soft set (G, A) is called generalized lower soft L (R, 2-sided)I of S₁ with respect to the aftersets if $({\underline{δ}}^{G}, A)$ is a soft L (R, 2-sided)Iof S₁.

Definition 8. Let (δ, A) be a soft compatible relation from S₁ to S₂ (S₁ and S₂ are semigroups). A soft set (L, A) over S₁ is said to be generalized lower soft SS of S₂ with respect to the foresets if $(L \underline{δ}, A)$ is a soft SS of S₂. The soft set (L, A) is called generalized lower soft L (R, 2-sided)I of S₂ with respect to the foresets if $(L \underline{δ}, A)$ is a soft L (R, 2-sided)I of S₂.

Example 8. Consider the semigroup and soft relations of Example 7 and Define (G, A) , a soft set over S₂ by G (e₁) ={ 1, 3, 5 } and G (e₂) = { 1, 2 } . Then (G, A) is a soft LI of S₂ but ${\underline{δ}}^{G} (e_{1}) = {a, c, e}$ is not a LI of S₁.

In the above example, we have shown that if (δ, A) is a soft compatible relation from a semigroup S₁ to a semigroup S₂ and (G, A) is a soft LI of S₂ even then $({\underline{δ}}^{L}, A)$ is not a LI of S₁. However, Theorem 10 is proceeded.

Theorem 10. Let (δ, A) be a soft complete relation with respect to the aftersets from S₁ to S₂ (S₁ and S₂ are semigroups). Then

(1) If (G, A) is a soft SS of S₂, then (G, A) is a generalized lower soft SS of S₁ with respect to the aftersets.

(2) If (G, A) is a soft L (R, 2-sided)I of S₂, then (G, A) is a generalized lower soft L (R, 2-sided)I of S₁ with respect to the aftersets.

Proof. (1) Suppose that (G, A) is a soft SS of S₂. If $φ \neq {\underline{δ}}^{G} (e)$ for e ∈ A. Then by Theorem 7 and Theorem 1(1), ${\underline{δ}}^{G} (e) . {\underline{δ}}^{G} (e) \subseteq {\underline{δ}}^{GG} (e) \subseteq {\underline{δ}}^{G} (e) .$ Therefore, $({\underline{δ}}^{G}, A)$ is a soft SS of S₁. Hence, (G, A) is a generalized lower soft SS of S₁ with respect to aftersets.

(2) Suppose that G is a soft LI of S₂. If $φ \neq {\underline{δ}}^{G} (e)$ for e ∈ A. Then by Theorem 7 and Theorem 1(1) , $S_{1} {\underline{δ}}^{G} (e) = {\underline{δ}}^{S_{2}} (e) . {\underline{δ}}^{G} (e) \subseteq {\underline{δ}}^{S_{2} G} (e)$

$\subseteq {\underline{δ}}^{G} (e) .$ Therefore, $({\underline{δ}}^{G}, A)$ is a soft LI of S₁. Hence, (G, A) is a generalized lower soft LI of S₁ with respect to the aftersets.

The rest of the cases can be demonstrated comparably. □

The proof of Theorem 11 is a routine work.

Theorem 11. Let (δ, A) be a soft complete relation with respect to the foresets from S₁ to S₂ (S₁ and S₂ are semigroups). Then

(1) If (L, A) is a soft SS of S₁, then (L, A) is a generalized lower soft SS of S₂ with respect to the foresets.

(2) If (L, A) is a soft L (R, 2-sided)I of S₁, then (L, A) is a generalized lower soft L (R, 2-sided)I of S₂ with respect to the foresets.

The converse of parts of Theorems 10 and 11 do not hold generally as shown by Example 9.

Example 9. Consider the semigroup of Example 5 and A = {e₁, e₂}. Define δ : A → P (S₁ × S₂) by $\begin{matrix} δ (e_{1}) & = & {(a, 2), (a, 3), (b, 2), (b, 3), (c, 2), \\ (c, 3), (d, 2), (d, 3)} and \\ δ (e_{2}) & = & {(a, 2), (b, 2), (c, 2), (d, 2)} . \end{matrix}$ Then (δ, A) is a soft complete relation with respect to the aftersets from the semigroup S₁ to the semigroup S₂. $\begin{matrix} a δ (e_{2}) & = & {2, 3}, b δ (e_{2}) = {2, 3}, \\ c δ (e_{2}) = {2, 3} and d δ (e_{2}) = {2, 3} . \\ Also, \\ a δ (e_{2}) & = & {2}, b δ (e_{2}) = {2}, c δ (e_{2}) = {2} \\ and d δ (e_{2}) = {2} . \end{matrix}$

Define (G, A) , a soft set over S₂ by G (e₁) ={ 2, 3, 4 } and G (e₂) = { 2, 4 } .

(1) (G, A) is not a soft SS of S₂ but ${\underline{δ}}^{G} (e_{1}) = {a, b, c, d}$ and ${\underline{δ}}^{G} (e_{2}) = {a, b, c, d},$ which show that $({\underline{δ}}^{G}, A)$ is a soft SS of S₁. Hence, (G, A) is a generalized lower soft SS of S₁ with respect to the aftersets.

(2) (G, A) is not a soft LI of S₂ but ${\underline{δ}}^{G} (e_{1}) = {a, b, c, d}$ and ${\underline{δ}}^{G} (e_{2}) = {a, b, c, d},$ which show that $({\underline{δ}}^{G}, A)$ is a soft LI of S₁. Hence, (G, A) is a generalized lower soft LI of S₁ with respect to the aftersets.

Now, Define δ : A → P (S₁ × S₂) by $\begin{matrix} δ (e_{1}) & = & {(d, 1), (d, 2), (d, 3), (d, 4)} and \\ δ (e_{2}) & = & {(a, 1), (a, 2), (a, 3), (a, 4), (d, 1), \\ (d, 2), (d, 3), (d, 4)} . \end{matrix}$ Then (δ, A) is a soft complete relation with respect to the foresets from the semigroup S₁ to the semigroup S₂. $\begin{matrix} δ (e_{2}) 1 & = & {d}, δ (e_{2}) 2 = {d}, δ (e_{2}) 3 = {d} \\ and δ (e_{2}) 4 = {d} . Also, \\ δ (e_{2}) 1 & = & {a, d}, δ (e_{2}) 2 = {a, d}, δ (e_{2}) 3 = \\ {a, d} and δ (e_{2}) 4 = {a, d} . \end{matrix}$

Define (L, A) , a soft set over S₁ by L (e₁) ={ b, c, d } and L (e₂) = { b, c, d } .

(1) (L, A) is not a soft SS of S₁ but $L \underline{δ} (e_{1}) = {1, 2, 3, 4}$ and $L \underline{δ} (e_{2}) = {1, 2, 3, 4},$ which show that $({\underline{δ}}^{G}, A)$ is a soft SS of S₂. Hence, (L, A) is a generalized lower soft SS of S₂ with respect to the foresets.

(2) (L, A) is not a soft LI of S₁ but $L \underline{δ} (e_{1}) = {1, 2, 3, 4}$ and $L \underline{δ} (e_{2}) = {1, 2, 3, 4},$ which show that $({\underline{δ}}^{G}, A)$ is a soft LI of S₂. Hence, (L, A) is a generalized lower soft LI of S₂ with respect to the foresets.

Theorem 12. Let (δ, A) be an SBR from S₁ to S₂ (S₁ and S₂ are semigroups). Then ${\bar{δ}}^{G_{1} G_{2}} \subseteq {\bar{δ}}^{G_{1}} \cap {\bar{δ}}^{G_{2}}$ for any soft RI (G₁, A) and soft LI (G₂, A) of S₂.

Proof. Suppose that (G₁, A) is a soft RI and (G₂, A) a soft LI of S₂, so by definition G₁ (e) ⊇ G₁ (e) S₂ ⊇ G₁ (e) G₂ (e) and G₂ (e) ⊇ S₂G₂ (e) ⊇ G₁ (e) G₂ (e) which implies that G₁ (e) ∩ G₂ (e) ⊇ G₁ (e) G₂ (e). It follows from Theorem 1 (parts (3) and (5)), ${\bar{δ}}^{G_{1} G_{2}} (e) \subseteq {\bar{δ}}^{G_{1} \cap G_{2}} (e) \subseteq {\bar{δ}}^{G_{1}} (e) \cap {\bar{δ}}^{G_{2}} (e)$ . Hence, ${\bar{δ}}^{G_{1} G_{2}} \subseteq {\bar{δ}}^{G_{1}} \cap {\bar{δ}}^{G_{2}}$ . □

The proof of Theorem 13 is a routine work.

Theorem 13. Let (δ, A) be an SBR from S₁ to S₂ (S₁ and S₂ are semigroups). Then $L_{1} L_{2} \bar{δ} \subseteq$ $L_{1} \bar{δ} \cap$ $L_{2} \bar{δ}$ for any soft RI (L₁, A) and soft LI (L₂, A) of S₁.

Theorem 14. Let (δ, A) be an SBR from S₁ to S₂ (S₁ and S₂ are semigroups). Then ${\underline{δ}}^{G_{1} G_{2}} \subseteq {\underline{δ}}^{G_{1}} \cap {\underline{δ}}^{G_{2}}$ for any soft RI (G₁, A) and soft LI (G₂, A) of S₂.

Proof. Suppose that (G₁, A) is a soft RI and (G₂, A) a soft LI of S₂, so by definition G₁ (e) G₂ (e) ⊆ G₁ (e) S₂ ⊆ G₁ (e) and G₁ (e) G₂ (e) ⊆ S₂G₂ (e) ⊆ G₂ (e) which implies that G₁ (e) G₂ (e) ⊆ G₁ (e) ∩ G₂ (e). It follows from Theorem 1 (parts (2) and (4)), ${\underline{δ}}^{G_{1} G_{2}} (e) \subseteq {\underline{δ}}^{G_{1} \cap G_{2}} (e) = {\underline{δ}}^{G_{1}} (e) \cap {\underline{δ}}^{G_{2}} (e)$ . Hence, ${\underline{δ}}^{G_{1} G_{2}} \subseteq {\underline{δ}}^{G_{1}} \cap {\underline{δ}}^{G_{2}}$ . □

The proof of Theorem 15 is a routine work.

Theorem 15. Let (δ, A) be an SBR from S₁ to S₂ (S₁ and S₂ are semigroups). Then $L_{1} L_{2} \underline{δ} \subseteq$ $L_{1} \underline{δ} \cap$ $L_{2} \underline{δ}$ for any soft RI (L₁, A) and soft LI (L₂, A) of S₂.

The next part is about II in semigroups.

Definition 9. Let (G, A) be a soft set over S₂ and (δ, A) an SBR from S₁ to S₂ (S₁ and S₂ are semigroups). Then (G, A) is said to be generalized lower (upper) soft II of S₁ with respect to the aftersets if $({\underline{δ}}^{G}, A)$

$(respectively ({\bar{δ}}^{G}, A))$ is a soft II of S₁.

Definition 10. Let (δ, A) an SBR from S₁ to S₂ (S₁ and S₂ are semigroups) and (L, A) be a soft set over S₁. Then (L, A) is said to be a generalized lower (upper) soft II of S₂ with respect to the foresets if $(L \underline{δ}, A) (respectively (L \bar{δ}, A))$ is a soft II of S₂.

Theorem 16. Let (δ, A) be a soft compatible relation from S₁ to S₂ (S₁ and S₂ are semigroups). If (G, A) is a soft II of S₂, then (G, A) is a generalized upper soft II of S₁ with respect to the aftersets.

Proof. As (G, A) is a soft II of S₂. It follows from Theorem 5 that $S_{1} {\bar{δ}}^{G} (e) S_{1} = {\bar{δ}}^{S_{2}} (e) . {\bar{δ}}^{G} (e) . {\bar{δ}}^{S_{2}} (e) \subseteq {\bar{δ}}^{S_{2} {GS}_{2}} (e) \subseteq {\bar{δ}}^{G} (e)$ . Therefore, $({\bar{δ}}^{G}, A)$ is a soft II of S₁. Hence, (G, A) is a generalized upper soft II of S₁ with respect to the aftersets. □

The example below describes that the counter part of above theorem is not valid generally.

Example 10. Consider the semigroups of Example 3 and A = {e₁, e₂}. Define δ : A → P (S₁ × S₂) by $\begin{matrix} δ (e_{1}) & = & {(1, a), (1, b), (1, c), (2, a), (2, b), \\ (3, a), (3, c)} and \\ δ (e_{2}) & = & {(1, a), (2, a), (2, b), (2, c), (3, c)} . \end{matrix}$ Then (δ, A) is a soft compatible relation from the semigroup S₁ to the semigroup S₂. Now, $\begin{matrix} 1 δ (e_{1}) & = & {a, b, c}, 2 δ (e_{1}) = {a, b} \\ and 3 δ (e_{1}) = {a, c}, \\ 1 δ (e_{2}) & = & {a}, 2 δ (e_{2}) = {a, b, c} \\ and 3 δ (e_{2}) = {c} . \end{matrix}$

Now, define (G, A) , a soft set over S₂ by G (e₁) ={ a } and G (e₂) = { a } , which is not a soft II of S₂ but ${\bar{δ}}^{G} (e_{1}) = {1, 2, 3}$ and ${\bar{δ}}^{G} (e_{2}) = {1, 2, 3},$ which show that $({\bar{δ}}^{G}, A)$ is a soft II of S₁. Hence, (G, A) is a generalized upper soft II of S₁ with respect to the aftersets.

The proof of Theorem 17 is a routine work.

Theorem 17. Let (δ, A) be a soft compatible relation from S₁ to S₂ (S₁ and S₂ are semigroups). If (L, A) is a soft II of S₁, then (L, A) is a generalized upper soft II of S₂ with respect to the foresets.

The example below describes that the converse part of above theorem is not valid generally.

Example 11. Consider the semigroups of Example 3 and soft relation of example 10, $\begin{matrix} δ (e_{1}) a & = & {1, 2, 3}, δ (e_{1}) b = {1, 2}, \\ δ (e_{1}) c = {1, 3}, \\ δ (e_{2}) a & = & {1, 2}, δ (e_{2}) b = {2} \\ and δ (e_{2}) c = {2, 3} . \end{matrix}$

Now, define (L, A) , a soft set over S₁ by L (e₁) ={ 1 } and L (e₂) ={ 2, 3 } which is not a soft II of S₁ but $L \bar{δ} (e_{1}) = {a, b, c}$ and $L \bar{δ} (e_{2}) = {a, b, c},$ which show that $(L \bar{δ}, A)$ is a soft II of S₂. Hence (L, A) is a generalized upper soft II of S₂ with respect to the foresets.

Theorem 18. Let (δ, A) be a soft complete relation from S₁ to S₂ (S₁ and S₂ are semigroups) with respect to the aftersets. If (G, A) is a soft II of S₂, then (G, A) is a generalized lower soft II of S₁ with respect to the aftersets.

Proof. As (G, A) is a soft II of S₂. Then by Theorem 1 (2) and Theorem 7, $S_{1} {\underline{δ}}^{G} (e) S_{1} = {\underline{δ}}^{S_{2}} (e) . {\underline{δ}}^{G} (e) . {\underline{δ}}^{S_{2}} (e) \subseteq {\underline{δ}}^{S_{2} {GS}_{2}} (e) \subseteq {\underline{δ}}^{G} (e)$ . Hence, $({\underline{δ}}^{G}, A)$ is a soft II of S₁. Thus, (G, A) is a generalized lower soft II of S₁ with respect to theaftersets. □

Example 12. Consider the semigroup of Example 5 and A = {e₁, e₂}. Define δ : A → P (S₁ × S₂) by $\begin{matrix} δ (e_{1}) & = & {(a, 2), (a, 3), (b, 2), (b, 3), (c, 2), \\ (c, 3), (d, 2), (d, 3)} and \\ δ (e_{2}) & = & {(a, 2), (b, 2), (c, 2), (d, 2)} . \end{matrix}$ Then (δ, A) is a soft complete relation from the semigroup S₁ to the semigroup S₂ with respect to the aftersets. $\begin{matrix} a δ (e_{2}) & = & {2, 3}, b δ (e_{2}) = {2, 3}, c δ (e_{2}) = \\ {2, 3} and d δ (e_{2}) = {2, 3} . Also, \\ a δ (e_{2}) & = & {2}, b δ (e_{2}) = {2}, c δ (e_{2}) = {2} \\ and d δ (e_{2}) = {2} . \end{matrix}$

Define (G, A) , a soft set over S₂ by G (e₁) ={ 2, 3, 4 } and G (e₂) ={ 2, 4 } which is not a soft II of S₂ but ${\underline{δ}}^{G} (e_{1}) = {a, b, c, d}$ and ${\underline{δ}}^{G} (e_{2}) = {a, b, c, d},$ which show that $({\underline{δ}}^{G}, A)$ is a soft II of S₁. Hence, (G, A) is a generalized lower soft II of S₁ with respect to the aftersets.

The proof of Theorem 19 is a routine work.

Theorem 19. Let (δ, A) be a soft complete relation from S₁ to S₂ (S₁ and S₂ are semigroups) with respect to the foresets. If (L, A) is a soft II of S₁, then (L, A) is a generalized lower soft II of S₂ with respect to the foresets.

Example 13. Consider the semigroup of Example 5 and A = {e₁, e₂}. Define δ : A → P (S₁ × S₂) by $\begin{matrix} δ (e_{1}) & = & {(d, 1), (d, 2), (d, 3), (d, 4)} and \\ δ (e_{2}) & = & {(a, 1), (a, 2), (a, 3), (a, 4), (d, 1), \\ (d, 2), (d, 3), (d, 4)} . \end{matrix}$ Then (δ, A) is a soft complete relation from the semigroup S₁ to the semigroup S₂ with respect to the foresets. $\begin{matrix} δ (e_{2}) 1 & = & {d}, δ (e_{2}) 2 = {d}, δ (e_{2}) 3 = {d} \\ and δ (e_{2}) 4 = {d} . Also, \\ δ (e_{2}) 1 & = & {a, d}, δ (e_{2}) 2 = {a, d}, \\ δ (e_{2}) 3 = {a, d} and δ (e_{2}) 4 = {a, d} . \end{matrix}$

Define (L, A) , a soft set over S₁ by L (e₁) ={ b, c, d } and L (e₂) ={ b, c, d } which is not a soft II of S₁ but $L \underline{δ} (e_{1}) = {1, 2, 3, 4}$ and $L \underline{δ} (e_{2}) = {1, 2, 3, 4}$ is an II of S₂. Hence, (L, A) is a generalized lower soft II of S₂ with respect to the foresets.

The next part is about BI in semigroups.

Definition 11. Let (δ, A) an SBR from S₁ to S₂ (S₁ and S₂ are semigroups) and (G, A) be a soft set over S₂. Then (G, A) is said to be generalized lower (upper) soft BI of S₁ with respect to the aftersets if $({\underline{δ}}^{G}, A) (respectively ({\bar{δ}}^{G}, A))$ is a soft BI of S₁.

Definition 12. Let (δ, A) an SBR from S₁ to S₂ (S₁ and S₂ are semigroups) and (L, A) be a soft set over S₁. Then (L, A) is said to be a generalized lower (upper) soft BI of S₂ with respect to the foresets if $(L \underline{δ}, A) (respectively (L \bar{δ}, A))$ is a soft BI of S₂.

Theorem 20. Let (δ, A) be a soft compatible relation from S₁ to S₂ (S₁ and S₂ are semigroups). Then every soft BI (G, A) of S₂ is a generalized upper soft BI of S₁ with respect to the aftersets.

Proof. Let (G, A) be a soft BI of S₂. It follows from Theorem 9 (1) , $({\bar{δ}}^{G}, A)$ is a soft SS of S₂. By Theorem 1 (3) and Theorem 5, ${\bar{δ}}^{G} (e) S_{1} {\bar{δ}}^{G} (e) = {\bar{δ}}^{G} (e) . {\bar{δ}}^{S_{2}} (e) . {\bar{δ}}^{G} (e) \subseteq {\bar{δ}}^{{GS}_{2} G} (e) \subseteq {\bar{δ}}^{G} (e)$ . Hence, $({\bar{δ}}^{G}, A)$ is a soft BI of S₁. Thus, (G, A) is a generalized upper soft BI of S₁. □

Example 14. Consider the semigroups and soft relations of Example 7. Define (G, A) , a soft set over S₂ by G (e₁) ={ 1, 2, 3 } and G (e₂) ={ 1, 2 } which is not a soft BI of S₂ but ${\bar{δ}}^{G} (e_{1}) = {a, b, c, d}$ and ${\bar{δ}}^{G} (e_{2}) = {a, b, d},$ which show that $({\bar{δ}}^{G}, A)$ is a soft BI of S₁. Hence, (G, A) is a generalized upper soft BI of S₁ with respect to the aftersets.

The proof of Theorem 21 is a routine work.

Theorem 21. Let (δ, A) be a soft compatible relation from S₁ to S₂ (S₁ and S₂ are semigroups). Then every soft BI (L, A) of S₁ is a generalized upper soft BI of S₂ with respect to the foresets.

Example 15. Consider the semigroups and soft relations of example 7. Define (L, A) , a soft set over S₁ by L (e₁) ={ a, b, c } and L (e₂) ={ a, b } which is not a soft BI of S₁ but $L \bar{δ} (e_{1}) = {1, 5, 3, 2}$ and $L \bar{δ} (e_{2}) = {1, 5, 3, 2},$ which show that $(L \bar{δ}, A)$ is a soft BI of S₂. Hence, (L, A) is a generalized upper soft BI of S₂ with respect to the foresets.

Theorem 22. Let (δ, A) be a soft complete relation from S₁ to S₂ (S₁ and S₂ are semigroups) with respect to aftersets. Then every soft BI (G, A) of S₂ is a generalized lower soft BI of S₁ with respect to the aftersets.

Proof. Let (G, A) be a soft BI of S₂. It is followed from Theorem 11 (2) , $({\underline{δ}}^{G}, A)$ is a soft SS of S₂. By Theorem 1(2) and Theorem 5, ${\underline{δ}}^{G} (e) . S_{1} . {\underline{δ}}^{G} (e) = {\underline{δ}}^{G} (e) . {\underline{δ}}^{S_{2}} (e) . {\underline{δ}}^{G} (e) \subseteq {\underline{δ}}^{{GS}_{2} G} (e) \subseteq {\underline{δ}}^{G} (e)$ . Hence, $({\underline{δ}}^{G}, A)$ is a soft BI of S₁. Hence, (G, A) is a generalized lower soft BI of S₁. □

Example 16. Consider the semigroup of Example 3 and A = {e₁, e₂}. Define δ : A → P (S₁ × S₂) by $\begin{matrix} δ (e_{1}) & = & {(a, 2), (a, 3), (b, 2), (b, 3), (c, 2), \\ (c, 3), (d, 2), (d, 3)} and \\ δ (e_{2}) & = & {(a, 2), (b, 2), (c, 2), (d, 2)} . \end{matrix}$ Then (δ, A) is a soft complete relation from the semigroup S₁ to the semigroup S₂ with respect to aftersets. Now, $\begin{matrix} a δ (e_{1}) & = & {2, 3}, b δ (e_{1}) = {2, 3}, c δ (e_{1}) \\ = {2, 3} and d δ (e_{1}) = {2, 3} . Also, \\ a δ (e_{2}) & = & {2}, b δ (e_{2}) = {2}, c δ (e_{2}) = {2} \\ and d δ (e_{2}) = {2} . \end{matrix}$ Then define (G, A) , a soft set over S₂ by G (e₁) ={ 2, 3, 4 } and G (e₂) ={ 2, 4 } which is not a soft BI of S₂ but ${\underline{δ}}^{G} (e_{1}) = {a, c, b, d}$ and ${\underline{δ}}^{G} (e_{2}) = {a, c, b, d},$ which show that $({\underline{δ}}^{G}, A)$ is a soft BI of S₁. Hence, (G, A) is a generalized lower soft BI of S₁ with respect to the aftersets.

The proof of Theorem 23 is a routine work.

Theorem 23. Let (δ, A) be a soft complete relation from S₁ to S₂ (S₁ and S₂ are semigroups) with respect to foresets. Then every soft BI (L, A) of S₁ is a generalized lower soft BI of S₂ with respect to the foresets.

Example 17. Consider the semigroup of Example 5 and A = {e₁, e₂}. Define δ : A → P (S₁ × S₂) by $\begin{matrix} δ (e_{1}) & = & {(d, 1), (d, 2), (d, 3), (d, 4)} and \\ δ (e_{2}) & = & {(a, 1), (a, 2), (a, 3), (a, 4), (d, 1), \\ (d, 2), (d, 3), (d, 4)} . \end{matrix}$ Then (δ, A) is a soft complete relation from the semigroup S₁ to the semigroup S₂ with respect to foresets. $\begin{matrix} δ (e_{2}) 1 & = & {d}, δ (e_{2}) 2 = {d}, δ (e_{2}) 3 = {d} \\ and δ (e_{2}) 4 = {d} . Also, \\ δ (e_{2}) 1 & = & {a, d}, δ (e_{2}) 2 = {a, d}, \\ δ (e_{2}) 3 = {a, d} and δ (e_{2}) 4 = {a, d} . \end{matrix}$

Define (L, A) , a soft set over S₁ by L (e₁) ={ b, d, c } and L (e₂) ={ b, d, c } which is not a soft BI of S₁ but $L \underline{δ} (e_{1}) = {1, 4, 3, 2}$ and $L \underline{δ} (e_{1}) = {1, 4, 3, 2},$ which show that $(L \underline{δ}, A)$ is a soft BI of S₂. Hence, (L, A) is a generalized upper soft BI of S₂ with respect to the foresets.

5 Conclusion

The soft set theory is considered as an effective theory and proved to have various applications in many areas. Soft binary relations are taken as a major tool in this paper to handle many things. Some fundamental ideas, operations and related properties are presented with respect to soft binary relations. This paper is engaged at studying the rough soft sets inside the context of Semigroup by soft compatible relations. In future, we will use a soft tolerance relation to handle this concept in a different way engaging any algebraic structure.

• The concept taken in this paper may be expanded under different conditions of soft binary relations.

• We are hopeful that in near future, the concept of roughness using soft binary relations will be expanded with other algebraic structures like Quantles and it will also be helpful in the additional examination of the semigroup theory.

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