A New Type of Soft Subincline of Incline

Abstract

Firstly, this paper presents the new concept of soft subincline. Then some equivalent conditions of it and two operations “RESTRICTED INTERSECT” and “AND” on it are discussed. After that the relationship between soft subincline and the dual of soft set based on the method of the dual of soft set are studied. In addition, the concepts and properties of maps between soft subincline are given. Finally, the chain condition of H which consists of all of the soft subinclines is introduced and obtain a necessary and sufficient condition for H is Artinian or Noetherian.

1 Introduction

As the fuzzy set theory was proposed, mathematical tools dealing with incomplete and uncertain problems were also presented. In particular, Pawlak and Atanassov proposed the rough set theory [1] and intuitionistic fuzzy set theory [2]. However, these mathematical theories are lack of parameter tools. Therefore, in order to solve this problem, Molodtsov gave the concept of soft sets innovatively in 1999 [3]. After that, Aktas and Cagman propose the definition of soft group in 2007 [4] which created a new field of soft algebra. A few years afterwards, many scholars had done a series of researches in soft algebra [5,6,7,8,9,10,11,12,13].

The notion of incline algebra was proposed by Cao in 1981 [14]. He also published a monograph about incline algebra with other two scholars [15]. In 2001, Jun applied fuzzy sets to incline algebra and proposed the concept of fuzzy subincline [16].

Liao applied soft sets to incline algebra and proposed the concept of soft incline in 2012 [17]. The concept of fuzzy soft incline and $( \in , \in \vee q) -$ fuzzy soft incline were proposed by Alshehri in 2012 [18]. The study of inclines and incline matrices is significant both in theory and in practice, they have good foreground of applications in many areas including automation theory, decision theory, cybernetics, graph theory and nervous system [15]. At present, the theories of incline algebras and incline matrices are highly utilized by computer science applications [19,20,21].

In 2008, Yuan and Wen introduced algebraic structures in parameter set and obtained a new algebraic structure of soft sets [22]. They introduced a soft algebra structure which can be reduced to L-fuzzy algebra by using the concept of dual soft sets, where L = P(X) (P (X) is the power set of the common universe X) is a Boolean algebra. In general the element in lattice L has no structure. However, the elements over L = P(X) is a set which can also have many elements and algebraic structure. Therefore, more meaningful results can be obtained than general L-fuzzy algebra.

In this paper, by using the idea above, we give the new concept of soft subincline. The difference between our new concept of soft subincline and the concept of soft incline proposed in literature [17] is that: the parameter set of a soft subincline is a fixed incline in this paper, while it is a subincline of a certain incline in literature [17]. Furthermore, we investigate some algebraic properties of our new type of soft subincline and introduced some properties of the new type of soft subincline of incline under the chain condition. These results enrich the theory of soft algebra.

2 Preliminary Notes

Definition 2.1

[14]. An inline (algebra) is a set $K$ with two binary operations denoted by “$\cdot$” and “$\cdot$” Satisfying the following axioms: for all $x,y,z \in K$,

• (1) $x + y = y + x$;

• (2) $(x + y) + z = x + (y + z)$;

• (3) $(xy)z = x(yz)$;

• (4) $x(y + z) = xy + xz$;

• (5) $(y + z)x = yx + zx$;

• (6) $x + x = x$;

• (7) $x + xy = x$;

• (8) $y + xy = y$.

For convenience, we pronounce “$+$” (resp.”$\cdot$”) as addition (resp. multiplication).

Every distributive lattice is an incline. An incline is a distributive lattice if and only if $xx = x$ for all $x \in K$.

Note that $x \le y \Leftrightarrow x + y = y$ for all $x,y \in K$.

A subincline of an incline $K$ is a non-empty subset $M$ of $K$ which is closed under addition and multiplication. $A$ subincline $M$ is said to be an ideal of an incline $K$ if $x \in M$ and $y \le x$ then $y \in M$. By a homomorphism of inclines we shall mean a mapping $f$ from an incline $K$ into an incline $L$ such that $f(x + y) = f(x)f(y)$ and $f(xy) = f(x)f(y)$ for all $x,y \in K$.

Definition 2.2

(Cartesian product). Let $A$ and $B$ be two non-empty set, then $A \times B = \{ (x,y)|x \in A,y \in B\}$ is called a Cartesian product over $A$ and $B$.

Theorem 2.1

[14]. Let $K_{1}$ and $K_{2}$ be incline algebras, then their Cartesian product is an incline algebra if for all $(x_{1} ,x_{2} ),(y_{1} ,y_{2} ) \in K_{1} \times K_{2}$:

$$ \begin{aligned} (x_{1} ,x_{2} ) + (y_{1} ,y_{2} ) & = (x_{1} + y_{1} ,x_{2} + y_{2} ), \\ (x_{1} ,x_{2} )(y_{1} ,y_{2} ) & = (x_{1} y_{1} ,x_{2} y_{2} ). \\ \end{aligned} $$

Definition 2.3

[14]. A pair $(F,A)$ is called a soft set (over $X$) if and only if $F$ is a mapping of $E$ into the set of all subsets of the set $X$.

Definition 2.4

[23] (Restricted intersection operation of two soft sets). Let $(F,A)$ and $(G,B)$ be two soft sets over a common universe $X$. If the soft set $(H,C)$ satisfy $C = A \cap B$ and for any $e \in C,H(e) = F(e) \cap G(e)$. We call $(H,C)$ is the restricted intersection of $(F,A)$ and $(G,B)$, and denote $(H,C){ = }(F,A) \cap (G,B)$.

Definition 2.5

[24] (AND operation on two soft sets). Let $(F,A)$ and $(G,B)$ be two soft sets, then “$(F,A)$ and $(G,B)$” denoted by $(F,A) \wedge (G,B)$ is defined to be $(F,A) \wedge (G,B) = (H,A \times B)$, where $H(\alpha ,\beta ) = F(\alpha ) \cap G(\beta ),\forall (\alpha ,\beta ) \in A \times B$.

Definition 2.6

[22] (The duality of soft sets)

$A_{H} :X \to P(E),x \mapsto A_{H} = \{ g|x \in H(g)\}$ is called the duality soft set of $H$ if $H:E \to P(X),g \mapsto H(g)$ is a soft set over $K$.

$H_{A} :E \to P(X),g \mapsto H_{(g)} = \{ x|g \in A(x)\}$ is called the duality soft set of $A$ if $A:X \to P(E)$ is a soft set over $X$.

Definition 2.7

[22] (The Extension Principle) let $X$ be the common universe. Let $f$ be defined by $f:K_{1} \to K_{2}$ and let $H_{1} :K_{1} \to P(X)$ and $H_{2} :K_{2} \to P(X)$ are soft sets over $K_{1}$ and $K_{2}$ respectively. Define soft sets $f(H_{1} )$ over $K_{1}$ and $f^{ - 1} (H_{2} )$ over $K_{2}$ by $\forall g_{2} \in K_{2}$, $f(H_{1} )(g_{2} ) = \left\{ {\begin{array}{*{20}c} {\mathop \cup \limits_{{f(g_{1} ) = g_{2} }} H_{1} (g_{1} )\;f^{ - 1} (g_{2} ) \ne \varnothing } \\ {\varnothing \quad f^{ - 1} (g_{2} ) = \varnothing } \\ \end{array} } \right.$ and $\forall g_{1} \in K_{1} ,f^{ - 1} (H_{2} )(g_{1} ) = H_{2}$. Then $f(H_{1} )$ is said to be the image of $H_{1}$ and $f^{ - 1} (H_{2} )$ is said to be the preimage of $K_{2}$.

Definition 2.8

[17]. Let $K$ be an incline algebra. A pair $(F,A)$ is called a soft incline over $K$ if $F(x)$ is a subincline of $K$ for all $x \in A$.

3 A New Type of Soft Subincline of Incline

Definition 3.1.

Let $K$ be an incline algebra and $X$ be the common universe, $H:K \to P(X)$ is a soft set. $H$ is called a new type of soft subincline of incline if it satisfies the following conditions: for all $g_{1} ,g_{2} \in K$,

• (1)
  $$ H(g_{1} + g_{2} ) \supseteq H(g_{1} ) \cap H(g_{2} ); $$
• (2)
  $$ H(g_{1} g_{2} ) \supseteq H(g_{1} ) \cap H(g_{2} ). $$

Example 3.1.

Let $K = \{ 0,a,b,1\}$ be an incline with the operation tables given in Table. Let $X = \{ 0,a,b,1\}$ and $H:K \to P(X)$ be a soft set defined $H(0) = \{ 0,a\}$ $H(a) = \{ 0,1\} ,H(b) = \{ a,b\} ,H(1) = \{ b\}$. Clearly, $H$ is a new type of soft subincline of incline over $K$ and it could be verified by Definition 2.1. Because $H(b) = \{ a,b\}$ is not a subincline of $K$, so $H$ is not a soft incline over $K$, then the new type of soft subincline is a new algebraic structure.

The operation tables of the incline

Theorem 3.1.

Let $K_{1}$ and $K_{2}$ be two subinclines of $K$ and let $H_{1}$ and $H_{2}$ be new type of soft subinclines of incline over $K_{1}$ and $K_{2}$ respectively, if $K_{1} \cap K_{2} \ne \varnothing$ and $(H_{1} ,K_{1} \cap K_{2} ) = (H,K_{1} ) \cap (H,K_{2} )$, then $H$ is a new type of soft incline of incline over $K_{1} \cap K_{2}$.

Proof.

Because of $K_{1}$ and $K_{2}$ be two subinclines of $K$ and $K_{1} \cap K_{2} \ne \varnothing$, it is easy to say that $K_{1} \cap K_{2}$ is also a subincline of $K$. For any $g_{1} ,g_{2} \in K_{1} \cap K_{2}$, we have

$$ \begin{aligned} H(g_{1} g_{2} ) & = H_{1} (g_{1} g_{2} ) \cap H_{2} (g_{1} g_{2} ) \supseteq [H_{1} (g_{1} ) \cap H_{1} (g_{2} )] \cap [H_{2} (g_{1} ) \cap H_{2} (g_{2} )] \\ & = [H_{1} (g_{1} ) \cap H_{2} (g_{1} )] \cap [H_{1} (g_{2} ) \cap H_{2} (g_{2} )] = H(g_{1} ) \cap H(g_{2} ). \\ \end{aligned} Z $$

Similarly, $H(g_{1} + g_{2} ) \supseteq H(g_{1} ) \cap H(g_{2} ),\forall g_{1} ,g_{2} \in K_{1} \cap K_{2}$.

Therefore, $H$ is a new type of soft subincline of incline of $K_{1} \cap K_{2}$.

Theorem 3.2.

Let $K_{1}$ and $K_{2}$ be two inclines and let $H_{1}$ and $H_{2}$ be new type of soft subinclines of incline over K₁ and K₂ respectively, if $K = K_{1} \times K_{2}$ and $(H,K) = (H_{1} ,K_{1} ) \wedge (H_{2} ,K_{2} )$, then $H$ is a new type of soft subincline of incline of $K$.

Proof.

Since $K_{1}$ and $K_{2}$ are two inclines, by Theorem 2.1, it is sufficient to show that $K_{1} \times K_{2}$ is also an incline. Then clearly $\begin{aligned} H[(x_{1} ,y_{1} )(x_{2} ,y_{2} )] & = H(x_{1} x_{2} ,y_{1} y_{2} ){ = }H_{1} (x_{1} x_{2} ) \cap H_{2} (y_{1} y_{2} ) \supseteq [H_{1} (x_{1} ) \cap H_{1} (x_{2} )] \cap [H_{2} (y_{1} ) \cap H_{2} (y_{2} )] \\ & = [H_{1} (x_{1} ) \cap H_{2} (y_{1} )] \cap [H_{1} (x_{2} ) \cap H_{2} (y_{2} )]{ = }H(x_{1} ,y_{1} ) \cap H(x_{2} ,y_{2} ) \\ \end{aligned}$ for all $(x_{1} ,y_{1} ),(x_{2} ,y_{2} ) \in K$.

Similarly, $H[(x_{1} ,y_{1} ) + (x_{2} ,y_{2} )] \supseteq H(x_{1} ,y_{1} ) \cap H(x_{2} ,y_{2} )$ for all $(x_{1} ,y_{1} )$, $(x_{2} ,y_{2} ) \in K$.

Therefore, $H$ is a new type of soft subincline of incline of $K$.

Theorem 3.3.

Let $K$ be an incline, then the following are equivalent:

• (i) $H$ is a new type of soft subincline of incline of $K$.

• (ii) For all $x \in X,A_{H} (x) \ne \emptyset$ is a subincline over $K$.

Proof.

$(i) \Rightarrow (ii)$ For all $g_{1} ,g_{2} (x) \in A_{H} (x)$, we have $x \in H(g_{1} )$ and $x \in H(g_{2} )$, therefore $x \in H(g_{1} ) \cap H(g_{2} )$. Since $H$ is a new type of soft subincline of incline of $K$, it follows that $H(g_{1} ) \cap H(g_{2} ) \subseteq H(g_{1} g_{2} )$, then $x \in H(g_{1} g_{2} )$. Hence $g_{1} g_{2} \in A_{H} (x)$.

Similarly, $g_{1} + g_{2} \in A_{H} (x)$ for all $g_{1} ,g_{2} \in A_{H} (x)$.

Therefore, $A_{H} (x)$ is a subincline over $K$ for all $x \in X$.

$(ii) \Rightarrow (i)$: For all $g_{1} + g_{2} \in K$, if $H(g_{1} ) \cap H(g_{2} ) = \varnothing$ , then $H(g_{1} ) \cap H(g_{2} ) = \varnothing \subseteq H(g_{2} )$; if $H(g_{1} ) \cap H(g_{2} ) \ne \varnothing$, assume that $x \in H(g_{1} ) \cap H(g_{2} )$, then $x \in H(g_{1} )$ and $x \in H(g_{2} )$, hence $g_{1} ,g_{2} \in A_{H} (x)$ Note that $A_{H} (x)$ is a subincline over $K$, then $g_{1} g_{2} \in A_{H} (x)$ and so $x \in H(g_{1} g_{2} )$. Thus $H(g_{1} ) \cap H(g_{2} ) \subseteq H(g_{1} g_{2} )$.

Similarly, $H(g_{1} ) \cap H(g_{2} ) \subseteq H(g_{1} + g_{2} )$ for all $x \in H(g_{1} ) \cap H(g_{2} )$.

Therefore, $H$ is a new type of soft subincline of incline of $K$.

Theorem 3.4.

Let $K$ be an incline, then the following are equivalent:

• (i) Let $A$ be defined by $A:X \to P(K)$, for all $x \in X,A_{H} (x) \ne \varnothing$ is a subincline over $K$.

• (ii) $H_{A}$ is a new type of soft subincline of incline over $K$.

Proof.

$(i) \Rightarrow (ii)$: Assume that $x \in H_{A} (g_{1} ) \cap H_{A} (g_{2} )$, then $g_{1} \in A(x)$ and $g_{2} \in A(x)$. Note that $A(x)$ is a subincline of $K$, then $g_{1} g_{2} \in A(x)$ and $g_{1} + g_{2} \in A(x)$. Clearly $x \in H_{A} (g_{1} g_{2} )$ and $x \in H_{A} (g_{1} + g_{2} )$. Thus $H_{A} (g_{1} ) \cap H_{A} (g_{2} ) \subseteq H_{A} (g_{1} g_{2} )$ and $H_{A} (g_{1} ) \cap H_{A} (g_{2} ) \subseteq H_{A} (g_{1} g_{2} )$.

Therefore, $H_{A}$ is a new type of soft subincline of incline over $K$.

$(ii) \Rightarrow (i)$: For any $g_{1} ,g_{2} \in A(x)$, we have $x \in H_{A} (g_{1} )$ and $x \in H_{A} (g_{2} )$, and so $x \in H_{A} (g_{1} ) \cap H_{A} (g_{2} )$. Since $H_{A}$ is a new type of soft subincline of incline over $K$, it follows that $H(g_{1} ) \cap H(g_{2} ) \subseteq H(g_{1} g_{2} )$ and $H(g_{1} ) \cap H(g_{2} ) \subseteq H(g_{1} + g_{2} )$, then $g_{1} g_{2} \in A(x)$ and $g_{1} + g_{2} \in A(x)$. Therefore,$A(x)$ is a subincline over $K$.

Theorem 3.5.

Let $K_{1}$ and $K_{2}$ be two inclines and let $X$ be the common universe. Let $f:K_{1} \to K_{2}$ be a hemimorphic mapping. Let $H_{1} :K_{1} \to P(X)$ and $H_{2} :K_{2} \to P(X)$ be soft sets over $K_{1}$ and $K_{2}$ respectively. Then $f(H_{1} )$ is a new type of soft incline of incline over $K_{2}$ if $H_{1}$ is a new type of soft subincline of incline over $K_{1}$.

Proof.

For all $g_{2} ,g_{2}^{{\prime }} \in K_{2}$.

Case1: Assume $f^{ - 1} (g_{2} ) \ne \varnothing$ and $f^{ - 1} (g_{2}^{{\prime }} ) \ne \varnothing$. if $f(H_{1} )(g_{2} ) \cap f(H_{1} )(g_{2}^{{\prime }} ) = \varnothing$, then $f(H_{1} )(g_{2} ) \cap f(H_{1} )(g_{2}^{{\prime }} ) \subseteq f(H_{1} )(g_{2} + g_{2}^{{\prime }} )$ if $f(H_{1} )(g_{2} ) \cap f(H_{1} )(g_{2}^{{\prime }} ) \ne \varnothing$ then $\forall x \in f(H_{1} )(g_{2} ) \cap f(H_{1} )(g_{2}^{{\prime }} )$ we have $x \in \mathop \cup \limits_{{f(g_{1} ) = g_{2} }} H_{1} (g_{1} )$ and $x \in \mathop \cup \limits_{{f(g^{'}_{1} ) = g^{'}_{2} }} H_{1} (g_{1}^{{\prime }} )$. Thus there exists $g_{1} \in K_{1}$ such that $x \in H_{1} (g_{1} )$ and $f(g_{1} ) = g_{2}$. There also exists $g_{1}^{{\prime }} \in K_{1}$ such that $x \in H_{1} (g_{1}^{{\prime }} )$ and $f(g_{1}^{{\prime }} ) = g_{2}^{{\prime }}$. Hence $x \in H_{1} (g_{1} ) \cap H_{1} (g_{1}^{{\prime }} )$. Since $H_{1}$ is a new type of soft subincline of incline over $K_{1}$, we can get $H_{1} (g_{1} ) \cap H_{1} (g_{1}^{{\prime }} ) \subseteq H_{1} (g_{1} + g_{1}^{{\prime }} )$, then $x \in H_{1} (g_{1} + g_{1}^{{\prime }} )$. Note that $f$ is a homomorphic mapping, so $f(g_{1} { + }g_{1}^{{\prime }} ) = f(g_{1} ) + f(g_{1}^{{\prime }} ){ = }g_{2} + g_{2}^{{\prime }}$ where $g_{1} + g_{1}^{{\prime }} \in K_{1}$. Therefore, $x \in H_{1} (g_{1} + g_{1}^{{\prime }} ) \subseteq \mathop \cup \limits_{{f(g) = g_{2} + g_{2}^{{\prime }} }} H_{1} (g) = f(H_{1} )(g_{2} + g_{2}^{{\prime }} )$.

Case2: If $f^{ - 1} (g_{2} ) = \varnothing$ or $f^{ - 1} (g^{\prime}_{2} ) = \varnothing$, then $f(H_{1} )(g_{2} ) = \varnothing$ or $f(H_{1} )(g_{2}^{{\prime }} ) = \varnothing$, and so $f(H_{1} )(g_{2} ) \cap f(H_{1} )(g_{2}^{{\prime }} ) = \emptyset \subseteq f(H_{1} )(g_{2} + g_{2}^{{\prime }} )$.

Similarly, $f(H_{1} )(g_{2} ) \cap f(H_{1} )(g_{2}^{{\prime }} ) \subseteq f(H_{1} )(g_{2} g_{2}^{{\prime }} ),\forall g_{2} ,g_{2}^{{\prime }} \in K_{2}$.

Therefore, $f(H_{1} )$ is a new type of soft incline of incline over $K_{2}$ .

Theorem 3.6.

Let $K_{1}$ and $K_{2}$ be two inclines and let $X$ be the common universe. Let be $f:K_{1} \to K_{2}$ be a homomorphic mapping. $H_{1} :K_{1} \to P(X)$ and $H_{2} :K_{2} \to P(X)$ are soft sets over $K_{1}$ and $K_{2}$ respectively. Then $f^{ - 1} (H_{2} )$ is a new type of soft subincline of incline of $K_{1}$, if $H_{2}$ is a new type of soft subincline of incline over $K_{2}$.

Proof.

For any $g_{1} ,g_{1}^{{\prime }} \in K_{1}$, we have

$$ \begin{aligned} f^{ - 1} (H_{2} )(g_{1} ) \cap f^{ - 1} (H_{2} )(g_{1}^{{\prime }} ) & = H_{2} (f(g_{1} )) \cap H_{2} (f(g_{1}^{{\prime }} )) \subseteq H_{2} (f(g_{1} ) + f(g_{1}^{{\prime }} )) \\ & = H_{2} (f(g_{1} + g_{1}^{{\prime }} )) = f^{ - 1} (H_{2} )(g_{1} { + }g_{1}^{{\prime }} ). \\ \end{aligned} $$

Similarly, $f^{ - 1} (H_{2} )(g_{1} ) \cap f^{ - 1} (H_{2} )(g_{1}^{{\prime }} ) \subseteq f^{ - 1} (H_{2} )(g_{1} g_{1}^{{\prime }} ),\forall g_{1} ,g_{1}^{{\prime }} \in K_{1}$.

Therefore, $f^{ - 1} (H_{2} )$ is a new type of soft subincline of incline of $K_{1}$.

Definition 3.2.

Let $K_{1}$ and $K_{2}$ be two inclines and let $H_{1}$ be a new type of soft subincline of incline over $K_{1}$. Let $f:K_{1} \to K_{2}$ be a map. For all $x,y \in K_{1}$, if $f(x) = f(y)$, we have $H_{1} (x) = H_{1} (y)$, then $H_{1}$ is said to be f-invariant.

Theorem 3.7.

Let $K_{1}$ and $K_{2}$ be two inclines and let $X$ be the common universe. If $f$ is a homomorphic mapping from $K_{1}$ to $K_{2}$, $H_{1} :K_{1} \to P(X)$ is a soft set over $K_{1}$ and $H_{1}$ is f-invariant. Then the following are equivalent:

• (i) $H_{1}$ is a new type of soft subincline of incline over $K_{1}$.

• (ii) $f(H_{1} )$ is a new type of soft subincline of incline over $K_{2}$.

Proof.

$(i) \Rightarrow (ii)$: Following Theorem 3.5, it is sufficient to show that the conclusion is correct.

$(ii) \Rightarrow (i)$: For any $g_{1} ,g_{1}^{{\prime }} \in K_{1}$ and $x \in H_{1} (g_{1} ) \cap H_{1} (g_{1}^{{\prime }} )$, we have $x \in H_{1} (g_{1} )$ and $x \in H_{1} (g_{1}^{{\prime }} )$. Assume that $f(g_{1} ) = g_{2}$ and $f(g_{1}^{{\prime }} ) = g_{2}^{{\prime }} \in K_{2}$, then $x \in \mathop \cup \limits_{{f(g) = g_{2} }} H_{1} (g) = f(H_{1} )(g_{2} )$ and $x \in \mathop \cup \limits_{{f(g^{{\prime }} ) = g_{2}^{{\prime }} }} H_{1} (g) = f(H_{1} )(g_{2}^{{\prime }} )$, and so $x \in f(H_{1} )(g_{2} ) \cap f(H_{1} )(g_{2}^{{\prime }} )$. since $f(H_{1} )$ is a new type of soft subincline of incline over $K_{2}$, hence $f(H_{1} )(g_{2} ) \cap f(H_{1} )(g_{2}^{{\prime }} ) \subseteq f(H_{1} )(g_{2} g_{2}^{{\prime }} )$, then $x \in f(H_{1} )(g_{2} g_{2}^{{\prime }} ) = \mathop \cup \limits_{{f(g) = g_{2} g_{2}^{{\prime }} }} H_{1} (g)$

So clearly there exists $g \in K_{1}$ such that $f(g) = g_{2} g_{2}^{{\prime }}$ and $x \in H_{1} (g)$. Since $f$ is a homomorphic mapping, then $f(g) = f(g_{1} )f(g_{1}^{{\prime }} ) = f(g_{1} g_{1}^{{\prime }} )$. Also note that $H_{1}$ is a f-invariant, then $H_{1} (g) = H_{1} (g_{1} g_{1}^{{\prime }} )$, so $x \in H_{1} (g_{1} g_{1}^{{\prime }} )$.

Hence $H_{1} (g_{1} ) \cap H_{1} (g_{1}^{{\prime }} ) \subseteq H_{1} (g_{1} g_{1}^{{\prime }} )$.

Similarly, $H_{1} (g_{1} ) \cap H_{1} (g_{1}^{{\prime }} ) \subseteq H_{1} (g_{1} + g_{1}^{{\prime }} ),\forall g_{1} ,g_{1}^{{\prime }} \in K$.

Therefore, $H_{1}$ is a new type of soft subincline of incline of $K_{1}$.

Theorem 3.8.

Let $K_{1}$ and $K_{2}$ be two inclines and let $X$ be the common universe. If $f$ is a homomorphic mapping from $K_{1}$ to $K_{2}$ and $H_{2} :K_{2} \to P(X)$ is a soft set over $K_{2}$. Then the following are equivalent:

• (i) $H_{2}$ is a new type of soft subincline of incline over $K_{2}$.

• (ii) $f^{ - 1} (H_{2} )$ is a new type of soft subincline of incline over $K_{1}$.

Proof.

$(i) \Rightarrow (ii)$: Following Theorem 3.6, it is sufficient to show that the conclusion is correct.

$(ii) \Rightarrow (i)$: For any $g_{2} ,g_{2}^{{\prime }} \in K_{2}$, note that $f$ is a homomorphic mapping, so there exists $g_{1} ,g_{1}^{{\prime }} \in K_{1}$ such that $g_{2} = f(g_{1} ),g_{2}^{{\prime }} = f(g_{1}^{{\prime }} )$ and $g_{2} g_{2}^{{\prime }} = f(g_{1} )f(g_{1}^{{\prime }} ) = f(g_{1} g_{1}^{{\prime }} )$, then $H_{2} (g_{2} ) \cap H_{2} (g_{2}^{{\prime }} ) = H_{2} (f(g_{1} )) \cap H_{2} (f(g_{1}^{{\prime }} )) = f^{ - 1} (H_{2} )(g_{1} ) \cap f^{ - 1} (H_{2} )(g_{1}^{{\prime }} )$ since $f^{ - 1} (H_{2} )$ is a new type of soft subincline of incline over $K_{1}$, we get $f^{ - 1} (H_{2} )(g_{1} ) \cap f^{ - 1} (H_{2} )(g_{1}^{{\prime }} ) \subseteq f^{ - 1} (H_{2} )(g_{1} g_{1}^{{\prime }} ) = H_{2} (f(g_{1} g_{1}^{{\prime }} )) = H_{2} (g_{1} g_{1}^{{\prime }} )$. Hence $H_{2} (g_{2} ) \cap H_{2} (g_{2}^{{\prime }} ) \subseteq H_{2} (g_{2} g_{2}^{{\prime }} )$.

Similarly, $H_{1} (g_{2} ) \cap H_{1} (g_{2}^{{\prime }} ) \subseteq H_{1} (g_{2} + g_{2}^{{\prime }} ),\forall g_{2} ,g_{2}^{{\prime }} \in K_{2}$.

Therefore, $H_{2}$ is a new type of soft subincline of incline over $K_{2}$.

4 The Chain Condition of Incline

Definition 4.1.

Let $K$ be an incline algebra and $H$ is the set of all new type of soft subincline of incline over $K$. Suppose $H_{1}$ and $H_{2}$ are elements of $H$. Define a binary relation “$\le$” over $H$ as follows: $H_{1} \le H_{2} \Leftrightarrow H_{1} (g) \subseteq H_{2} (g),\forall g \in K$.

Definition 4.2.

Let $K$ be an incline algebra, $H$ is the set of all new type of soft subincline of incline over $K,H_{1}$ and $H_{2}$ are elements of $H$. Define a binary relation “$=$” over $H$ as follows: $H_{1} = H_{2} \Leftrightarrow H_{1} (g) = H_{2} (g),\forall g \in K$.

Theorem 4.1.

$(H, \le )$ is a partially ordered set.

Proof.

For any $H_{1} \in H$, we have $H_{1} (g) \in H_{1} (g)$ for all $g \in K$, hence $H_{1} \le H_{1}$.

For any $H_{1} ,H_{2} ,H_{3} \in H$, assume that $H_{1} \le H_{2}$ and $H_{2} \le H_{3}$. Clearly for any $g \in K$ we have $H_{1} (g) \subseteq H_{2} (g)$ and $H_{2} (g) \subseteq H_{3} (g)$, then $H_{1} (g) \subseteq H_{3} (g)$. Hence $H_{1} \le H_{3}$.

For any $H_{1} ,H_{2} \in H$, assume that $H_{1} \le H_{2}$ and $H_{2} \le H_{1}$. Clearly for any $g \in K$, we have $H_{1} (g) \subseteq H_{2} (g)$ and $H_{2} (g) \subseteq H_{1} (g)$, then $H_{2} (g) = H_{1} (g)$.

Hence $H_{2} = H_{1}$.

Therefore, $(H, \le )$ is a partially ordered set.

Theorem 4.2.

$K$ is an incline, $H_{1}$ and $H_{2}$ are new type of soft subinclines of incline over $K$. Then $H_{1} \le H_{2}$ if and only if $A_{{H_{1} }} (x) \subseteq A_{{H_{2} }} (x)$ for all $x \in X$.

Proof.

Necessity: For any $g \in A_{{H_{1} (x)}}$, we have $H_{1} (g) \subseteq H_{1} (g)$. Since $H_{1} \le H_{2}$, then $H_{1} (g) \subseteq H_{2} (g)$. This implies that $x \in H_{2} (g)$, so that $g \in A_{{H_{2} }} (x)$. Therefore $A_{{H_{1} }} (x) \subseteq A_{{H_{2} }} (x)$.

Sufficiency: For any $g \in K$ and $x \in H_{1} (g)$, we have $g \in A_{{H_{1} }} (x)$. Since $A_{{H_{1} }} (x) \subseteq A_{{H_{2} }} (x)$ then $g \in A_{{H_{2} }} (x)$. This implies that $x \in H_{2} (g)$, so that $H_{2} (g) \subseteq H_{2} (g)$. Therefore $H_{1} \le H_{2}$.

Corollary 4.1.

$K$ is an incline, $H_{1}$ and $H_{2}$ are new type of soft subinclines of incline over $K$. Then $H_{1} = H_{2}$ if and only if $A_{{H_{1} }} (x) = A_{{H_{2} }} (x)$ for all $x \in X$.

Definition 4.3.

$K$ is an incline, $\Omega (K)$ is a subincline family of $K$. For any ascending chain of $\Omega (K),K_{1} \subseteq K_{2} \subseteq \cdots \subseteq K_{n} \subseteq \cdots$, if there exists a positive integer n such that $K_{n} = K_{m}$ for all m > n, $\Omega (K)$ is called Noetherian. $K$ is called a Noetherian incline if $\Omega (K)$ is the set of all subinclines over $K$. The number min $\{ i|K_{i} = K_{i + 1} ,i = 1,2, \cdots \}$ is called the stabilize index of Noetherian and denoted by $m_{{\{ K_{i} \} }}$.

Definition 4.4.

$K$ is an incline, $\Omega (K)$ is a subincline family of $K$. For any descending chain of $\Omega (K),K_{1} \supseteq K_{2} \supseteq \cdot \cdot \cdot \supseteq K_{n} \supseteq \cdots$, if there exists a positive integer n such that $K_{n} = K_{m}$ for all m > n, $\Omega (K)$ is called Artinian. $K$ is called a Artinian incline if $\Omega (K)$ is the set of all subinclines over $K$. The number min $\{ i|K_{i} = K_{i + 1} ,i = 1,2, \cdots \}$ is called the stabilize index of Artinian and denoted by $n_{{\{ K_{i} \} }}$.

Definition 4.5.

$K$ is an incline, ${\sum }$ is the set of all subinclines over $K$. $K$ is said to satisfy the maximal condition if every nonempty set over ${\sum }$ has a maximal element.

Definition 4.6.

$K$ is an incline, ${\sum }$ is the set of all subinclines over $K$. $K$ is said to satisfy the minimal condition if every nonempty set over ${\sum }$ has a minimal element.

Theorem 4.3.

$K$ is an incline, $K$ is a Noetherian incline if and only if $K$ satisfies the maximal condition.

Proof.

Necessity: Let $\sum$ be the set of all subinclines over $K$. Assume that nonempty subset $\sum^{{\prime }}$ over $\sum$ without maximal element, then for any $K_{i} \in \sum^{{\prime }}$, there exists $K_{i + 1}$

such that $K_{i} \subseteq K_{i + 1}$. Hence there exists an infinite ascending chain $K_{1} \subseteq K_{2} \subseteq \cdots \subseteq K_{i} \subseteq K_{i + 1} \subseteq \cdots$, a contradiction. Therefore, $K$ satisfies the maximal condition.

Theorem 4.4.

$K$ is an incline, $K$ is an Artinian incline if and only if $K$ satisfy the minimal condition.

Proof.

The proof of this theorem is similar to the proof of Theorem 4.3 and so is omitted.

Definition 4.7.

$K$ is an incline, let $A$ be a subincline over $K$. $A$ is said to be reducible if there exist $B$ and $C$, which are subincline over $K$, properly including $A$, such that $A = B \cap C$. If $A = B \cap C$, there must have $A = B$ or $A = C$, then $A$ is said to be irreducible.

Theorem 4.5.

$K$ is a Noetherian incline, then every subincline over $K$ can be expressed as intersectiob of finite number of subinclines which are irreducible.

Proof.

Let ${\sum }_{1}$ be the set of subinclines over $K$ which are cannot be expressed as intersection of a finite number of irreducible subinclines. Assume that $\sum_{1} \ne \varnothing$. Since $K$ is a Noetherian incline, clearly we know $K$ satisfies the maximal condition. Hence there exists a maximal element in $\sum_{1}$ and denoted by $a$. Because the elements in ${\sum }_{1}$ are reducible, there exist $b$ and $c$, which are subinclines over $K$, such that $a = b \cap c$ where $a \subset b$ and $a \subset c$. For $a$ is the maximal element, then $b,c \notin \sum_{1}$, and so $b,c$ can be expressed as intersection of a finite number of irreducible subinclines, denoted by $b = b_{1} \cap \cdots \cap b_{m} ,c = c_{1} \cap \cdots \cap c_{n}$ (the $b_{i} ,c_{j}$ are irreduclble). Clearly, $a = b \cap c = b_{1} \cap \cdots \cap b_{m} \cap c_{1} \cap \cdots \cap c_{n}$. $a$ can be expressed as intersection of a finite number of irreducible subinclines, a contradiction. So we have ${\sum }_{1} = \emptyset$, hence every subincline over $K$ can be expressed as intersection of finite number of subinclines which are irreducible.

Definition 4.8.

$K$ is a Noetherian incline, $H$ is the set of all new type of soft subinclines of incline over $K$. For any ascending chain of new type of soft subinclines of incline $H_{1} \le H_{2} \le H_{3} \le \cdots$, if there exists a positive integer $n$ such that $H_{m} = H_{n}$ for all $m > n,H$ is said to have the ascending chain condition, or we say $H$ is Noetherian.

Definition 4.9.

$K$ is an incline, $H$ is the set of all new type of soft subinclines of incline over $K$. For any descending chain of new type of soft subinclines of incline $H_{1} \ge H_{2} \ge H_{3} \ge \cdots$, if there exists a positive integer $n$ such that $H_{m} = H_{n}$ for all $m > n,H$ is said to have the descending chain condition, or we say $H$ is Artinian.

Theorem 4.6.

$K$ is an incline, $H$ is the set of all new type of soft subinclines of incline over $K$. $H$ is Noetherian if and only if $\Omega (K)(x)\mathop = \limits^{\Delta } \{ A_{{H_{i} }} (x)|H_{i} \in H\}$ is Noetherian for all $x \in X$ and ${ \sup }\{ m_{{\{ A_{{H_{i} (x)}} \} }} |x \in X\}$ is finited.

Proof.

Necessity: Following Theorem 3.3, it is sufficient to show that $A_{{H_{i} }} (x)$ is a subincline of $K$. Then $\Omega (K)(x)$ is a subincline family. Let $A_{{H_{1} }} (x) \subseteq A_{{H_{2} }} (x) \subseteq \cdots A_{{H_{n} }} (x) \subseteq \cdots$ be an ascending chain over $\Omega (K)(x)$, according to Theorem 4.2, it follows that $H_{1} \le H_{2} \le \cdots \le H_{n} \le \cdots$ For $H$ is Noetherian, so there exists a positive integer $n$ such that $H_{m} = H_{n}$ for all $m > n$. According to Corollrry 4.1, it follows that $A_{{H_{m} }} (x) = A_{{H_{n} }} (x)$ for all $x \in X$.

Consequently, we infer that $\Omega (K)(x)$ is Noetherian for all $x \in X$ and $\sup \{ m_{{\{ A_{{H_{i} (x)}} \} }} |x \in X\}$ is finited.

Sufficiency: Let $H_{1} \le H_{2} \le \cdots \le H_{n} \le \cdot \cdot \cdot$ be an ascending chain over $H$, according to Theorem 4.2, we have that $A_{{H_{1} }} (x) \subseteq A_{{H_{2} }} (x) \subseteq \cdots A_{{H_{n} }} (x) \subseteq \cdots$ for all $x \in X$. Let $\sup \{ m_{{\{ A_{{H_{i} (x)}} \} }} |x \in X\} = n$, since $\Omega (K)(x)$ is Noetherian, so $A_{{H_{m} }} (x) = A_{{H_{n} }} (x)$ for all $x \in X$ if $m > n$. According to Corollary 4.1, it follows that $H_{m} { = }H_{n}$ if $m > n$. Therefore $H$ is Noetherian.

Theorem 4.7.

$K$ is an incline, $H$ is the set of all new type of soft subinclines of incline over $K$. $H$ is Artinian if and only if $\Omega (K)(x)\mathop = \limits^{\Delta } \{ A_{{H_{i} }} (x)|H_{i} \in H\}$ is Artinian for all $x \in X$ and $\sup \{ n_{{\{ A_{{H_{i} (x)}} \} }} |x \in X\}$ is finited.

Proof.

The proof of this theorem is similar to the proof of Theorem 4.6 and so is omitted.