Non-commutative residuated lattices based on soft sets

Abstract

Molodtsov introduced the concept of soft set as a new mathematical tool for dealing with uncertainties. Recently, Cagman and Enginoglu [16] provided new definitions and operations on soft set theory. The paper applies new definitions and operations of soft sets to non-commutative residuated lattices. The notion of soft non-commutative residuated lattices is introduced. We give some specific examples to show the existence of soft non-commutative residuated lattices. The union, intersection, ∧-product, ∨-product and difference operations of soft non-commutative residuated lattices are investigated. Finally, we study the homomorphism properties of soft non-commutative residuated lattices. It is pointed out that the soft non-commutative residuated lattices are so general, all results in this paper also hold in most of soft non-commutative logic algebras and soft logic algebras.

Keywords

Soft set soft non-commutative residuated lattice union intersection ∧–product ∨–product

1 Introduction

In the real world, there are a lot of inaccurate, incomplete or not fully reliable information (collectively referred to as uncertain information). Therefore, people need to deal with a lot of uncertainty information. There are some theories, such as the theory of probability, the theory of fuzzy sets, and the theory of rough sets which people can consider as mathematical tools for dealing with uncertainties. But, all these theories have their own difficulties. Molodtsov [4] suggested that one reason for these difficulties may be due to the inadequacy of the parametrization tool of the theory. To overcome these difficulties, he [4] 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. Molodtsov pointed out several directions for the applications of soft sets. At present, works on the soft set theory are progressing rapidly. Maji et al. [18] described the application of soft set theory to a decision making problem. Maji et al. [19] also studied several operations on the theory of soft sets. Ali et al. [14] pointed out some problems in the operations proposed in paper [19], and gave some new operations in soft set theory. Chen et al. [3] presented a new definition of soft set parametrization reduction, and compared this definition to the related concept of attributes reduction in rough set theory. In recent years, Aktas et al. [7] initiated to apply the soft set theory to the group structure, defined the soft group and discussed the related properties. Since then, some authors applied soft set theory to other kinds of algebraic structure, such as soft BCK/BCI-algebras [30], soft BL-algebras [10], soft semirings [5], soft rings [22], soft Lie algebras [13], etc. Recently, Cagman and Enginoglu [16] provided new definitions and some results on soft set theory.

Non-commutative residuated lattices, introduced by Dilworth in [20], are as non-commutative extension of the residuated lattices [15]. Some non-commutative logic algebras, such as pseudo MV-algebras [6], pseudo BL-algebras [1], pseudo MTL-algebras [17], pseudo NM-algebras [3, 33], pseudo weak R₀-algebras [32], pseudo R₀-algebras [32] are all subclasses of non-commutative residuated lattices. Non-commutative residuated lattice and its subclasses have been studied extensively by many authors [8 , 35]. In this paper, we apply Cagman and Enginoglu’s new definitions and operations of soft set to non-commutative residuated lattices. The notion of soft non-commutative residuated lattices is introduced. We give some specific examples to show the existence of soft non-commutative residuated lattices. The union, intersection, ∧-product, ∨-product and difference operations of soft non-commutative residuated lattices are investigated. The homomorphism properties of soft non-commutative residuated lattices are studied. All results in this paper also hold in most of soft non-commutative logic algebras and soft logicalgebras.

2 Preliminaries

Definition 2.1. [1, 20] A non-commutative residuated lattice is a structure L = (L, ∨ , ∧ , ⊙ , → , ↪ , 0, 1) of type (2, 2, 2, 2, 2, 0, 0) satisfying the following conditions:

(L1) (L, ∨ , ∧ , 0, 1) is a bounded lattice;

(L2) (L, ⊙ , 1) is a monoid, i.e., ⊙ is associative with 1 as unit element;

(L3) x ⊙ y ≤ z iff x ≤ y → z iff y ≤ x ↪ z for all x, y, z ∈ L .

If a non-commutative residuated lattice L satisfies commutative law, that is x ⊙ y = y ⊙ x for all x, y ∈ L, then L is a residuated lattice. In what follows, we denote by L a non-commutative residuated lattice (unlessotherwise specified). For x ∈ L we define x ^- = x → 0 and x ^∼ = x ↪ 0. The following main rules of calculus in a non-commutative residuated lattice L can be found, for example, in [1 , 33]:

(1) x ↪ x = x → x = 1;

(2) 1 ↪ x = 1 → x = x;

(3) x ⊙ y ≤ x ∧ y;

(4) y ⊙ x ≤ x ∧ y;

(5) x → (y ↪ z) = y ↪ (x → z);

(6) x⊙ (x ↪ y) ≤ y ≤ x ↪ (x ⊙ y) ; x ⊙ (x ↪ y) ≤ x ≤ y ↪ (y ⊙ x);

(7) (x → y) ⊙ x ≤ x ≤ y → (x ⊙ y); (x→ y) ⊙ x ≤ y ≤ x → (y ⊙ x) ;

(8) If x ≤ y, then x ⊙ z ≤ y ⊙ z, z ⊙ x ≤ z ⊙ y;

(9) If x ≤ y, then z ↪ x ≤ z ↪ y, z → xleqz → y;

(10) If x ≤ y, then y ↪ z ≤ x ↪ z, y → zleqx → z;

(11) x ≤ y iff x ↪ y = 1 iff x → y = 1;

(12) x↪ y ≤ (z ↪ x) ↪ (z ↪ y) ; x→ y ≤ (z → x) → (z → y) ;

(13) x↪ y ≤ (y ↪ z) → (x ↪ z) ; x→ y ≤ (y → z) ↪ (x → z) ;

(14) (x⊙ y) → z = x → (y → z) ; (y⊙ x) ↪ z = x ↪ (y ↪ z) ;

(15) x→ (y ∨ z) = (x → y) ∨ (x → z) ; x↪ (y ∨ z) = (x ↪ y) ∨ (x ↪ z) ;

(16) (x∧ y) → z = (x → z) ∨ (y → z) ; (x ∧ y) ↪z = (x ↪ z) ∨ (y ↪ z);

(17) 1^∼ = 1^- = 0;

(18) 0^∼ = 0^- = 1;

(19) x ⊙ x ^∼ = x ^- ⊙ x = 0;

(20) If x ≤ y, then y ^- ≤ x ^-, y ^∼ ≤ x ^∼.

The following definitions come from [16], but we have some different expression, especially in the union, intersection, ∧-product, ∨-product and difference operations, in order to make the concepts more clear and easy to use.

From now on, let U be an initial universe set, E be a set of parameters, A, B, C ⊂ E and P (U) denote the power set of U.

Definition 2.2. [4, 16] A soft set f _A over U is defined as a set-valued mapping f _A : E → P (U) such that f _A (x) =∅ if x ∉ A. Here, f _A is also called an approxi-mate function.

A soft set over U can be represented by the set of ordered pairs f _A = {(x, f _A (x)) |x ∈ E, f _A (x) ∈ P (U)}. It is clear to see that a soft set is a parameterized family of subsets of U. Note that the set of all soft sets over U will be defined by S (U).

Definition 2.3. [16] Let f _A, h _B ∈ S (U). Then, f _A is called a soft subset of h _B denoted by $f_{A} \tilde{\subset} h_{B}$ if f _A (x) ⊂ h _B (x) for all x ∈ E. f _A and h _B are said to be soft equal, denoted by f _A = h _B, if $f_{A} \tilde{\subset} h_{B}$ and $h_{B} \tilde{\subset} f_{A}$ .

Definition 2.4. [16] Let f _A, h _B ∈ S (U). Then, the union of f _A and h _B denoted by $f_{A} \tilde{\cup} h_{B}$ is defined by $f_{A} \tilde{\cup} h_{B} = k_{C}$ , where C = A ∪ B and k _C (x) = f _A (x) ∪ h _B (x) for all x ∈ E.

Definition 2.5. [16] Let f _A, h _B ∈ S (U). Then, the intersection of f _A and h _B denoted by $f_{A} \tilde{\cap} h_{B}$ is defined by $f_{A} \tilde{\cap} h_{B} = k_{C}$ , where C = A ∩ B and k _C (x) = f _A (x) ∩ h _B (x) for all x ∈ E.

Definition 2.6. [16] If f _A, h _B ∈ S (U), then ∧-product of two soft sets f _A and h _B denoted by f _A ∧ h _B is defined by f _A ∧ h _B = k _C, where C = A × B and k _C (x, y) = f _A (x) ∩ h _B (y) for all (x, y) ∈ E × E.

Definition 2.7. [16] If f _A, h _B ∈ S (U), then ∨-product of two soft sets f _A and h _B denoted by f _A ∨ h _B is defined by f _A ∨ h _B = k _C, where C = A × B and k _C (x, y) = f _A (x) ∪ h _B (y) for all (x, y) ∈ E × E.

Definition 2.8. [16] Let f _A, h _B ∈ S (U). Then, the difference of f _A and h _B denoted by $f_{A} \tilde{∖} h_{B}$ is defined by $f_{A} \tilde{∖} h_{B} = k_{C}$ , where C = A ∩ B and k _C (x) = f _A (x) ∖ h _B (x) for all x ∈ E.

3 Non-commutative residuated lattices based on new definitions and operations of soft sets

Definition 3.1. Let L be a non-commutative residuated lattice and S a nonempty subset of L with 0, 1 ∈ S. Then S is called a non-commutative residuated sublattice of L if S is a non-commutative residuated lattice on the operations of L.

Proposition 3.2. Let L be a non-commutative residuated lattice and S a nonempty subset of L with 0, 1 ∈ S. Then S is a non-commutative residuated sublattice of L iff ∀x, y ∈ S, x ∨ y, x ∧ y, x ⊙ y ∈ S.

Proof. Straightforward.

In what follows, let U = L, where L is a non-commutative residuated lattice, be an initial universe set.

Definition 3.3. Let f _A be a soft set over L with A≠ ∅. Then f _A is called a soft non-commutative residuated lattice over L if f _A (x) is either empty or a non-commutative residuated sublattice of L for all x ∈ A.

Definition 3.4. Let f _A and h _B be two soft non-commutative residuated lattices over L. If ∀x ∈ A, f _A (x) is a non-commutative residuated sublattice of h _B (x), then f _A is called a soft non-commutative residuated sublattice of h _B. In this case, we write $f_{A} \tilde{<} h_{B}$ .

Let f _A and f _B be two soft non-commutative residuated lattices over L. If A ⊂ B, then $f_{A} \tilde{<} f_{B}$ .

Example 3.5. Let L = {0, a, b, c, 1}, where 0 < a < b < c < 1. Consider the operations ⊙, → , ↪ given by the following Cayley tables:

⊙	a	b	c	1
0	0	0	0	0
a	0	0	0	a
b	0	0	0	b
c	0	a	a	c
1	a	b	c	1

→	0	a	b	c	1
0	1	1	1	1	1
a	c	1	1	1	1
b	b	c	1	1	1
c	b	c	c	1	1
1	0	a	b	c	1

↪	0	a	b	c	1
0	1	1	1	1	1
a	c	1	1	1	1
b	c	c	1	1	1
c	a	c	c	1	1
1	0	a	b	c	1

Then L = (L, ∨ , ∧ , ⊙ , → , ↪ , 0, 1) is a non-commutative residuated lattice [11]. Let E = {x ₁, x ₂, x ₃, x ₄, x ₅}, A = {x ₁, x ₂, x ₃, x ₄} and for all x ∈ E,

$f_{A} (x) = {\begin{matrix} L, & x = x_{1}, \\ {0, a, b, 1}, & x = x_{2}, \\ {0, a, c, 1}, & x = x_{3}, \\ {0, b, 1}, & x = x_{4}, \\ \emptyset, & x = x_{5} . \end{matrix}$

Then f _A : E → P (L) is a set-valued mapping, and so f _A is a soft set over L. By routine calculations, {0, b, 1} , {0, a, b, 1} , {0, a, c, 1} , L are all non-commutative residuated sublattices of L. Hence, f _A is a soft non-commutative residuated lattice over L. Let B = {x ₁, x ₂, x ₃} and for all x ∈ E,

$h_{B} (x) = {\begin{matrix} {0, a, c, 1}, & x = x_{1}, \\ {0, a, 1}, & x = x_{2}, \\ {0, 1}, & x = x_{3}, \\ \emptyset, & x = x_{4}, x_{5} . \end{matrix}$ Then h _B is also a soft non-commutative residuated lattice over L. ∀x ∈ B, h _B (x) is a non-commutative residuated sublattice of f _A (x), By Definition 3.4, we have $h_{B} \tilde{<} f_{A}$ .

Example 3.5. shows that there exists a set-valued mapping f _A : E → P (L) such that the soft set f _A is a soft non-commutative residuated lattice over L.

Example 3.6. Let L = {0, a, b, c, 1}, where 0 < a < b, c < 1 and b, c are incomparable. Consider the operations ⊙, → , ↪ given by the following Cayley tables:

⊙	a	b	c	1
0	0	0	0	0
a	0	0	a	a
b	a	b	a	b
c	0	0	c	c
1	a	b	c	1

→	0	a	b	c	1
0	1	1	1	1	1
a	c	1	1	1	1
b	c	c	1	c	1
c	0	b	b	1	1
1	0	a	b	c	1

↪	0	a	b	c	1
0	1	1	1	1	1
a	b	1	1	1	1
b	0	c	1	c	1
c	b	b	b	1	1
1	0	a	b	c	1

Then L = (L, ∨ , ∧ , ⊙ , → , ↪ , 0, 1) is a non-commutative residuated lattice [11]. Let E = L, A = {0, a, b} and for all x ∈ A, f _A (x) = {y∈ L|x ↪ y = 1} , f _A (x) = ∅ if x ∉ A. Clearly, f _A is a soft set over L. By routine calculations, f _A (0) = {0, a, b, c, 1} is a non-commutative residuated sublattice of L, but f _A (a) = {a, b, c, 1} and f _A (b) = {b, 1} are not both non-commutative residuated sublattices of L as 0 ∉ f _A (a) and f _A (b). This shows that there exists a set-valued mapping f _A : E → P (L) such that the soft set f _A is not a soft non-commutative residuated lattice over L.

Theorem 3.7. Let f_A be a soft non-commutative residuated lattice over L. If B ⊂ A, then f_B is a soft non-commutative residuated lattice over L.

Proof. Straightforward.

Next, we investigate the operation properties of soft non-commutative residuated lattices.

Proposition 3.8. Let f_A and h_B be two soft non-commutative residuated lattices over L. Then the union $f_{A} \tilde{\cup} h_{B}$ may not be a soft non-commutative residuated lattice over L.

It is shown by the following two examples.

Example 3.9. Let L = {0, a, b, c, d, 1}, where 0 < a < b, c < d < 1 and b, c are incomparable. Consider the operations ⊙, → , ↪ given by the following Cayley tables:

⊙	a	b	c	d	1
0	0	0	0	0	0
a	0	a	0	a	a
b	0	b	0	b	b
c	a	a	c	c	c
d	a	b	c	d	d
1	a	b	c	d	1

→	0	a	b	c	d	1
0	1	1	1	1	1	1
a	b	1	1	1	1	1
b	0	c	1	c	1	1
c	b	b	b	1	1	1
d	0	a	b	c	1	1
1	0	a	b	c	d	1

↪	0	a	b	c	d	1
0	1	1	1	1	1	1
a	c	1	1	1	1	1
b	c	c	1	c	1	1
c	0	b	b	1	1	1
d	0	a	b	c	1	1
1	0	a	b	c	d	1

Then L = (L, ∨ , ∧ , ⊙ , → , ↪ , 0, 1) is a non-commutative residuated lattice. Let E = {x ₁, x ₂, x ₃, x ₄, x ₅, x ₆}, A = {x ₁, x ₂, x ₃, x ₄} , B = {x ₃, x ₄, x ₅, x ₆} and for all x ∈ E,

$f_{A} (x) = {\begin{matrix} {0, 1}, & x = x_{1}, \\ {0, a, 1}, & x = x_{2}, \\ {0, b, 1}, & x = x_{3}, \\ {0, a, c, 1}, & x = x_{4}, \\ \emptyset, & x = x_{5}, x_{6} . \end{matrix}$ $h_{B} (x) = {\begin{matrix} \emptyset, & x = x_{1}, x_{2}, \\ {0, c, 1}, & x = x_{3}, \\ {0, d, 1}, & x = x_{4}, \\ {0, a, b, 1}, & x = x_{5}, \\ {0, c, d, 1}, & x = x_{6} . \end{matrix}$

Then f _A and h _B are two soft sets over L. By routine calculations, {0, 1} , {0, a, 1} , {0, b, 1} , {0, a, c, 1}, {0, c, 1} , {0, d, 1} , {0, a, b, 1} , {0, c, d, 1} are all non-commutative residuated sublattices of L, and so f _A and h _B are two soft non-commutative residuatedlattices over L. Writing the union $f_{A} \tilde{\cup} h_{B} = k_{C}$ , where C = A ∪ B = {x ₁, x ₂, x ₃, x ₄, x ₅, x ₆} and ∀x ∈ E,

$k_{C} (x) = {\begin{matrix} {0, 1}, & x = x_{1}, \\ {0, a, 1}, & x = x_{2}, \\ {0, b, c, 1}, & x = x_{3}, \\ {0, a, c, d, 1}, & x = x_{4}, \\ {0, a, b, 1}, & x = x_{5}, \\ {0, c, d, 1}, & x = x_{6} . \end{matrix}$

When x = x ₃, k _C (x ₃) = f _A (x ₃) ∪ h _B (x ₃) = {0, b, c, 1} is not a non-commutative residuated sublattice of L, and so the union k _C is not a soft non-commutative residuated lattice over L. In fact, c ⊙ b = a ∉ {0, b, c, 1}.

Example 3.10. Let L = {0, a, b, c, d, e, 1}, where 0 < a < b, c < d < e < 1 and b, c are incomparable. Consider the operations ⊙, → , ↪ given by the following Cayley tables:

⊙	a	b	c	d	e	1
0	0	0	0	0	0	0
a	a	a	a	a	a	a
b	a	a	a	a	a	b
c	a	a	c	c	c	c
d	a	a	c	c	c	d
e	a	b	c	d	e	e
1	a	b	c	d	e	1

→	0	a	b	c	d	e	1
0	1	1	1	1	1	1	1
a	0	1	1	1	1	1	1
b	0	d	1	d	1	1	1
c	0	b	b	1	1	1	1
d	0	b	b	d	1	1	1
e	0	b	b	d	d	1	1
1	0	a	b	c	d	e	1

↪	0	a	b	c	d	e	1
0	1	1	1	1	1	1	1
a	0	1	1	1	1	1	1
b	0	e	1	e	1	1	1
c	0	b	b	1	1	1	1
d	0	b	b	e	1	1	1
e	0	a	b	c	d	1	1
1	0	a	b	c	d	e	1

Then L = (L, ∨ , ∧ , ⊙ , → , ↪ , 0, 1) is a non-commutative residuated lattice. Let E = L, A = {0, a, b, c},B = {b, c, d, e} and for all x ∈ E, $f_{A} (x) = {\begin{matrix} {0, a, b, c, e, 1}, & x = 0, \\ {0, a, b, c, d, 1}, & x = a, \\ {0, e, 1}, & x = b, \\ {0, a, c, 1}, & x = c, \\ \emptyset, & other . \end{matrix}$

$h_{B} (x) = {\begin{matrix} {0, c, 1}, & x = b, \\ {0, a, e, 1}, & x = c, \\ {0, c, d, e, 1}, & x = d, \\ L, & x = e, \\ \emptyset, & other . \end{matrix}$

Then f _A and h _B are two soft sets over L. By routine calculations, {0, a, b, c, e, 1} , {0, a, b, c, d, 1}, {0, e, 1}, {0, a, c, 1} , {0, c, 1} , {0, a, e, 1} , {0, c, d, e, 1} and L are all non-commutative residuated sublattices of L, and so f _A and h _B are two soft non-commutative residuated lattices over L. Writing the union $f_{A} \tilde{\cup} h_{B} = k_{C}$ , where C = A ∪ B = {0, a, b, c, d, e} and ∀x ∈ E,

$k_{C} (x) = {\begin{matrix} {0, a, b, c, e, 1}, & x = 0, \\ {0, a, b, c, d, 1}, & x = a, \\ {0, c, e, 1}, & x = b, \\ {0, a, c, e, 1}, & x = c, \\ {0, c, d, e, 1}, & x = d, \\ L, & x = e, \\ \emptyset, & x = 1 . \end{matrix}$

Because {0, c, e, 1} , {0, a, c, e, 1} are all non-commutative residuated sublattices of L, and so the union k _C is a soft non-commutative residuated lattice over L.

We have following:

Theorem 3.11. If f_A and h_B are two soft non-commutative residuated lattices over L with A∩ B = ∅, then the union $f_{A} \tilde{\cup} h_{B}$ is a soft non-commutative residuated lattice over L.

Proof. Let $f_{A} \tilde{\cup} h_{B} = k_{C}$ , where C = A ∪ B and k _C (x) = f _A (x) ∪ h _B (x) for all x ∈ E. Since f _A and h _B are both soft residuated lattices over L, then A≠ ∅ and B≠ ∅, and so C = A∪ B ≠ ∅. ∀x ∈ C = A ∪ B, since A∩ B = ∅, then x ∈ A or x ∈ B. If x ∈ A, then k _C (x) = f _A (x) ∪ ∅ = f _A (x) is a residuated sublattice of L. If x ∈ B, then k _C (x) = h _B (x) ∪ ∅ = h _B (x) is also a non-commutative residuated sublattice of L.

Therefore the union k _C is a soft non-commutative residuated lattice over L.

Theorem 3.12. Let f_A and h_B be two soft non-commutative residuated lattices over L with A∩ B ≠ ∅. Then the intersection $f_{A} \tilde{\cap} h_{B}$ is a soft non-commutative residuated lattice over L.

Proof. Let $f_{A} \tilde{\cap} h_{B} = k_{C}$ , where C = A ∩ B and ∀x ∈ C, k _C (x) = f _A (x) ∩ h _B (x). Since f _A is a soft non-commutative residuated lattice over L, then ∀x ∈ A, f _A (x) is a non-commutative residuated sublattice of L. Similarly, ∀x ∈ B, h _B (x) is a non-commutative residuated sublattice of L. Thus, ∀x ∈ C, k _C (x) = f _A (x) ∩ h _B (x) is a non-commutative residuated sublattice of L since the intersection of two non-commutative residuated sublattices of L is also a non-commutative residuated sublattice of L. Therefore $f_{A} \tilde{\cap} h_{B} = k_{C}$ is a soft non-commutative residuated lattice over L.

Theorem 3.13. Let f_A and h_B be two soft non-commutative residuated lattices over L. Then the ∧-product f_A ∧ h_B is a soft non-commutative residuated lattice over L.

Proof. By Definition 2.6, we know that f _A ∧ h _B = k _C, where C = A × B and k _C (α, β) = f _A (α) ∩ h _B (β) for all (α, β) ∈ E × E. Since ∀ (α, β) ∈ C, f _A (α) and h _B (β) are non-commutative residuated sublattices of L, the intersection f _A (α) ∩ h _B (β) is also a non-commutative residuated sublattice of L. Hence k _C (α, β) is a non-commutative residuated sublattice of L for all (α, β) ∈ C, and therefore f _A ∧ h _B = k _C is a soft non-commutative residuated lattice over L.

Proposition 3.14. Let f_A and h_B be two soft non-commutative residuated lattices over L. Then the ∨-product f_A ∨ h_B may not be a soft non-commutative residuated lattice over L.

It is shown by the following two examples.

Example 3.15. Let L = {0, a, b, c, 1}, where 0 < a < b < c < 1. Consider the operations ⊙, → , ↪ given by the following Cayley tables:

⊙	a	b	c	1
0	0	0	0	0
a	0	0	0	a
b	0	b	b	b
c	a	b	c	c
1	a	b	c	1

→	0	a	b	c	1
0	1	1	1	1	1
a	b	1	1	1	1
b	a	a	1	1	1
c	a	a	b	1	1
1	0	a	b	c	1

↪	0	a	b	c	1
0	1	1	1	1	1
a	c	1	1	1	1
b	a	a	1	1	1
c	0	a	b	1	1
1	0	a	b	c	1

Then L = (L, ∨ , ∧ , ⊙ , → , ↪ , 0, 1) is a non-commutative residuated lattice [11]. Let E = {α, β, γ, δ, η}, A = {α, β}, B = {γ, δ} and for all x ∈ E,

$f_{A} (x) = {\begin{matrix} {0, a, b, 1}, & x = α, \\ {0, c, 1}, & x = β, \\ \emptyset, & other . \end{matrix}$

$h_{B} (x) = {\begin{matrix} {0, b, c, 1}, & x = γ, \\ {0, a, 1}, & x = δ, \\ \emptyset, & other . \end{matrix}$

Then f _A and h _B are two soft sets over L. By routine calculations, {0, a, b, 1} , {0, c, 1} , {0, b, c, 1}, {0, a, 1} are all non-commutative residuated sublattices of L, and so f _A and h _B are two soft non-commutative residuated lattices over L. Writing the ∨-product f _A ∨ h _B = k _C, where C = A × B and ∀ (x, y) ∈ E × E,

$k_{C} (x, y) = {\begin{matrix} L, & x = α, y = γ, \\ {0, a, b, 1}, & x = α, y = δ, \\ {0, b, c, 1}, & x = β, y = γ, \\ {0, a, c, 1}, & x = β, y = δ, \\ \emptyset, & other . \end{matrix}$

Because {0, a, b, 1} , {0, b, c, 1} , {0, a, c, 1} and L are all non-commutative residuated sublattices of L, and so the ∨-product k _C is a soft non-commutative residuated lattice over L.

Example 3.16. In Example 3.9, f _A and h _B are two soft non-commutative residuated lattices over L. Consider the ∨-product f _A ∨ h _B = k _C, where C = A × B and k _C (x, y) = f _A (x) ∪ h _B (y) for all (x, y) ∈ E × E. When x = x ₃, y = x ₃, then k _C (x ₃, x ₃) = f _A (x ₃) ∪ h _B (x ₃) = {0, b, c, 1} is not a non-commutative residuated sublattice of L as c ⊙ b = a ∉ {0, b, c, 1}, and so f _A ∨ h _B = k _C is not a soft non-commutative residuated latticeover L.

Proposition 3.17. Let f_A and h_B be two soft non-commutative residuated lattices over L with A∩ B ≠ ∅. Then the difference $f_{A} \tilde{∖} h_{B}$ is not a soft non-commutative residuated lattice over L.

Proof. Let $f_{A} \tilde{∖} h_{B} = k_{C}$ , where C = A ∩ B and k _C (x) = f _A (x) ∖ h _B (x) for all x ∈ E. Since f _A and h _B are both soft non-commutative residuated lattices over L, then ∀x ∈ C, f _A (x) and h _B (x) are both non-commutative residuated sublattices of L, and so 0, 1 ∈ h _B (x). Since k _C (x) = f _A (x) ∖ h _B (x), 0, 1 ∉ k _C (x). It means that k _C (x) is not a non-commutative residuated sublattice of L. Therefore the difference $f_{A} \tilde{∖} h_{B} = k_{C}$ is not a soft non-commutative residuated lattice over L.

Next, we investigate the homomorphism properties of soft non-commutative residuated lattices.

Let X and Y be two non-commutative residuated lattices. A mapping φ : X → Y is called a homomorphism, if φ (0) =0, φ (1) =1, φ (a ∧ b) = φ (a) ∧φ (b) , φ (a∨ b) = φ (a) ∨ φ (b) , φ (a ⊙ b) = φ (a) ⊙ φ (b) , φ (a → b) = φ (a) → φ (b) and φ (a ↪ b) = φ (a) ↪ φ (b) for all a, b ∈ X. A homomorphism φ : X → Y is called a isomorphism, if φ is bijection.

Let φ : X → Y be a mapping of non-commutative residuated lattices. For a soft set f _A over X, φ (f _A) is a soft set over Y, where φ (f _A) : E → P (Y) is defined by φ (f _A) (x) = φ (f _A (x)) for all x ∈ E.

Theorem 3.18. Let φ : X → Y be a homomorphism of non-commutative residuated lattices. If f_A is a soft non-commutative residuated lattice over X, then φ (f_A) is a soft non-commutative residuated lattice over Y.

Proof. For each x ∈ A, we have φ (f _A) (x) = φ (f _A (x)) is a non-commutative residuated sublattice of Y since f _A (x) is a non-commutative residuated sublattice of X and its homomorphic image is also a non-commutative residuated sublattice of Y. Hence φ (f _A) is a soft non-commutative residuated lattice over Y.

Theorem 3.19. Let φ : X → Y be a homomorphism of non-commutative residuated lattices. If f_A and h_B are two soft non-commutative residuated lattice over X with $f_{A} \tilde{<} h_{B}$ , then $φ (f_{A}) \tilde{<} φ (h_{B})$ .

Proof. Since $f_{A} \tilde{<} h_{B}$ , then ∀x ∈ A, f _A (x) is a non-commutative residuated sublattice of h _B (x). And so φ (f _A) (x) = φ (f _A (x)) is a non-commutative residuated sublattice of φ (h _B (x)) = φ (h _B) (x) as φ is a homomorphism. By Theorem 3.18, φ (f _A) and φ (h _B) are both soft non-commutative residuated lattices over Y. Using Definition 3.4, $φ (f_{A}) \tilde{<} φ (h_{B})$ .

Definition 3.20. A soft non-commutative residuated lattice f _A over L is said to be whole if f _A (x) = L for all x ∈ A.

Proposition 3.21. Let φ : X → Y be a homomorphism of non-commutative residuated lattices and f_A a soft non-commutative residuated lattice over X. If φ is onto and f_A is whole, then φ (f_A) is the whole soft non-commutative residuated lattice over Y.

Proof. Suppose that φ is onto and f _A is whole. Then, f _A (x) = X for all x ∈ A, and so φ (f _A) (x) = φ (f _A (x)) = φ (X) = Y for all x ∈ A. It follows from Theorem 3.18 and Definition 3.20 that φ (f _A) is the whole soft non-commutative residuated lattice over Y.

4 Conclusion

Non-commutative residuated lattices are as non-commutative extension of the residuated lattices. So, it is a common structure among algebras associated with logic systems. In this paper, we initiated to apply Cagman and Enginoglu’s new definitions and operations of soft sets to non-commutative residuated lattice structure, defined the soft non-commutative residuated lattices and discussed the related properties. All results in this paper also hold in most of soft logic algebras. In the future, we can apply the method in this paper to other algebraic structure such as groups, rings and BCK/BCI-algebras etc. We also can study soft filter theory of soft non-commutative residuated lattices, and investigate relations between various kinds soft filters of soft non-commutative residuated lattices.

Acknowledgments

The authors are very grateful to referees for their valuable comments and suggestions for improving this paper.

This work was supported by the National Natural Science Foundation of China (Grant No. 11461025, 61175055) and the Fujian Province Natural Science Foundations of China (Grant No. 2013J01017).

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0	0	0	0	0	0	0
a	a	a	a	a	a	a
b	a	a	a	a	a	b
c	a	a	c	c	c	c
d	a	a	c	c	c	d
e	a	b	c	d	e	e
1	a	b	c	d	e	1

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0	1	1	1	1	1	1	1
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c	0	b	b	1	1	1	1
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0	1	1	1	1	1	1	1
a	0	1	1	1	1	1	1
b	0	e	1	e	1	1	1
c	0	b	b	1	1	1	1
d	0	b	b	e	1	1	1
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0	0	0	0	0	0	0
a	a	a	a	a	a	a
b	a	a	a	a	a	b
c	a	a	c	c	c	c
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1	a	b	c	d	e	1

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0	1	1	1	1	1	1	1
a	0	1	1	1	1	1	1
b	0	d	1	d	1	1	1
c	0	b	b	1	1	1	1
d	0	b	b	d	1	1	1
e	0	b	b	d	d	1	1
1	0	a	b	c	d	e	1

↪	0	a	b	c	d	e	1
0	1	1	1	1	1	1	1
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0	0	0	0	0	0	0
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b	a	a	a	a	a	b
c	a	a	c	c	c	c
d	a	a	c	c	c	d
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1	a	b	c	d	e	1

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0	1	1	1	1	1	1	1
a	0	1	1	1	1	1	1
b	0	d	1	d	1	1	1
c	0	b	b	1	1	1	1
d	0	b	b	d	1	1	1
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1	0	a	b	c	d	e	1

↪	0	a	b	c	d	e	1
0	1	1	1	1	1	1	1
a	0	1	1	1	1	1	1
b	0	e	1	e	1	1	1
c	0	b	b	1	1	1	1
d	0	b	b	e	1	1	1
e	0	a	b	c	d	1	1
1	0	a	b	c	d	e	1