# A New Approach to Concavity Fuzzification

Authors: Ibtesam Alshammari; Azza M. Alghamdi; A. Ghareeb
Journal: Journal of Mathematics
 (2021)

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## Abstract

In this paper, we introduce a more general approach to the fuzzification of fuzzy concavity. More specifically, the degree ofL,M-fuzzy concavity is introduced and characterized as a generalization of L-concave structure and L,M-fuzzy concave structure. Based on that, the degree of L,M-fuzzy concavity preserving and L,M-fuzzy concave-to-concave of a function are defined. Some properties and relationships between the degree of L,M-fuzzy concavity preserving and L,M-fuzzy concave-to-concave functions are discussed.


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## Body

## 1. Introduction

The concept of convexity is a fundamentally important geometrical property that plays a significant role in pure and applied mathematics. It can be sorted into concrete and abstract convexity. In this paper, we are mainly focusing on the abstract convexity. Fuzzification of the abstract convexity was initiated by Rosa [1–3] in 1994. He introduced fuzzy convex to convex and fuzzy convexity preserving functions and discussed some of their properties. To be worth mentioning, he developed the theory of fuzzy convexity by introducing its subspace, product, and quotient structures.Many researchers have been involved in extending the notion of fuzzy convexity to the broader frame work of lattice-valued setting. In [4], Maruyama extended fuzzy convexity to L-fuzzy setting framework, where L is a completely distributive lattice. Jin and Li [5] proposed two functors between the categories of convex spaces and stratified L-convex spaces, where L is a continuous lattice. Both of the functors have been used to prove the embedding of convex spaces in stratified L-convex spaces as a reflective and coreflective subcategory, where L satisfies the multiplicative condition. Also, stratified L-convex spaces, convex-generated L-convex spaces, weakly induced L-convex spaces, and induced L-convex spaces are introduced and their relationships are discussed category-theoretically by Pang and Shi [6]. In 2016, Pang and Zhao [7] introduced the concept of L-concave spaces, concave L-neighborhood systems, and concave L-interior operators which are the dual concepts of L-convex spaces, convex L-neighborhood systems, and convex L-interior operators. The isomorphism between these categories and the category of L-convex spaces are discussed and studied when L is a completely distributive lattice with an order-reversing involution.Shi and Xiu [8] initiated the concept of M-fuzzifying convex structure as a new approach to convexity fuzzification. Recently, Shi and Li [9] generalized the classical restricted hull operators to M-fuzzifying restricted hull operators and used it to characterize M-fuzzifying convex structures. In [10], Wu and Bai discussed some properties of hull operator and introduced M-fuzzifying JHC property and M-fuzzifying Peano property. Further, Xiu and his colleagues [11–14] verified the categorical relationship between M-fuzzifying convexity and other related spatial structures. Later, Shi and Xiu [15] introduced and characterized the notion of L,M-fuzzy convexity as an extension to L-convexity and M-fuzzifying convexity. In the new structure, each L-fuzzy subset can be regarded as an L-convex set to some degree. M-fuzzifying convex spaces and L,M-fuzzy convex spaces have also been investigated in many studies [16–21].In 1992, Šostak [22] introduced the concept of fuzzy category. In a fuzzy category, the potential objects and morphisms could be such to some degrees. Later, Kubiak and Šostak [23] fuzzified the category of M-valued L-fuzzy topological spaces. In [24], Zhong and Shi characterized the degree to which a function T:LX⟶M is an L,M-fuzzy topology. Moreover, the degree to which an L-subset is an L-open set with respect to T is studied. Further, the degrees of continuity, openness, closeness, and quotient of a function f:X,TX⟶Y,TY with respect to the L,M-fuzzy topologies TX and TY, are given and their properties are characterized. Further, several kinds of continuity, compactness, and connectedness are generalized with their elementary properties to L,M-fuzzy topological spaces setting based on graded concepts (see [25–30]). In [31], Ghareeb et al. studied the L,M-fuzzy measurability in view of degree. Firstly, they generalized L,M-fuzzy σ-algebra by presenting the degree of an L,M-fuzzy σ-algebra with respect to a mapping σ:LX⟶M. Moreover, they defined and discussed some special degrees such as the degree of L,M-fuzzy measurable mapping, L,M-fuzzy measurable-to-measurable mapping, isomorphic mapping, and quotient mapping with respect to mappings between two L,M-fuzzy measurable spaces in detail.The aim of this paper is to present the degree ofL,M-fuzzy concavity as an extension of L,M-fuzzy concave structure. Moreover, we present the degree of L,M-fuzzy concavity preserving functions and L,M-fuzzy concave-to-concave functions. Some properties and relationship between the degree of L,M-fuzzy concavity preserving and L,M-fuzzy concave-to-concave functions are characterized.


## 2. Preliminaries

This section begins with some introductory material onL,M-fuzzy convexity. In the sequel, X refers to a finite set, both L and M denote completely distributive lattices. The zero and the unit elements in L and M are symbolized by ⊥L, ⊥M and ⊤L, ⊤M, respectively. By λii∈Ωdir⊆LX (resp. λii∈Ωcdir⊆LX), we refer to the directed (resp. codirected) subfamily λii∈Ω of LX. For a completely distributive lattice M, there exists a residual implication ⟼:M×M⟶M which is right adjoint for meet operation ∧ and given by(1)r⟼s=∨t∈M|r∧t≤s.Moreover, the operation⟷ is defined by(2)r⟷s=r⟼s∧s⟼r.The following lemma lists some properties of implication operation.Lemma 1.
(see [32]). For any r, s, t∈M, rii∈Ω, and sii∈Ω∈M, we have the following statements:(1)
⊤M⟼r=r(2)
t≤r⟼s⟺r∧t≤s(3)
r⟼s=⊤M⟺r≤s(4)
r⟼∧i∈Ωri=∧i∈Ωr⟼ri, hence r⟼s≤r⟼t whenever s≤t(5)
∨i∈Ωri⟼s=∨i∈Ωri⟼s, hence r⟼t≥s⟼t whenever r≤s(6)
r⟼t∧t⟼s≤r⟼s
An elementr∈M is said to be a prime element if r≤s∧t leads to r≤s or r≤t. Also, r∈M is said to be coprime if r≥s∨t leads to r≥s or r≥t. The collection of all nonunit prime and nonzero coprime elements in M are symbolized by PM and JM, respectively.The binary relation≪ on M is defined as follows: for r, s∈M, r≪s if and only if for every subset B⊆M, the relation s≤supB always leads to the existence of t∈B with r≤t. The family r∈M:r≪s is called the greatest minimal family of s, symbolized by βs, and β∗s=βs∩JM. Moreover, for s∈M, we define αs=r∈M:r′≪s′ and α∗s=αs∩PM. In a completely distributive lattice L, there exist αs and βs for each s∈M, s=∧αs=∨βs, and α⊤M=β⊥M=∅ (see [33–35] for more details).In [7], Pang and Zhao introduced the concept of L-concave spaces, which is a dual concept of L-convex spaces as follows.Definition 1.
(see [7]). An L-concave structure ℭ on a nonempty set X is a subset of LX such that(L1)
⊤¯LX, ⊥¯LX∈ℭ(L2)
∧i∈Ωλi∈ℭ for each λii∈Ωcdir⊆LX(L3)
∨in∈Ωλi∈ℭ for each λii∈Ω⊆LX
The pairX,ℭ is called an L-concave space. A function f:X,ℭX⟶Y,ℭY is called L-concavity preserving if ν∈ℭY implies that fL←ν∈ℭX, and f is called concave-to-concave function if fL⟶λ∈ℭY for each λ∈LX.
The following definition extendsL-concavity to L,M-fuzzy setting.Definition 2.
A functionC:LX⟶M is called an L,M-fuzzy concavity on X if C satisfies the following statements:(C1)
C⊤¯LX=C⊥¯LX=⊤M(C2)
C∧i∈Ωλi≥∧i∈ΩCλi, for every ∅≠λii∈Ωcdir⊆LX(C3)
C∨i∈Ωλi≥∧i∈ΩCλi, for every ∅≠λii∈Ω⊆LX
The pairX,C is called an L,M-fuzzy concave structure. For all λ∈LX, the value Cλ represents the degree to which λ is concave L-subset. For any two L,M-fuzzy concavities C and D on X, we say C is coarser than D (i.e., D is finer than C) if and only if Cλ≤Dλ, for every λ∈LX. For any two L,M-fuzzy concave structures X,CX and Y,CY, the function f:X,CX⟶Y,CY is said to be(1)
AnL,M-fuzzy concavity preserving function if CXfL←ν≥CYν for any ν∈LY(2)
AnL,M-fuzzy concave-to-concave function if CYfL⟶λ≥CXλ for any λ∈LX
Obviously, a2,M-fuzzy concave structure can be viewed as an M-fuzzifying concave structure, where 2=⊤,⊥. Moreover, an L,2-fuzzy concave structure is called an L-concave structure [7]. Further, when L=M=0,1, the L,M-fuzzy concave structure is called a 0,1-fuzzy concave structure. A crisp convex structure can be regarded as a 2,2-fuzzy convex structure.Example 1.
If anL,M-fuzzy cotopology O:LX⟶M (see [36, 37]) satisfies(3)O∨i∈Ωλi≥∧i∈ΩOλi,for every λii∈Ω⊆LX, then O is called saturated L,M-fuzzy cotopology. The pair X,O is called an Alexandroff L,M-fuzzy cotopological space. It can be easily verified that Alexandroff L,M-fuzzy cotopological space is an L,M-fuzzy concave structure.
For eachC:LX⟶M and r∈M, we have the following two cut sets:(4)Cr=λ∈LX:Cλ≥r,Cr=λ∈LX:r∉αCλ.Theorem 1.
LetC:LX⟶M be a function. Then, the following statements are equivalent:(1)
X,C is an L,M-fuzzy concave structure(2)
For anyr∈M⊥M, Cr is an L-concavity on X(3)
For anyr∈α⊥M, Cr is an L-concavity on X
In the following theorem, we assume the existence of an order-reversing involution “′” with the completely distributive latticeM,∨,∧, i.e., M,∨,∧,′ is a completely distributive DeMorgan algebra.Theorem 2.
(see [20, 38, 39]) .The closure (resp. hull) operator coC:LX⟶MJLX of an L,M-fuzzy concave structure X,C is defined by(5)coCλxr=∧xr≰μ≥λCμ′′,∀λ∈LX,∀xr∈JLX.
Then, for everyλ,μ∈LX and xr∈JLX, coC achieves the following statements:(H1)
coC⊥¯LXxr=⊥M(H2)
Ifxr≤λ, then coCλxr=⊤M(H3)
Ifλ≤μ, then coCλ≤coCμ(H4)
coCλxr=∧xr≤μ≥λ∨ys≤μcoCμys
Conversely, letco:LX⟶MJLX be an operator achieving (H1)–(H4) and the function Cco:LX⟶M is given by(6)Ccoλ=∧xr≰λ′coλ′xr′,∀λ∈LX.
Then,Cco is an L,M-fuzzy concave structure.


## 3. The Degree ofL,M-Fuzzy Concavity

In this section, we present and characterize the degree ofL,M-fuzzy concavity. Moreover, M-fuzzifying concavity degree is presented and its characterizations are introduced.Definition 3.
LetC:LX⟶M be a function. Then, CocC defined by(7)CocC=C⊤¯LX∧C⊥¯LX∧∧λii∈Ωcdir⊆LX∧i∈ΩCλi⟼C∧i∈Ωλi∧∧λii∈Ω⊆LX∧i∈ΩCλi⟼C∨i∈Ωλi,is called an L,M-fizzy concavity degree (i.e., the degree to which C is an L,M-fuzzy concavity on X).Remark 1.
(1)
IfCocC=⊤M, then C⊤¯LX=C⊥¯LX=⊤M, ∧i∈ΩCλi≤C∧i∈Ωλi for each λii∈Ωcdir⊆LX and ∧i∈ΩCλi≤C∨i∈Ωλi for each λii∈Ω⊆LX. It is exactly the definition of L,M-fuzzy concave structure on X (see Definition 2). Moreover, C is an L,M-fuzzy concavity if and only if CocC=⊤M.(2)
IfL=⊥,⊤ in Definition 3, then CocC is called an M-fuzzifying concavity degree of C.Example 2.
IfC:LX⟶0,1 is a function such that Cλ=1/2 for any λ∈LX. By Definition 3, we get CocC=1/2≠1.Lemma 2.
LetC:LX⟶M be a function. For any r∈M, r≤CocC if and only if r≤C⊤¯LX, r≤C⊥¯LX, ∧i∈ΩCλi∧r≤C∧i∈Ωλi for each λii∈Ωcdir⊆LX and ∧i∈ΩCλi∧r≤C∨i∈Ωλi for each λii∈ω⊆LX.
The following theorem can be proved using the previous lemma.Theorem 3.
For the functionC:LX⟶M, we have(8)CocC=∨r∈Mr≤C⊤¯LX,r≤C⊥¯LX,∧i∈ΩCλi∧r≤C∧i∈Ωλi,∀λii∈Ωcdir⊆LX,∧i∈ΩCλi∧r≤C∨i∈Ωλi,∀λii∈ω⊆LX..Theorem 4.
LetC:LX⟶M be a function. Then,(9)CocC=∨r∈M|∀s≤r,Csis anL−concavity.Proof.
Letr≤C⊤¯LX, r≤C⊥¯LX, and ∧i∈ΩCλi∧r≤C∧i∈Ωλi for every λii∈Ωcdir⊆LX and ∧i∈ΩCλi∧r≤C∨i∈Ωλi for every λii∈Ω⊆LX. For s≤r, λii∈Ω⊆Cs, and λii∈Ωcdir⊆Cs, we have C⊤¯LX≥s, C⊥¯LX≥s, C∧i∈Ωλi≥∧i∈ΩCλi∧r≥s, and C∨i∈Ωλi≥∧i∈ΩCλi∧r≥s. Therefore, ⊤¯LX, ⊥¯LX∈Cs, ∧i∈Ωλi∈Cs, and ∨i∈Ωλi∈Cs and hence CocC≤R, where R denotes the right hand side of the equality.
Conversely, suppose thatCs is an L-concavity on X for any s≤r. Let s=r, then ⊤¯LX, ⊥¯LX∈Cs, which means C⊤¯LX≥r and C⊥¯LX≥r. For any λii∈Ωcdir⊆LX, let s=∧i∈ΩCλi∧r, then s≤r and λii∈Ωcdir⊆Cs. Thus, ∧i∈Ωλi∈Cs, i.e., C∧i∈Ωλi≥∧i∈ΩCλi∧r. For any λii∈Ω⊆LX, let s=∧i∈ΩCλi∧r, then s≤r and λii∈Ω⊆Cs. Thus, ∨i∈Ωλi∈Cs, i.e., C∨i∈Ωλi≥∧i∈ΩCλi∧r. Hence, CocC≥R.Theorem 5.
LetC:LX⟶M be a function. Then,(10)CocC=∨r∈M|∀s∉αr,Csis anL−concavity.Proof.
Letr≤C⊤¯LX, r≤C⊥¯LX, and ∧i∈ΩCλi∧r≤C∧i∈Ωλi for each λii∈Ωcdir⊆LX and ∧i∈ΩCλi∧r≤C∨i∈Ωλi for each λii∈Ω⊆LX. For any s∉αr, λii∈Ω⊆Cs, and λii∈Ωcdir⊆Cs, we have s∉αr∪∪i∈ΩαCλi. Then,(11)αr∪∪i∈ΩαCλi=αr∧∧i∈ΩCλi⊇αC∧i∈Ωλi,αr∪∪i∈ΩαCλi=αr∧∧i∈ΩCλi⊇αC∨i∈Ωλi.
Thus,s∉αC∧i∈Ωλi and s∉αC∨i∈Ωλi. This shows ∧i∈Ωλi∈Cs and ∨i∈Ωλi∈Cs. Since αr⊇αC⊤¯LX and s∉αC⊥¯LX, we know s∉αC⊤¯LX and s∉αC⊥¯LX, which proves ⊤¯LX, ⊥¯LX∈Cs. Hence, CocC≤R, where R denotes the right hand side of the equality.
Conversely, suppose that for anys∉αr, Cs is an L-concavity and ⊤¯LX, ⊥¯LX∈Cs, i.e., s∉αC⊤¯LX and s∉αC⊥¯LX. Then, αC⊤¯LX⊆αr and αC⊥¯LX⊆αr, i.e., r≤C⊤¯LX and r≤C⊥¯LX. For λii∈Ωcdir⊆LX, take any s∉α∧i∈ΩCλi∧r. By α∧i∈ΩCλi∧r=∪i∈ΩαCλi∪αr, we have s∉αr, s∉αCλi, for all i∈Ω, which means λii∈Ωcdir⊆Cs. Since Cs is an L-concavity, we have ∧i∈Ωλi∈Cs, i.e., s∉αC∧i∈Ωλi. Then, αC∧i∈Ωλi⊆α∧i∈ΩCλi∧r, i.e., ∧i∈ΩCλi∧r≤C∧i∈Ωλi. Similarly, we can show that ∧i∈ΩCλi∧r≤C∨i∈Ωλi for each λii∈Ω⊆LX. Hence, CocC≥R.
The following corollary gives similar characterization ofM-fuzzifying concavity.Corollary 1.
LetC:2X⟶M be a function. Then,(1)
CocC=∨r∈M,r≤CX,r≤C∅,∧i∈ΩCAi∧r≤C∩i∈ΩAi,∀Aii∈Ωcdir⊆2X,∧i∈ΩCAi∧r≤C∪i∈ΩAi,∀Aii∈ω⊆2X(2)
CocC=∨r∈M|∀s≤r,Csis a concavity(3)
CocC=∨r∈M|∀s∉αr,Csis  a  concavity
AnL,M-fuzzy concavity degree CocC can also be treated as a function Coc:MLX⟶M given by C⟶CocC. This function has the following property.Theorem 6.
LetCj|Cj:LX⟶Mj∈J be a family of functions. Then,(12)∧j∈JCocCj≤Coc∧j∈JCj.Proof.
From Definition3, we have(13)Coc∧j∈JCj=∧j∈JCj⊤¯LX∧∧j∈JCj⊥¯LX∧∧λii∈Ωcdir⊆LX∧i∈Ω∧j∈JCjλi⟼∧j∈JCj∧i∈Ωλi∧∧λii∈Ω⊆LX∧i∈Ω∧j∈JCjλi⟼∧j∈JCj∧i∈Ωλi=∧j∈JCj⊤¯LX∧∧j∈JCj⊥¯LX∧∧λii∈Ωcdir⊆LX∧j∈J∧i∈Ω∧j∈JCjλi⟼Cj∧i∈Ωλi∧∧λii∈Ω⊆LX∧j∈J∧i∈Ω∧j∈JCjλi⟼Cj∨i∈Ωλi≥∧j∈JCj⊤¯LX∧∧j∈JCj⊥¯LX∧∧j∈J∧λii∈Ωcdir⊆LX∧i∈ΩCjλj⟼Cj∧i∈Ωλi∧∧j∈J∧λii∈Ω⊆LX∧i∈ΩCjλj⟼Cj∨i∈Ωλi=∧j∈JCocCj.


## 4. The Degree of Concavity Preserving and Concave-to-Concave Functions

This section presents the degree ofL,M-fuzzy concavity preserving and L,M-fuzzy concave-to-concave functions and discuss their properties.Definition 4.
Given a functionC:LX⟶M, L,M-fuzzy concavity degree CocC of C, for any λ∈LX, we define PCocλ by PCocλ=CocC∧Cλ is called the degree to which λ is an L-concave set with respect to C (or the L-concave set degree of λ with respect to C).Remark 2.
IfCocC=⊤M, which means that C is an L,M-fuzzy concavity, then PCocλ=Cλ, which can be regarded as a generalization of Cλ.Proposition 1.
Given a functionC:LX⟶M and L,M-fuzzy concavity degree CocC of C, for each λ∈LX, PCocλ denotes the L-concave set degree of λ with respect to C. Then,(1)
∧i∈ΩPCocλi≤PCoc∧i∈Ωλi, ∀λii∈Ωcdir⊆LX(2)
∧i∈ΩPCocλi≤PCoc∨i∈Ωλi, ∀λii∈Ω⊆LX
That is, ifPCocλ is regarded as a function PCoc:LX⟶M defined by λ⟼PCocλ, then PCoc is an L,M-fuzzy concavity on X.Proof.
(1)
Based on Definition4, it suffices to prove that CocC∧∧i∈ΩCλi≤CocC and CocC∧∧i∈ΩCλi≤C∧i∈Ωλi. By Definition 3, we have(14)CocC∧∧i∈ΩCλi≤∧i∈ΩCλi⟼C∧i∈Ωλi∧∧i∈ΩCλi≤C∧i∈Ωλi.(2)
It is similar to (3).
The following theorem characterizesL-concave set degree.Theorem 7.
LetC:LX⟶M be a function and CocC be L,M-fuzzy concavity degree of C. For each λ∈LX, PCocλ denotes the L-concave set degree of λ with respect to C. Then,(1)
PCocλ=∨r∈M,r≤Cλ,r≤C⊤¯LX,r≤C⊥¯LX,∧i∈ΩCλi∧r≤C∧i∈Ωλi,∀λii∈Ωcdir⊆LX,∧i∈ΩCλi∧r≤C∨i∈Ωλi,∀λii∈ω⊆LX(2)
PCocλ=∨r∈M|∀s≤r,Csis anL−concavity andλ∈Cs(3)
PCocλ=∨r∈M|∀s∉αr,Csis anL−concavity andλ∈CsProof.
The proofs are similar to those of Theorems3–5.Definition 5.
Given two functionsCX:LX⟶M and CY:LY⟶M, let f:X,CX⟶Y,CY be a function. Then,(1)
The concavity preserving degreep of f with respect to CX and CY is defined by(15)pf=∧ν∈LYPCocYν⟼PCocXfL←ν.(2)
The concave-to-concave degreec of f with respect to CX and CY is defined by(16)cf=∧λ∈LXPCocXλ⟼PCocYfL⟶λ,where CocX=CocCX and CocY=CocCY.Theorem 8.
LetCX:LX⟶M, CY:LY⟶M be two functions and let CocCX, CocCY refer to the L,M-fuzzy concavity degrees of CX and CY, respectively. Then,(1)
pf=∧r∈M|CocCY∧CYν∧r≤CocCX∧CXfL←ν,∀ν∈LY(2)
pf=∧r∈M|∀s≤CocCY∧r,∀ν∈CYs,s≤CocCX,fL←ν∈CXs(3)
pf=∧r∈M|∀s∉αCocCY∧r,∀ν∈CYss∉αCocCX,fL←ν∈CXsProof.
(1)
Since for eachr∈M and ν∈LY, we have(17)r≤pf⟺r∧PCocYν≤PCocXfL←ν⟺CocCY∧CYν∧r≤CocCX∧CXfL←ν.The proof of (1) is clear.(2)
Suppose thatCocCY∧CYν∧r≤CocCX∧CXfL←ν for each ν∈LY. For any s≤CocCY∧r and ν∈CYs, i.e., s≤CYν, we have s≤CocCX∧CXfL←ν. Thus, s≤CocCX and s≤CXfL←ν, i.e., fL←ν∈CXs. By (1), hence pf≤R, where R refers to the right hand side of equality.Conversely, lets≤CocCY∧CYν∧r. Then, s≤CocCY∧r and s≤CYν, i.e., ν∈CYs. Thus, CocCX≥s and fL←ν∈CXs, i.e., CXfL←ν≥s. This implies CocCX∧CXfL←ν≥s. Since s is arbitrary, we have CocCY∧CYν∧r≤CocCX∧CXfL←ν. From this result together with (1), we have pf≥R.(3)
Suppose thatCocCY∧CYν∧r≤CocCX∧CXfL←ν for each ν∈LY. For any s∉αCocCY∧r and ν∈CYs, i.e., s∉αCYν, we have s∉αCYν∪αCocCY∧r. Since α is a ∧-∪ map, it follows that(18)αCYν∪αCocCY∧r=αCocCY∧CYν∧r⊇αCocCX∧CXfL←ν=αCocCX∪αCXfL←ν.
This impliess∉αCocCX and s∉αCXfL←ν, i.e., fL←ν∈CXs. By (1), we have pf≤R, where R refers to the right hand side of the equality.
Conversely, take anys∉αCocCY∧CYν∧r. By(19)αCocCY∧CYν∧r=αCocCY∧r∪αCYν,we have s∉αCocCY∧r and s∉αCYν, i.e., ν∈CYs. Thus, s∉αCocCX and fL←ν∈CXs, i.e., s∉αCXfL←ν. This implies(20)s∉αCocCX∪αCXfL←ν=αCocCX∧CXfL←ν.
Sinces is arbitrary, we have CocCY∧CYν∧r≤CocCX∧CXfL←ν. From this result together with (1), we get pf≥R.Theorem 9.
Given two functionsCX:LX⟶M and CY:LY⟶M, if CocCX and CocCY denote the L,M-fuzzy concavity degrees of CX and CY, respectively, then(1)
cf=∧r∈M|CocCX∧CXλ∧r≤CocCY∧CYfL⟶λ,∀λ∈LX(2)
cf=∧r∈M|∀s≤CocCX∧r,∀λ∈CXss≤CocCY,fL⟶λ∈CYs(3)
cf=∧r∈M|∀s∉αCocCX∧r,∀λ∈CXs,s∉αCocCY,fL⟶λ∈CYsProof.
Similar to the proof of Theorem8.Proposition 2.
IfM is a complete distributive DeMorgan algebra and f:X,CX⟶Y,CY is a function between two L,M-fuzzy concavity structures X,CX and Y,CY, then(21)pf=∧xr∈JLX∧ν∈LYcoCYν′fxr′⟼coCXfL←ν′xr′,where coCX and coCY are the corresponding hull operators.Proof.
Straightforward.Proposition 3.
Given five functionsCX:LX⟶M, CY:LY⟶M, CZ:LZ⟶M, f:X,CX⟶Y,CY, and g:Y,CY⟶Z,CZ, then(1)
pf∧pg≤pg°f(2)
cf∧cg≤cg°fProof.
(1)
Sinceg°fL←ξ=fL←gL←ξ for each ξ∈LZ, by Definition 5, we have(22)pf∧pg=∧ν∈LYPCocYν⟼PCocXfL←ν∧∧γ∈LZPCocZγ⟼PCocYgL←γ≤∧ξ∈LZPCocYgL←ξ⟼PCocXfL←gL←ξ∧∧γ∈LZPCocZγ⟼PCocYgL←γ=∧ξ∈LZPCocYgL←ξ⟼PCocXg°fL←ξ∧∧γ∈LZPCocZγ⟼PCocYgL←γ=∧γ∈LZPCocYgL←γ⟼PCocXg°fL←γ∧PCocZγ⟼PCocYgL←γ≤∧γ∈LZPCocZγ⟼PCocXg°fL←γ=pg°f.(2)
similar to the proof of (1).Proposition 4.
Given three functionsCX:LX⟶M, CY:LY⟶M, and CZ:LZ⟶M, let f:X,CX⟶Y,CY and g:Y,CY⟶Z,CZ. Then,(1)
Iff is surjective, then cg°f∧pf≤cg.(2)
Ifg is injective, then cg°f∧pg≤cf.Proof.
(1)
Sincef is surjective, we know fL⟶fL←ν=ν for all ν∈LY. Then,(23)g°fL⟶fL←ν=gL⟶fL⟶fL←ν=gL⟶ν.Hence,(24)cg°f∧pf=∧λ∈LXPCocXλ⟼PCocZg°fL⟶λ∧∧ν∈LYPCocYν⟼PCocXfL←ν≤∧γ∈LYPCocXfL←γ⟼PCocZg°fL⟶fL←γ∧∧ν∈LYPCocYν⟼PCocXfL←ν=∧γ∈LYPCocXfL←γ⟼PCocZgL⟶γ∧∧ν∈LYPCocXν⟼PCocXfL←ν=∧ν∈LYPCocXfL←ν⟼PCocZgL⟶ν∧PCocYν⟼PCocXfL←ν≤∧ν∈LYPCocYν⟼PCocZgL⟶ν=cg.(2)
Sinceg is injective, we have gL←gL⟶ν=ν for each ν∈LY. Then, gL←g°fL⟶μ=fL⟶μ for all μ∈LX. Hence,(25)cg°f∧pg=∧λ∈LXPCocXλ⟼PCocZg°fL⟶λ∧∧ξ∈LZPCocZξ⟼PCocYgL←ξ≤∧λ∈LXPCocXλ⟼PCocZg°fL⟶λ∧∧γ∈LXPCocZg°fL⟶γ⟼PCocYgL←g°fL⟶γ=∧λ∈LXPCocXλ⟼PCocZgL⟶fL⟶λ∧∧γ∈LXPCocZgL⟶fL⟶γ⟼PCocYfL⟶γ=∧λ∈LXPCocXλ⟼PCocZgL⟶fL⟶λ∧PCocZgL⟶fL⟶λ⟼PCocYfL⟶λ≤∧λ∈LXPCocXλ⟼PCocYfL⟶λ=cf.Definition 6.
For any two functionsCX:LX⟶M and CY:LY⟶M, if the function f:X,CX⟶Y,CY is bijective, then the isomorphism degree isof of the function f is given by isof=pf∧pf−1.Theorem 10.
For any two functionsCX:LX⟶M and CY:LX⟶M, if the function f:X,CX⟶Y,CY is bijective, then pf−1=cf and isof=pf∧pf−1=pf∧cf.Proof.
From the bijectivity of the functionf, we have f−1L←λ=fL⟶λ for any λ∈LX. Thus,(26)pf−1=∧λ∈LXPCocXλ⟼PCocYf−1L←λ=∧λ∈LXPCocXλ⟼PCocYfL⟶λ=cf.
Therefore,isof=pf∧pf−1=pf∧cf. This completes the proof.Proposition 5.
Given a functionCX:LX⟶M, if id:X,CX⟶X,CX is the identity function, then(27)isoid=pid=cid=⊤M.Proof.
Straightforward.Proposition 6.
Given three functionsCX:LX⟶M, CY:LY⟶M, and CZ:LZ⟶M, let f:X,CX⟶Y,CY and g:Y,CY⟶Z,CZ be two bijective functions, then isof∧isog≤isog°f.Proof.
Straightforward.Lemma 3.
For any two functionsCX:LX⟶M and CY:LY⟶M, if the function f:X,CX⟶Y,CY is bijective, then(1)
pf=∧λ∈LXPCocYfL⟶λ⟼PCocXλ(2)
cf=∧ν∈LYPCocXfL←ν⟼PCocYνProof.
(1)
From the bijectivity of the functionf, we get fL←fL⟶λ=λ for any λ∈LX and fL⟶fL←ν=ν for any ν∈LY. Thus,(28)∧λ∈LXPCocYfL⟶λ⟼PCocXλ=∧λ∈LXPCocYfL⟶λ⟼PCocXfL←fL⟶λ≥∧λ∈LXPCocYν⟼PCocXfL←ν=pf=∧ν∈LXPCocYfL⟶fL←ν⟼PCocXfL←ν≥∧λ∈LXPCocYfL⟶λ⟼PCocXλ.Therefore,(29)pf=∧λ∈LXPCocYfL⟶λ⟼PCocXλ.(2)
similar to (1).Theorem 11.
For any two functionsCX:LX⟶M and CY:LY⟶M, if the function f:X,CX⟶Y,CY is bijective, then(1)
isof=∧λ∈LXPCocXλ⟷PCocYfL⟶λ(2)
isof=∧ν∈LYPCocYν⟷PCocXfL←ν


## 5. The Quotient Degree of Functions betweenL,M-Fuzzy Concave Structures

In this section, we endow the quotient functions with some degree and discuss the relationship with the degree ofL,M-fuzzy concavity preserving functions and the degree of L,M-fuzzy concave-to-concave functions.Definition 7.
For any two functionsCX:LX⟶M and CY:LY⟶M, if the function f:X⟶Y is surjective, then the quotient degree of the function f with respect to CX and CY, denoted by qf, is given by(30)qf=∧ν∈LYPCocYν⟷PCocXfL←ν.
The proof of the following theorem is similar to the proof of Theorem8.Theorem 12.
For any two functionsCX:LX⟶M and CY:LY⟶M, if CocX and CovY refer to the L,M-fuzzy concavity degree of CX and CY, respectively, and f:X,CX⟶Y,CY is a surjective function, then(1)
qf=∧r∈M|CocY∧CYν∧r≤CocX∧CXfL←ν,CocX∧CXfL←ν∧r≤CocY∧CYν,∀ν∈LY(2)
qf=∧r∈M|∀s≤CocY∧r,∀ν∈CYss≤CocX,fL←ν∈CXs,∀t≤CocX∧r,∀fL←ν∈CXtt≤CocY,ν∈CYt(3)
qf=∧r∈M|∀s∉αCocY∧r,∀ν∈CYss∉αCocX,fL←ν∈CXs,∀t∉αCocX∧r,∀fL←ν∈CXtt∉αCocY,ν∈CYtTheorem 13.
For any two functionsCX:LX⟶M and CY:LY⟶M, if f:X,CX⟶Y,CY is a surjective function, then qf≤pf.Proof.
StraightforwardTheorem 14.
For any two functionsCX:LX⟶M and CY:LY⟶M, if the function f:X,CX⟶Y,CY is surjective, then pf∧cf≤qf.Proof.
From the surjectivity off, fL⟶fL←γ=γ for each γ∈LY. Then,(31)pf∧cf=∧ν∈LYPCocYν⟼PCocXfL←ν∧∧λ∈LXPCocXλ⟼PCocYfL⟶λ≤∧ν∈LYPCocYν⟼PCocXfL←ν∧∧γ∈LYPCocXfL←γ⟼PCocYfL⟶fL←γ=∧ν∈LYPCocYν⟼PCocXfL←ν∧∧γ∈LYPCocXfL←γ⟼PCocYγ=∧ν∈LYPCocYν⟼PCocXfL←ν∧PCocXfL←ν⟼PCocYν=∧ν∈LYPCocYν⟷PCocXfL←ν=qf.Theorem 15.
For any three functionsCX:LX⟶M, CY:LY⟶M, and CZ:LZ⟶M, if the functions f:X,CX⟶Y,CY and g:Y,CY⟶Z,CZ are surjective, then(1)
qf∧qg≤qg°f(2)
qg°f∧pf∧pg≤qgProof.
(1)
(32)qf∧qg=∧ν∈LYPCocYν⟷PCocXfL←ν∧∧γ∈LZPCocZγ⟷PCocYgL←γ≤∧ξ∈LZPCocYgL←ξ⟷PCocXfL←gL←ξ∧∧γ∈LZPCocZγ⟷PCocYgL←γ=∧ξ∈LZPCocYgL←ξ⟷PCocXg°fL←ξ∧∧γ∈LZPCocZγ⟷PCocYgL←γ=∧γ∈LZPCocYgL←γ⟷PCocXg°fL←γ∧PCocZγ⟷PCocYgL←γ≤∧γ∈LZPCocXγ⟷PCocXg°fL←γ=qg∘f.(2)
Since(33)qg°f=∧γ∈LZPCocZγ⟷PCocXg°fL←γ≤∧γ∈LZPCocXg°fL←γ⟼PCocZγ,pf=∧ν∈LYPCocYν⟼PCocXfL←ν≤∧ξ∈LZPCocYgL←ξ⟼PCocXfL←gL←ξ=∧ξ∈LZPCocYgL←ξ⟼PCocXg°fL←ξ,then(34)qg∘f∧pf≤∧ξ∈LZPCocYgL←ξ⟼PCocZξ.
Therefore,(35)qg°f∧pf∧pg≤∧ξ∈LZPCocYgL←ξ⟼PCocZξ∧∧γ∈LZPCocZγ⟼PCocYgL←γ=∧γ∈LZPCocYgL←γ⟼PCocZγ∧PCocZγ⟼PCocYgL←γ=∧γ∈LZPCocZγ⟷PCocYgL←γ=qg.
This completes the proof.


## 6. Conclusion

In this paper, we presented the degree to which a functionC:LX⟶M is an L,M-fuzzy concavity on a nonempty set X. Moreover, the degree to which an L-subset is an L-concave set with respect to C was considered. Also, we defined the concavity preserving, concave-to-concave degree, and quotient degree for functions between L,M-fuzzy concave structures. Their characterizations were given and the relationships among them were discussed. We think our results will be useful to consider many properties of concave structures under degree of L,M-fuzzy concavity. We think that studying the topological properties of such degree is the most abstract and generalization of these properties, which leads to the results of previous studies as soon as the degree equals to ⊤M. Thus, the fuzzification theory would be applied in a better and more generalized way.



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