# A Novel Concept of Fuzzy Subsemigroup (Ideal)

Authors: Yu-Hong Che; Qi Liu
Journal: Journal of Mathematics
 (2022)

---

## Abstract

This paper focuses on a generalized definition of fuzzy subsemigroup (ideal) on semigroup. LetL be a completely distributive lattice; we introduce the definition of L-fuzzy ideal and also the novel concept of subsemigroup (ideal) on semigroup. Then, we discuss the necessary and sufficient conditions of L-fuzzy subsemigroup (ideal) measure using the four level cuts of an L-fuzzy set. Moreover, we study the properties of L-fuzzy subsemigroup (ideal) measure. As an application of L-fuzzy subsemigroup (ideal) measure, we obtain the L-fuzzy convexities on a semigroup and bijective semigroup homomorphic mapping is an L-fuzzy isomorphism.


---

## Body

## 1. Introduction

Zadeh introduced fuzzy set theory [1]. Then, Van de Vel [2] gave a systematic study of the theory of convexity in 1993. Rosa [3] extended the concept of convexity to the fuzzy situation in 1994, which is said to be an L-convex structure later, and the relevant results are shown in [4–8]. In addition, Maruyama [9] studied the concept of convexity on a completely distributive lattice in 2009. As a matter of fact, there is a convex space in many other mathematical structures [10–13].Table 1
Values of the semigroup operator∘.
s1s2s1∘s2lnpqlllllnnlllpllnlqllllMany academics studied on generalized semigroups (ideal) [14–19]. Zhao [19] introduced the concept of L-fuzzy subsemigroup. To be worth mentioning, Shi and Xin opened up a prospect of inducing the L-fuzzy convexity [20]. In recent years, mathematicians have constructed the L-fuzzy convexities on a series of algebraic structures. Li and Shi [21] constructed an L-fuzzy convexity according to L-convex fuzzy sublattice measure. Furtherly, they obtained L,M-fuzzy convexity spaces. In particular, an L,M-fuzzy convexity became known simply as L-fuzzy convexity if L=M, where M is a completely distributive lattice [22]. In 2017, Wen et al. [23] defined an L-fuzzy convexity by the L-convex measure on vector spaces. Two years later, Han and Shi [24] introduced an L-fuzzy convexity according to L-convex ideal measure on a lattice. At the same year, Zhong et al. [25] constructed an L-fuzzy convexity according to L-fuzzy convex subgroup measure on an ordered group. The study by Mehmood et al. [26] constructed an L-fuzzy convexity on a ring in 2020. In 2021, Zeng et al. (preprint [27]) constructed an L-fuzzy convexity according to an L-fuzzy subfield degree on a field.As a continuation of [19], we will extend fuzzy subsemigroup (ideal) on semigroup to general cases on the basis of the above researches. It recalled some necessary concepts and theorems of general semigroup, fuzzy semigroup, and L-fuzzy convexity. Following that, the L-fuzzy ideal is studied and its characterizations are given. Then, an L-fuzzy subsemigroup (ideal) measure is presented, and at the same time, the characterizations and properties are studied. Then, an L-fuzzy convexity is generated by an L-fuzzy subsemigroup (ideal) measure on a semigroup. What is more, the related L-fuzzy convex preserving mappings, L-fuzzy convex to convex mappings, and L-fuzzy isomorphism mappings are discussed. The findings show that a semigroup homomorphic mapping can be considered as L-fuzzy convexity preserving (L-fuzzy convex-to-convex) mapping and bijective semigroup homomorphic mapping is an L-fuzzy isomorphism.


## 2. Preliminaries

To begin with, we give some useful materials for necessary. From [28], we know the operation of AC is defined as AC=xy∈S:x∈Aandy∈C, where S is a semigroup and A,C⊆S.Definition 1 (see [28]).
LetS be a semigroup. A nonemptysubset C of S is said to be a subsemigroup (ideal) of S if
(1)CC⊆CSC⊆CandCS⊆C.Moreover, the intersection of any collection of subsemigroup ofS is a subsemigroup of S [28]. Beside, a semigroup S is said to be a commutative semigroup if the operation of S satisfies the commutative law.Mordeson et al. extended Definition1 to the fuzzy situation.Definition 2 (see [28]).
LetS be a semigroup. An L-fuzzy subset ν∈LS is called to be
(1)
a fuzzy subsemigroup ofS if ∀t1,t2∈S such that νt1t2≥νt1∧μt2(2)
a fuzzy left (right) ideal ofS if ∀t1,t2∈S such that νt1t2≥νt2νt1t2≥νt1(3)
a fuzzy ideal ofS if it is a fuzzy left ideal and it is a fuzzy right ideal of SThroughout this paper,⊥ and Τ denote the minimum element and the maximum element in L, respectively. We denote A as a subset of S by A⊆S, the set of all L-fuzzy subsets on S by LS, and an L-fuzzy subset ν of S by ν∈LS. LS is a completely distributive lattice if it maintains the lattice L via defining ≤ point wisely, naturally. In addition, symbols χ∅ and χS denote the minimum element and the maximum element in LS, respectively. Moreover, ∧∅=Τ and ∨∅=⊥. Let x,y, and z be three elements in L. Then, x is said to be a coprime if x is less than or equal to y∨z leading to x less than or equal toz or x less than or equal to y. Moreover x is said to be prime if y∧z is less than or equal to x leading to either y less than or equal to x or z less than or equal to x. In addition, PL and JL denote the collection of all nonunit prime and nonzero coprime elements, respectively. ∀x,y∈L and D⊆L, y less than or equal to ∨D, x≺y means that x is less than or equal to d for some d∈D. Then, βy is equal to x∈L:x≺y and β∗y is equal to βy∩JL mainly based on [29]. Moreover, ∀x,y∈L, y≺opx, in another way, there exists d in D, by D⊆L, ∧D less than or equal to y; we can have d which is less than or equal to x; then, we say αx is less than or equal to y∈M:x≺opy and α∗y is less than or equal to αy∩PL [29]. Moreover, ∀x,y∈L and ∀s1∈PLs1∈JL, it means x≤y⇔x≰s1s1≤x⇒y≰s1s1≤y [23].The four cut sets play an important part in fuzzy theory. Supposingν∈LS and ∀x,y∈L, according to the definition of cut sets and the fact that an implication operator ⟶ in correspondence to ∧ in [23], we infer x⟶y is equal to ∨z∈L:x∧z≤y and [23]
(2)νx=s1∈S:x≤μs1,νx=s1∈S:νs1≰x,νx=s1∈S:x∈βμs1,νx=s1∈S:x∉ανs1.We denoteL-fuzzy convexity and L-fuzzy convexity preserving mapping (L-fuzzy convex-to-convex mapping) with LFC and LFCPMLFCCM, respectively.From the definition ofL,M-fuzzy convexity [22], we know the following.Definition 3 (see ([23], [28]).
A mappingS:LS⟶L is called to be an LFC on S if the following conditions are true:
(1)
Sχ∅=SχS=Τ(2)
∧i∈ΩSνi≤S∧i∈Ωνi if νi∣i∈Ω⊆LS is nonempty(3)
∧i∈ΩSνi≤S∨i∈Ωνi if νi∣i∈Ω⊆LS is nonempty and totally orderedFrom [23], we know the following.Definition 4 (see ([23]).
LetS,C and S1,D be LFC spaces. A mapping f:S⟶S1 is called to be an LFCPM (LFCCM) if
(3)DE≤CfL⟵EforallE∈LS1CG≤DfL⟶GforallG∈LS.
Then,f:X⟶Y is called to be an L-fuzzy isomorphism if f is bijective, LFCPM, and LFCCM.In this paper, we aim at a novel definition of fuzzy subsemigroup (ideal) on semigroup. According to it, any fuzzy set can be considered as a fuzzy subsemigroup (ideal) generalized. This paper is arranged as follows: first, we will treat the novel definition of fuzzy subsemigroup (ideal) on the basis of the notion of anL-fuzzy subsemigroup (ideal) on semigroup, their properties, and their equivalent characterizations. Then, we will obtain L-fuzzy convexities induced by L-fuzzy subsemigroup (ideal) measure and a conclusion that bijective semigroup homomorphic mapping is an L-fuzzy isomorphism and so on.


## 3. A Novel Definition of anL-Fuzzy Subsemigroup (Ideal)

In this section, the novel definitions ofL-fuzzy subsemigroup ideal are introduced and their characterizations are given. First, we will extend Definition 2 to the L-fuzzy situation.
(1)
LetA=l,n and B=l,p. We define β,α∈LS as follows:(4)βl=βn=0.7,βp=0.6,βq=0.5,(5)αl=αn=0.7,αp=0.8,αq=0.9Definition 5.
LetS be a semigroup, ν∈LS; then, ν is called to be
(1)
anL-fuzzy subsemigroup of S, if νt1t2≥νt1∧νt2∀t1,t2∈S [19](2)
anL-fuzzy left (right) ideal of S, if νt1t2≥νt2νt1t2≥νt1∀t1,t2∈S(3)
anL-fuzzy ideal of S, if it is an L-fuzzy left ideal and it is an L-fuzzy right ideal of SExample 1.
LetS=l,n,p,q be a semigroup with the following table.From Definition6 (1), we can infer that β is an L-fuzzy subsemigroup of S, but α is not.
(2)
Then, we defineμ∈LS as follows:(6)μl=μn=0.7,μp=0.5,μq=⊥L,and ν∈LS as follows:
(7)νl=νn=0.7,νp=0.8,νq=0.9From Definition5 (2), we get μ is an L-fuzzy ideal of S, but ν is not.According to the definition of anL-fuzzy subsemigroup in [19], an L-subset is either an L-fuzzy subsemigroup or not. We discuss the measure to which an L-subset is an L-fuzzy subsemigroup using the implication operator of L. Moreover, we define an L-fuzzy ideal and study the measure to which an L-subset is an L-fuzzy ideal.Definition 6.
LetS be a semigroup, ν∈LS; then,
(1)
Sν is called to be the subsemigroup measure of ν as(8)Sν=∧t1,t2∈Sνt1∧νt2↦νt1t2(2)
Lν is called to be the ideal measure of ν as(9)Lν=∧t1,t2∈Sνt1↦νt1t2∧νt2↦νt1t2Obviously,ν is an L-fuzzy subsemigroup (ideal) of S if and only if Sν=ΤLν=Τ, and ν always is an L-fuzzy subsemigroup (ideal) to the measure SνLν.From Example1, we know βμ is an L-fuzzy subsemigroup (ideal) of S, but αν is not. We will explain L-fuzzy subsemigroup (ideal) measure according to the following example, which indicates that any L-subset can be regarded as an L-fuzzy subsemigroup (ideal) to some measure.Example 2.
On the basis of Example1, we define L-fuzzy subsemigroup (ideal) measure of αν as
(10)Sα=∧t1,t2∈Sαt1∧αt2↦αt1t2=0.7↦0.7∧0.8↦0.7∧0.9↦0.7.=0.7.According to Definition6 (1), this means that α is an L-fuzzy subsemigroup to the measure Sα=0.7. In addition,
(11)Lν=∧t1,t2∈Sνt1↦νt1t2∧νt2↦νt1t2=0.7↦0.7∧0.8↦0.7∧0.9↦0.7=0.7.From Definition6 (2), we can say that ν is an L-fuzzy ideal to the measure Lν=0.7.By Definition6, we can infer the following conclusion.Theorem 7.
LetS be a semigroup and ν∈LS, then Lν≤Sν.Inspired by the conclusionz≤x⟶y if and only if x∧z≤y [23] ∀x,y,z∈L, the following lemma is clear.Lemma 8.
LetS be a semigroup and ν∈LS; then,
(1)
Sν≤x if and only if ∀t1,t2∈S,νt1∧νt2∧x≤νt1t2∀x∈L(2)
Lν≤x if and only if ∀t1,t2∈S,νt1∧x≤νt1t2, and νt2∧x≤νt1t2∀x∈LBy Lemma8, we can easily obtain the other characterizations of the subsemigroup measure of an L-fuzzy subset as follows.Theorem 9.
LetS be a semigroup and ν∈LS; then,
(1)
Sν=∨x∈Lνt1∧νt2∧x≤νt1t2,∀t1,t2∈S(2)
Lν=∨x∈Lνt1∧x≤νt1t2,νt2∧x≤νt1t2,∀t1,t2∈SNext, we characterize the subgroup (ideal) measure of anL-fuzzy set by means of its four levels in the following theorem.Theorem 10.
LetS be a semigroup, ν∈LS; then,
(1)
Sν=∨x∈L:∀y≤x,νyisasubsemigroup(2)
Sν=∨x∈L:∀y∉αx,νyisasubsemigroup(3)
Sν=∨x∈L:∀y∈PL,x≤y,νyisasubsemigroup(4)
Sν=∨x∈L:∀y∈βx,νyisasubsemigroup(5)
Lν=∨x∈L:∀y≤x,νyisanideal(6)
Lν=∨x∈L:∀y∉αx,νyisanideal(7)
Lν=∨x∈L:∀y∈PL,x≤y,νyisanideal(8)
Lν=∨x∈L:∀y∈βx,νyisanidealProof.
(1)
First, we prove(12)Sν≤∨x∈L:∀y≤x,νyisasubsemigroup
∀t1,t2∈S, let νt1∧νt2∧x≤νt1t2; then, ∀y≤x, ∀t1,t2∈νy; we have
(13)νt1t2≥νt1∧νt2∧x≥νt1∧νt2∧y=y.
This showst1t2∈νy; therefore, νy is a subsemigroup of S. Thus,
(14)Sν≤∨x∈L:∀y≤x,νyisasubsemigroup.
Next, we prove(15)∨x∈L:∀y≤x,νyisasubsemigroup≤Sν.
Letνy be a subsemigroup of S, ∀y≤x, x∈L. We want to show
(16)∀t1,t2∈S,νt1∧νt2∧x≤νt1t2.
Supposey=νt1∧νt2∧x, we can infer y≤x and t1,t2∈νy, then t1t2∈νy if and only if νt1t2≥y=νt1∧νt2∧x. Thus,
(17)∨x∈L:∀y≤x,νyisasubsemigroup≤Sν.
Now, we get(18)∨x∈L:∀y≤x,νyisasubsemigroup=Sν.(2)
First, we prove(19)Sν≤∨x∈L:∀y∉αx,νyisasubsemigroup
We take anyt1,t2∈S, if νt1∧νt2∧x≤νt1t2, then ∀t1,t2∈νy, ∀y∉αx if and only if y∉ανt1,y∉ανt2; we can obtain
(20)y∉ανt1∪ανt2∪αx=ανt1∧νt2∧x.
Byνt1∧νt2∧x≤μt1t2, then
(21)ανt1t2⊆ανt1∧νt2∧x,we can infer y∉ανt1t2 if and only if t1t2∈νy. This means νy is a subsemigroup of S, and
(22)x∈x∈L:∀y∉αx,μyisasubsemigroup.
This demonstrates(23)Sν=∨x∈L:νt1∧νt2∧x≤νt1t2,∀t1,t2∈S≤∨x∈L∣∀y∉αx,νyisasubsemigroup.
Next, we prove(24)∨x∈L:∀y∉αx,νyisasubsemigroup≤Sν.
Supposeνy is a subsemigroup of S for any y∉αx,x∈L. We want to show
(25)νt1∧νt2∧x≤νt1t2foranyt1,t2∈S.
Supposey∉ανt1∧νt2∧x=ανt1∪ανt2∪αx, then y∉αx and t1,t2∈νy; thus, t1t2∈νy if and only if y∉ανt1t2. This shows νt1∧νt2∧x≤νt1t2. It is proven that
(26)Sν=∨x∈L:νt1∧νt2∧x≤νt1t2,∀t1,t2∈S≥∨x∈L:∀y∉αx,νyisasubsemigroup.
Thus, (2) is true.(3)
First, we prove(27)Sν≤∨x∈L:∀x≰y,νyisasubsemigroup
Supposingx∈L, it satisfies νt1∧νt2∧x≤νt1t2∀t1,t2∈S. Let y∈PL, x≰y, and t1,t2∈νy; now we prove t1t2∈νy. Assume that t1t2∉νy. According to the fact t1t2∉νy if and only if νt1t2≤y, we can infer νt1∧νt2∧x≤νt1t2≤y. By y∈PL and t1,t2∈νy, now, we can get x≤y, which contradicts x≰y. Hence, t1t2∈νy. This shows that νy is a subsemigroup of S. Therefore,
(28)Sν=∨x∈L:νt1∧νt2∧x≤νt1t2,∀t1,t2∈S≤∨x∈L:∀x≰y,νyisasubsemigroup.
Next, we prove(29)∨x∈L:∀x≰y,νyisasubsemigroup≤Sν.
Supposingx∈L, it satisfies νyisasubsemigroupofS∀x≰y. Now, we want to prove
(30)νt1∧νt2∧x≤νt1t2,∀t1,t2∈S.
Supposey∈PL and νt1∧νt2∧x≰y, then νt1≰y, νt2≰y, and x≰y if and only if t1,t2∈νy. For a subsemigroup νy, it means t1t2∈νy if and only if νt1t2≰y. This shows that νt1∧νt2∧x≤νt1t2. It is proved that ∀t1,t2∈S,
(31)Sν=∨x∈L:∀x≰y,νyisasubsemigroup≥∨x∈L:νt1∧νt2∧x≤νt1t2.
It is now obvious that (3) holds.(4)
First, we prove(32)Sν≤∨x∈L:νyisasubsemigroup
We assume∀t1,t2∈S, x∈x∈L:νt1∧νt2∧x≤νt1t2, then ∀y∈βx, t1,s2∈νy; it holds that
(33)y∈βνt1∩βνt2∩βx=βνt1∧νt2∧x⊆βνt1t2.
Then,t1t2∈νy. This shows that νy is a subsemigroup of S. This means that
(34)Sν=∨x∈L:νt1∧νt2∧x≤νt1t2,∀t1,t2∈S≤∨x∈L:∀y∈βx,νyisasubsemigroup.
Next, we prove(35)∨x∈L:,νyisasubsemigroup≤Sν.
Supposingx∈L, it satisfies νyisasubsemigroup∀y∈βx. Now, we prove that
(36)νt1∧νt2∧x≤νt1t2,∀t1,t2∈S.
Lety∈βνt1∧νt2∧x. By βνt1∧νt2∧x=βνt1∩βνt2∩βx, we know that t1,t2∈νy, and y∈βx. Since νy is a subsemigroup of S, it holds that t1t2∈νy if and only if y∈βνt1t2. This shows that νt1∧νt2∧x≤νt1t2. Therefore,
(37)Sν=∨x∈L:νt1∧νt2∧x≤νt1t2,∀t1,t2∈S≥∨x∈L:∀y∈βx,νyisasubsemigroup.
It is now obvious that (4) holds.
(5)-(7) By the fact thatνx, νx, and νx are the ideals of S for any x∈L and ν∈LS, then the statements are easy to be proved with Lemma 8 (2) and are omitted
(8)
Supposeνt1∧x≤νt1t2 and νt2∧x≤νt1t2∀t1,t2∈S. It is a fact that y∈βνt2∩βx=βνt2∧x⊆βνt1t2 if and only if t1t2∈νy∀y∈βx and t2∈νy. In addition, this shows that νy is an ideal of S. This means that ∀t1,t2∈S for any t1∈νy, t1t2∈νy,(38)Lν=∨x∈L∣νt1∧x≤νt1t2andνt2∧x≤νt1t2,∀t1,t2∈S≤∨x∈L:∀y∈βx,νyisanideal
Conversely, supposeνy is an ideal of S∀y∈βx. Now, we prove that
(39)νt1∧x≤νt1t2,νt2∧x≤νt1t2,∀t1,t2∈S.
Lety∈βνt2∧x; by βνt2∧x=βνt2∩βx, we know that t2∈νy and y∈βx. For an ideal νy of S, it holds that t1t2∈νy if and only if y∈βνt1t2; it shows that νt2∧x≤νt1t2. We can obtain νt1∧x≤νt1t2 in a similar way. Therefore,
(40)Lν≥∨x∈L:∀y∈βx,νyisanideal.
It is now obvious that (8) holds.Theorem 11.
Letνii∈Ω be a family of L-fuzzy subsets in a semigroup S. Then,
(1)
∧i∈ΩSνi≤S∧i∈Ωνi(2)
∧i∈ΩLνi≤L∧i∈ΩνiProof.
Supposeνii∈Ω is a family of L-fuzzy subsets in a semigroup S∀t1,t2∈S,x∈L.
(1)
Supposex≤∧i∈ΩSνi, then ∧i∈Ωνit1∧∧i∈Ωνit2∧x≤∧i∈Ωνit1t2. We can infer x≤S∧i∈Ωνi(2)
Supposex≤∧i∈ΩLνi, then νit2∧x≤νit1t2; it is easy to obtain ∧i∈Ωνit2∧x≤∧i∈Ωνit1t2, and by νit1∧x≤νit1t2, we can obtain(41)∧i∈Ωνit1∧x≤∧i∈Ωνit1t2
We can inferx≤L∧i∈Ωνi.The following result is obvious.Letf:S0⟶S1 be a mapping and ν and η be two L-fuzzy subsets in S0 and S1, respectively. Then, we have
(1)
fS⟶fS⟵η=ηfL⟶fL⟵η=η if f is surjective(2)
fS⟵fS⟶ν=νfL⟵fL⟶ν=ν if f is injectivewhere fS⟶:LS0⟶LS1fL⟶:LS0⟶LS1 and fS⟵:LS1⟵LS0fL⟵:LS1⟵LS0 are defined by
(42)fS⟶νs′=∨νs:fs=s′,fL⟶νs′=∨νs:fs=s′,fS⟵η=η∘f,fL⟵η=η∘f.Now, we investigate the subsemigroup (ideal) measures of homomorphic image and preimage ofL-fuzzy subset.Theorem 12.
Supposef:S⟶S′ is a semigroup homomorphism. Then,
(1)
ifν∈LS, then SfS⟶ν≥Sν. And if f is injective, then SfS⟶ν=Sν(2)
ifη∈LS′, then SfS⟵η≥Sη. And if f is surjective, then SfS⟵η=Sη(3)
ifν∈LS, then LfL⟶ν≥Lν. And if f is injective, then LfL⟶ν=Lν(4)
ifη∈LS′, then LfL⟵η≥Lη. And if f is surjective, then LfL⟵η=LηProof.
(1)
We complete theproof by combining the fact(43)SfL⟶ν=∨x∈L:fL⟶νt1′∧fL⟶νt2′∧x≤fL⟶νt1′t2′,∀t1′,t2′∈S.=∨x∈L:∨ft1=t1′νt1∧∨ft2=t2′νt2∧x≤∨fz=t1′t2′νz,∀t1′,t2′∈S≥∨x∈L:νt1∧νt2∧x≤νt1t2,∀t1,t2∈S=Sν
Iff is injective, the above ≤ can be replaced by =, i.e., Sν=SfL⟶ν. Now, (1) is true.
(2)
We complete the proof by combining the fact(44)SfL⟵η=∧t1,t2∈SfL⟵ηt1∧fL⟵ηt2↦fL⟵ηt1t2=∧t1,t2∈Sηft1∧ηft2↦ηft1ft2≥∧t1′,t2′∈S′ηt1′∧ηt2′↦ηt1′t2′=Sη
Iff is surjective, the above ≤ can be replaced by =, i.e., Sν=SfL⟵η. Now, (2) is true.
(3)
We complete the proof by combining the fact
For anyt1,t2∈S,t1′,t2′∈S′,(45)LfL⟶ν=∨fL⟶νt1′∧x≤fL⟶νt1′t2′x∈LfL⟶νt2′∧x≤fL⟶νt1′t2′∀t1′,t2′∈S′=∨x∈L∨ft1=t1′νt1∧x≤∨fz=t1′t2′νz∨ft2=t2′νt2∧x≤∨fz=t1′t2′νz∀t1′,t2′∈S′≥∨x∈Lνt1∧x≤νt1t2,νt2∧x≤νt1t2,∀t1,t2∈S=Lν.
Iff is injective, the above ≤ can be replaced by =, i.e., Lν=LfL⟶ν. Now, (3) is true.
(4)
We complete the proof by combining the fact(46)LfL⟵η=∧t1,t2∈SfL⟵ηt1↦fL⟵ηt1t2∧fL⟵ηt2↦fL⟵ηt1t2=∧t1,t2∈Sηft1↦ηft1ft2∧ηft2↦ηft1ft2≥∧t1′,t2′∈S′ηt1′∧ηt2′↦ηt1′t2′=Lη
Iff is surjective, the above ≤ can be replaced by =, i.e., Lη=LfL⟵η. It is now obvious that the theorem holds.On the basis of the definition of the Cartesian product of theL-fuzzy subset [20], we discuss the subsemigroup ideal measure of the product of L-fuzzy subsets of a semigroup.
(2)
The proof is simple, and we omit itTheorem 13.
LetSii=1n be a family of semigroup and ∏i=1nνi be the product of νii=1n, where νi∈LSi. Then,
(1)
S∏i=1nνi≤∧i=1nSνi(2)
L∏i=1nνi≤∧i=1nLνiProof.
(1)
For∏i=1nνi=∧i=1nPi−1νi, let Pi:∏i=1nSi⟶Si be projections i=1,2,⋯,n; then, Pi is semigroup homomorphic mappings; we can infer(47)SPi−1νi≥Sνi
Then,(48)S∏i=1nνi=S∧i=1nPi−1νi≥∧i=1nSPi−1νi≥∧i=1nSνi.Definition 14.
LetS be a semigroup, x∈PL, and ν,η∈LS, then ν∘η∈LS defined by
(49)ν∘ηz=∨z=t1t2νt1∧ηt2.
It is obvious thatν∘ηx=νx∘ηx.Example 3.
LetS=l,n,p,q be a semigroup with the same operation as Example 1. We define ν,η∈LS as follows:
(50)νl=0.4,νn=0.7,νp=0.6,νq=0.5,ηl=0.3,ηn=0.7,ηp=0.8,ηq=0.9.
From Definition14, we get
(51)ν∘ηl=∨l=t1t2νt1∧ηt2=0.7,ν∘ηn=∨n=t1t2νt1∧ηt2=0.6.Theorem 15.
LetS be a semigroup, ν,η∈LS. If ν∘η=η∘ν, then
(52)Sν∘η≥Sν∧Sη.Proof.
. LetSν∧Sη≥x for any x∈PL, then Sν≥x and Sη≥x. Then, ∀z≱x, by Theorem 10 (3), νz and ηz are subsemigroups, respectively. By ν∘ηz=νz∘ηz, then ν∘ηz is a subsemigroup of S. We can infer Sν∘η≥z. It is now obvious that the theorem holds.Theorem 16.
LetS be a commutative semigroup, ν, η∈LS, then
(53)Lν∘η≥Lν∧Lη.Proof.
SupposeLν∧Lη≥x for any x∈PL; we can infer Lν≥x and Lη≥x. Then, ∀z≱x; by Theorem 10 (7), νz and ηz are an ideal, respectively. By ν∘ηz=μz∘ηz, then ν∘ηz is an ideal of S. We can infer Lν∘η≥x. It is now obvious that the theorem holds.


## 4. TheL-Fuzzy Convexities and Their Properties

In the following, we will use theL-fuzzy subsemigroup (ideal) measures, we will construct an L-fuzzy convexity on a semigroup. After that, we will study the relationship between LFC and LFCPM (LFCCP).∀ν∈LS, Sν (Lν) can be reasonably considered as a mapping S: LS⟶LL: LS⟶L specified by ν↦Sνν↦Lν. It will indicate that SL is still an LFC on a semigroup S.The following convention is that the null set is a subsemigroup (ideal) of any semigroup. It is necessary to discuss the next theorem.Theorem 17.
LetS be a semigroup, ν∈LS. Then,
(1)
the mappingS: LS⟶L specified by ν↦Sν is an LFC induced via L-fuzzy subsemigroup measure on S(2)
the mappingL: LS⟶L specified by ν↦Lν is an LFC induced via L-fuzzy ideal measure on SProof.
(1) We will prove (1) in accordance with the three axioms of Definition3 as follows:
(i)
It indicatesSχ∅=SχS=Τ(ii)
Theorem11 has proven ∧i∈ΩSνi≤S∧i∈Ωνi if νii∈Ω, a family of L-fuzzy subsets in a semigroup S, is nonempty(iii)
Supposeνi:i∈Ω≠∅, νi:i∈Ω⊆LS is a chain. We want to prove(54)∧i∈ΩSνi≤S∨i∈ΩνiIt needs to show that
(55)x≤S∨i∈Ωνi∀x≤∧i∈ΩSνi.
By Lemma8( 2), ∀i∈Ω,t1,t2∈S, we know
(56)νit1∧νit2∧x≤νit1t2.
Assumey∈JL; it satisfies
(57)y≺∨i∈Ωνit1∧∨i∈Ωνit2∧x.
Then, we have(58)y≺∨i∈Ωνis1,y≺∨i∈Ωνit2,y≤x.
So there existsi,j∈Ω; it satisfies
(59)y≤νit1,y≤νjt2,y≤x.
Forνi:i∈Ω, suppose νj≤Ai; it follows that y≤νit1∧νit2∧x. By νit1∧νit2∧x≤νit1t2, we obtain y≤νit1t2. Hence, y≤∨i∈Ωνit1t2. Then, we can infer
(60)∨i∈Ωνit1∧∨i∈Ωνit2∧x≤∨i∈Ωνit1t2.
Combining Lemma8 (2), we have x≤S∨i∈Ωνi. Then, we obtain∧i∈ΩSνi≤S∨i∈Ωνi. Therefore, it is proved.
(2) There is something in common shown in Theorem17 (1), and it is omitted here.In the rest of this paper, we will study the properties ofLFC on semigroup. We denote L-convexity with LC. From [22, 23], we get the equivalent characterization between an LC and LFC. It is found that SL is an LFC if and only if SxLx is an LC for any x∈SL\⊥ or SxLx is an LC for any x∈α⊥. Then, we get the next theorem.Theorem 18.
LetS be a semigroup; then, the following characterizations are equivalent:
(1)
SL is the LFC induced by SνLν, where ν∈LS(2)
∀x∈JL, SxLx is an LC on S(3)
∀x∈α⊥, SxLx is an LC on SCombining Definition3 and Theorem 12, we can get the next theorem.Theorem 19.
Letf:S0⟶S1 be a semigroup homomorphic mapping and SS0LS0 and SS1LS1 be the L-fuzzy convexities induced by L-fuzzy subsemigroup (ideal) measures on S0 and S1, respectively. Then,
(1)
f:S0,SS0⟶S1,SS1 is an LFCPM(2)
f:S0,LS0⟶S1,LS1 is an LFCPM(3)
f:S0,SS0⟶S1,SS1 is an LFCCM(4)
f:S0,LS0⟶S1,LS1 is an LFCCMProof.
To prove (1) and (2) hold, combining Theorem12 (2) and (4) with the conditions f:S0⟶S1 is a semigroup homomorphic mapping and SS0LS0 and SS1LS1 are the L-fuzzy convexities induced by L-fuzzy subsemigroup (ideal) measures on S0 and S1, we can get
(61)SS0fS⟵ν≥SS1νLfL⟵ν≥Lνforanyη∈LS′.
Then, we can infer (1) and (2) hold according to Definition4.
Similarly, we can prove (3) and (4) hold.Combining Theorem13 and Theorem 19 with Definition 4, we can get the next theorem in a similar way of Theorem 19.Theorem 20.
LetSii=1n be a family of semigroup and Pii=1n be a family of projection by Pi:∏i=1nSi⟶Sii=1,2,⋯,n. Then,
(1)
Pi:∏i=1nSi,S∏i=1nSi⟶Si,SSi (Pi:∏i=1nSi,L∏i=1nSi⟶Si,LSi) is an LFCPM(2)
Pi:∏i=1nSi,S∏i=1nSi⟶Si,SSi (Pi:∏i=1nSi,L∏i=1nSi⟶Si,LSi) is an LFCCMFrom Definition4 and Theorem 19 again, we can infer the following theorem.Theorem 21.
Letf be a semigroup homomorphic mapping and a bijective by f:S0⟶S1. Then
(1)
f:S0,SS0⟶S1,SS1 is an L-fuzzy isomorphism(2)
f:S0,LS0⟶S1,LS1 is an L-fuzzy isomorphism


## 5. Conclusions

It is presented in this paper that anyL-fuzzy subsets can be considered as an L-fuzzy subsemigroup (ideal) on semigroup to some measure. By this, we obtained L-fuzzy convexities and studied their properties. Next, we will research the related definitions and properties on order semigroup.



---

*Source: 5830590-2022-09-30.xml*