On Some NeutroHyperstructures

2021·0.26 MB·

by  Madeleine Al-Tahan

SS
symmetry
Article
On Some NeutroHyperstructures
MadeleineAl-Tahan1 ,BijanDavvaz2 ,FlorentinSmarandache3,* andOsmanAnis4
1 DepartmentofMathematics,LebaneseInternationalUniversity,Bekaa1803,Lebanon;
[email protected]
2 DepartmentofMathematics,YazdUniversity,Yazd89139,Iran;[email protected]
3 MathematicsandScienceDivision,GallupCampus,UniversityofNewMexico,705GurleyAve.,
Gallup,NM87301,USA
4 DepartmentofMathematics,LebaneseInternationalUniversity,Akkar35111,Lebanon;
[email protected]
* Correspondence:[email protected]
Abstract: Neutrosophy,thestudyofneutralities,isanewbranchofPhilosophythathasapplications
inmanydifferentfieldsofscience.InspiredbytheideaofNeutrosophy,Smarandacheintroduced
NeutroAlgebraicStructures(orNeutroAlgebras)byallowingthepartialityandindeterminacytobe
includedinthestructures’operationsand/oraxioms.Theaimofthispaperistocombinetheconcept
ofNeutrosophywithhyperstructurestheory.Inthisregard,weintroduceNeutroSemihypergroupsas
wellasNeutroHv-Semigroupsandstudytheirpropertiesbyprovidingseveralillustrativeexamples.
Keywords:NeutroHypergroupoid;NeutroSemihypergroup;NeutroHv-semigroup;NeutroHyper-
ideal;NeutroStrongIsomorphism
(cid:1)(cid:2)(cid:3)(cid:1)(cid:4)(cid:5)(cid:6)(cid:7)(cid:8)(cid:1) MSC:03A99;03G99;20N20
(cid:1)(cid:2)(cid:3)(cid:4)(cid:5)(cid:6)(cid:7)
Citation: Al-Tahan,M.;Davvaz,B.;
Smarandache,F.;Anis,O. OnSome
1. Introduction
NeutroHyperstructures.Symmetry
2021,13, 535. https://doi.org/ In1995andinspiredbytheexistenceofneutralities,SmarandacheintroducedNeu-
10.3390/sym13040535 trosophyasanewbranchofPhilosophythatdealswithindeterminacy. Duringthepast,
ideaswereviewedas“True”or“False”;however, ifweviewanideafromaneutrosophic
AcademicEditor:PalleE.T. point of view, it will be “True”, “False”, or “Indeterminate”. The indeterminacy is the
Jorgensen key that distinguishes Neutrosophy from other approaches. In the past twenty years,
this field demonstrated important progress in which it grabbed the attention of many
Received:20February2021
researchersanddifferentworksweredonefrombothatheoreticalpointofviewandfrom
Accepted:23March2021
anapplicativeview. Unlikeourrealworldthatisfullofimperfectionsandpartialities,
Published:25March2021
abstractsystemsareconstructedonagivenperfectspace(set),wheretheoperationsare
totallywell-definedandtheaxiomsaretotallytrueforallspacialelements. Startingfrom
Publisher’sNote:MDPIstaysneutral
thelatteridea,Smarandache[1–3]introducedNeutroAlgebra,whoseoperationsarepar-
withregardtojurisdictionalclaimsin
tiallywell-defined,partiallyindeterminate,andpartiallyouter-defined,andtheaxiomsare
publishedmapsandinstitutionalaffil-
partiallytrue,partiallyindeterminate,andpartiallyfalse. Manyresearchersworkedon
iations.
specialtypesofNeutroAlgebrasbyapplyingthemtodifferenttypesofalgebraicstructures
suchasgroups,rings,BE-Algebras,BCK-Algebras,etc. Formoredetails,wereferto[4–10].
On the other hand, hyperstructure theory is a generalization of classical algebraic
structuresandwasintroducedin1934attheeighthCongressofScandinavianMathemati-
Copyright: © 2021 by the authors.
cians by Marty [11]. Marty generalized the notion of groups by defining hypergroups.
Licensee MDPI, Basel, Switzerland.
Theclassofalgebraichyperstructuresislargerthanthatofalgebraicstructureswherethe
This article is an open access article
operationontwoelementsinthelatterisagainanelement,whereasthehyperoperationof
distributed under the terms and
twoelementsinthefirstclassisanon-voidset. Fordetailsabouthyperstructuretheoryand
conditionsoftheCreativeCommons
itsapplications,werefertothearticles[12–15]andthebooks[16–18]. Ageneralizationof
Attribution(CCBY)license(https://
algebraichyperstructures,knownasweakhyperstructures(H -structures),wasintroduced
creativecommons.org/licenses/by/ v
4.0/).
Symmetry2021,13,535.https://doi.org/10.3390/sym13040535 https://www.mdpi.com/journal/symmetry
Symmetry2021,13,535 2of12
in 1994 by Vougiouklis [19]. The axioms in the latter are weaker than that of algebraic
hyperstructures. FordetailsaboutH -structures,wereferto[19–22].
v
AsanaturalextensionofNeutroAlgebraicStructure,NeutroHyperstructurewasde-
finedrecently[23,24]whereIbrahimandAgboola[23]definedNeutroHypergroupsand
studiedaspecialtype. OurpaperisconcernedaboutsomeNeutroHyperstructuresandis
organizedasfollows: Section2presentssomebasicpreliminariesrelatedtohyperstructure
theory. Section3definesNeutroSemihypergroups,NeutroH -Semigroups,andsomere-
v
latednewconceptsandillustratesthesenewconceptsviaexamples. Moreover,westudy
somepropertiesoftheirsubsetsunderNeutroStrongHomomorphism.
2. AlgebraicHyperstructures
In this section, we present some definitions and examples about (weak) algebraic
hyperstructuresthatareusedthroughoutthepaper. Formoredetailsabouthyperstructure
theory,wereferto[16–20].
Definition1([16]). LetHbeanon-emptysetandP∗(H)bethefamilyofallnon-emptysubsets
ofH. Then,amapping◦ : H×H → P∗(H)iscalledabinaryhyperoperationon H. Thecouple
(H,◦)iscalledahypergroupoid.
If AandBaretwonon-emptysubsetsofHandh ∈ H,thenwedefine:
A◦B = (cid:83) a◦b, h◦A = {h}◦Aand A◦h = A◦{h}.
a∈A
b∈B
Ahypergroupoid(H,◦)iscalledasemihypergroupiftheassociativeaxiomissatisfied.
i.e.,foreveryx,y,z ∈ H,x◦(y◦z) = (x◦y)◦z. Inotherwords,
(cid:83) x◦u = (cid:83) v◦z.
u∈y◦z v∈x◦y
Anelementhinahypergroupoid(H,◦)iscalledidempotentifh◦h = h.
Example 1. Let H be any non-empty set and define “(cid:63)” on H as follows. For all x,y ∈ H,
x(cid:63)y = {x,y}. Then(H,(cid:63))isasemihypergroup.
Example2. LetH = {e,b,c}and(H ,+)bedefinedbythefollowingtable.
0 0
+ e b c
e e {e,b} {e,c}
b e {e,b} {e,c}
c e {e,b} {e,c}
Then(H ,+)isasemihypergroupandeisanidempotentelementinH .
0 0
Asageneralizationofalgebraichyperstructures,Vougiouklis[19,20]introduced H -
v
structures. Weakaxiomsin H -structuresreplacesomeaxiomsofclassicalalgebraichyper-
v
structures.
Definition2([19,20]). Ahypergroupoid(H,◦)iscalledanH -semigroupiftheweakassociative
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axiomissatisfied. i.e.,(x◦(y◦z))∩((x◦y)◦z) (cid:54)= ∅forallx,y,z ∈ H.
Example3. Let H = {0,1,2,3} and“+”bethehyperoperationon H definedbythefollow-
1 1
ingtable.
+ 0 1 2 3
0 0 1 {0,2} 3
1 1 2 3 0
2 2 3 0 1
3 3 0 1 2
Symmetry2021,13,535 3of12
Then(H ,+)isanH -semigroup.
1 v
Remark1. EverysemigroupisasemihypergroupandeverysemihypergroupisanH -semigroup.
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Definition3([17]). Let(H,◦)beasemihypergroup(H -semigroup)and M (cid:54)= ∅ ⊆ H. Then M
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isa
1. subsemihypergroup(H -subsemigroup)ofHif(M,◦)isasemihypergroup(H -semigroup).
v v
2. lefthyperidealofHif Misasubsemihypergroup(H -subsemigroup)ofHandh◦a ⊆ Mfor
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allh ∈ H.
3. righthyperidealofHif Misasubsemihypergroup(H -subsemigroup)ofHanda◦h ⊆ Mfor
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allh ∈ H.
4. hyperidealofHif Misboth: alefthyperidealofHandarighthyperidealofH.
Remark2. Let(H,◦)beasemihypergroup(H -semigroup)and M (cid:54)= ∅ ⊆ H. Toprovethat M
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issubsemihypergroup(H -subsemigroup)ofH,itsufficestoshowthata◦b ⊆ Mforalla,b ∈ M.
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3. NeutroHyperstructures
Inthissection,wedefineNeutroSemihypergroupsandNeutroH -Semigroups,present
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some illustrative examples, and study several properties of some important subsets of
NeutroSemihypergroupsandNeutroH -Semigroups.
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Definition4. Let Abeanynon-emptysetand“·”beahyperoperationon A. Then“·”iscalled
a NeutroHyperoperation on A if some (or all) of the following conditions hold in a way that
(T,I,F) ∈/ {(1,0,0),(0,0,1)}.
1. Thereexistx,y ∈ Awithx·y ⊆ A. (Thisconditioniscalleddegreeoftruth,“T”).
2. Thereexistx,y ∈ Awithx·y(cid:42) A. (Thisconditioniscalleddegreeoffalsity,“F”).
3. There exist x,y ∈ A with x·y is indeterminate in A. (This condition is called degree of
indeterminacy,“I”).
Definition5. Let Abeanynon-emptysetand“·”beahyperoperationon A. Then“·”iscalledan
AntiHyperoperationon Aifx·y(cid:42) Aforallx,y ∈ A.
Definition6. Let Abeanynon-emptysetand“·”beahyperoperationon A. Then“·”iscalled
NeutroAssociative on A if there exist x,y,z,a,b,c,e, f,g ∈ A satisfying some (or all) of the
followingconditionsinawaythat(T,I,F) ∈/ {(1,0,0),(0,0,1)}.
1. x·(y·z) = (x·y)·z;(Thisconditioniscalleddegreeoftruth,“T”).
2. a·(b·c) (cid:54)= (a·b)·c;(Thisconditioniscalleddegreeoffalsity,“F”).
3. e·(f ·g) is indeterminate or (e· f)·g is indeterminate or we cannot find if e·(f ·g) and
(e· f)·gareequal. (Thisconditioniscalleddegreeofindeterminacy,“I”).
Definition7. Let Abeanynon-emptysetand“·”beahyperoperationon A. Then“·”iscalled
AntiAssociativeon Aifa·(b·c) (cid:54)= (a·b)·cforalla,b,c ∈ A.
Definition8. Let Abeanynon-emptysetand“·”beahyperoperationon A. Then“·”iscalleda
NeutroWeakAssociativeon Aifthereexistx,y,z,a,b,c,e, f,g ∈ Asatisfyingsome(orall)ofthe
followingconditionsinawaythat(T,I,F) ∈/ {(1,0,0),(0,0,1)}.
1. [x·(y·z)]∩[(x·y)·z] (cid:54)= ∅;(Thisconditioniscalleddegreeoftruth,“T”).
2. [a·(b·c)]∩[(a·b)·c] = ∅;(Thisconditioniscalleddegreeoffalsity,“F”).
3. e·(f ·g) is indeterminate or (e· f)·g is indeterminate or we cannot find if e·(f ·g) and
(e· f)·ghavecommonelements. (Thisconditioniscalleddegreeofindeterminacy,“I”).
Definition9. Let Abeanon-emptysetand“·”beahyperoperationon A. Then(A,·)iscalleda
1. NeutroHypergroupoidif“·”isaNeutroHyperoperation.
2. NeutroSemihypergroupif“·”isNeutroAssociativebutnotanAntiHyperoperation.
Symmetry2021,13,535 4of12
3. NeutroH -Semigroupif“·”isNeutroWeakAssociativebutnotanAntiHyperoperation.
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Example4. Let A = {0,1}and(A,+)bedefinedbythefollowingtable.
+ 0 1
0 {0,1} 0
1 1 0
Then(A,+)isaNeutroSemihypergroupandNeutroH -Semigroup. Thisisclearas
v
0+(0+0) = {0,1} = (0+0)+0and(1+1)+1=0(cid:54)=1=1+(1+1).
Example5. LetRbethesetofrealnumbersanddefine“(cid:63)”onRasfollows.

[[yx,,xy]] iiffyx<<xy;;
x(cid:63)y =
01 iiffxx == yy =(cid:54)=00;.
x
Then(R,(cid:63))isaNeutroSemihypergroup. Thisisclearas(1(cid:63)1)(cid:63)1 = 1 = 1(cid:63)(1(cid:63)1)and
(1(cid:63)2)(cid:63)2= {1}∪[1,2] (cid:54)= [1,1] =1(cid:63)(2(cid:63)2).
2 2
Example6. Let M = {m,a,d}and(M,·)bedefinedbythefollowingtable.
· m a d
m m m m
a m {m,a} d
d m d d
Then(M,·)isaNeutroSemihypergroup. Thisisclearasm·(m·m) = m = (m·m)·mand
a·(a·d) = d (cid:54)= {m,d} = (a·a)·d.
Remark3. Itiswellknowninclassicalalgebraichyperstructuresthateverysemihypergroupis
a hypergroupoid. This may fail to occur in NeutroHyperstructures. In Example 6, (M,·) is a
NeutroSemihypergroupthatisnotaNeutroHypergroupoid.
Proposition1. Every H -semigroupthatisnotasemihypergroupandhasanidempotentelement
v
isaNeutroSemihypergroup.
Proof. Let(H,◦)beanH -semigroupwithh2 = hforsomeh ∈ H. Thenh◦(h◦h) = h =
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(h◦h)◦h. Since(H,◦)isnotasemihypergroup,itfollowsthatthereexistx,y,z ∈ Hwith
x◦(y◦z) (cid:54)= (x◦y)◦z. Therefore,(H,◦)isaNeutroSemihypergroup.
Example7. Let M = {m,a,d}and(M,(cid:5))bedefinedbythefollowingtable.
(cid:5) m a d
m m {a,d} d
a {a,d} d m
d d m a
Then (M,(cid:5)) is an H -semigroup having m as an idempotent element and hence, it is a
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NeutroSemihypergroup.
Symmetry2021,13,535 5of12
Remark 4. It is well known in algebraic hyperstructures that every semihypergroup is an H -
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semigroup. ThismaynotholdinNeutroHyperstructures. i.e.,ANeutroSemihypergroupmaynot
beaNeutroH -Semigroup.
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The H -semigroup(M,(cid:5))inExample7isaNeutroSemihypergroupthatisnotNeutroH -
v v
Semigroup.
Example8. LetZbethesetofintegersanddefine“⊕”onZ2asfollows. Forallm,n,p,q ∈Z,
(m,0)⊕(0,0) = (0,0)⊕(m,0) = {(0,0),(m,0)},
(0,n)⊕(0,0) = (0,0)⊕(0,n) = {(0,0),(0,n)},
andif(n,p,q) (cid:54)= (0,0,0),(m,p,q) (cid:54)= (0,0,0)
(m,n)⊕(p,q) = (p,q)⊕(m,n) = (m+p,n+q).
Then(Z2,⊕)isaNeutroSemihypergroup. Thisisclearas
[(1,2)⊕(1,3)]⊕(1,4) = (3,9) = (1,2)⊕[(1,3)⊕(1,4)]
and
[(1,0)⊕(1,0)]⊕(0,0) = {(2,0),(0,0)} (cid:54)= {(2,0),(1,0),(0,0)} = (1,0)⊕[(1,0)⊕(0,0)].
Example9. LetZbethesetofintegersanddefine“(cid:12)”onZ2asfollows. Forallm,n,p,q ∈Z,

(mp,nq) if(m,n) (cid:54)= (1,1)and(p,q) (cid:54)= (1,1);

(m,n)(cid:12)(p,q) = {(p,q),(1,1)} if(m,n) = (1,1);
{(m,n),(1,1)} if(p,q) = (1,1).
Then(Z2,(cid:12))isaNeutroSemihypergroup. Thisisclearas
[(1,2)(cid:12)(1,3)](cid:12)(1,4) = (1,24) = (1,2)(cid:12)[(1,3)(cid:12)(1,4)]
and
(1,1)(cid:12)[(2,2)(cid:12)(3,3)] = {(1,1),(6,6)} (cid:54)= {(1,1),(3,3),(6,6)} = [(1,1)(cid:12)(2,2)](cid:12)(3,3).
Z (cid:1) Z
Example10. Let bethesetofintegersunderadditionmodulo6anddefine“ ”on asfollows.
6 6
x(cid:1)y = (x+y) mod 6forall(x,y) ∈/ {(0,3),(0,5)},
0(cid:1)3= {0,3}, and0(cid:1)5= {0,5}.
Then(Z ,(cid:1))isaNeutroSemihypergroup. Thisisclearas0(cid:1)(0(cid:1)0) = 0 = (0(cid:1)0)(cid:1)0
6
and0(cid:1)(1(cid:1)2) = {0,3} (cid:54)=3= (0(cid:1)1)(cid:1)2.
Example11. Let M = {m,a,d}and(M,•)bedefinedbythefollowingtable.
• m a d
m a a d
a {m,a} m d
d d d m
Then(M,•)isaNeutroH -Semigroup. Thisisclearas
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[m•(m•m)]∩[(m•m)•m] = {a}∩{m,a} (cid:54)= ∅
Symmetry2021,13,535 6of12
and
[m•(d•d)]∩[(m•d)•d] = {a}∩{m} = ∅.
Moreover,(M,•)isaNeutroSemihypergroupasd•(d•d) = (d•d)•d.
Remark5. EveryNeutroSemigroupisboth:aNeutroSemihypergroupandaNeutroH -Semigroup.
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So,theresultsrelatedtoNeutroSemihypergroups(NeutroH -Semigroups)aremoregeneralthan
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thatrelatedtoNeutroSemigroupsandasaresult,wecandealwithNeutroSemigroupsasaspecial
caseofNeutroSemihypergroups(NeutroH -Semigroups).
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Example12. LetS = {s,a,m}and(S ,· )bedefinedbythefollowingtable.
1 1 1
· s a m
1
s s m s
a m a m
m m m m
In[6],Al-Tahanetal. provedthat (S ,· ) isaNeutroSemigroup. Thus, (S ,· )isaNeu-
1 1 1 1
troSemihypergroup.
Theorem1. Let(H,◦)beaNeutroSemihypergroup(NeutroH -Semigroup)and“(cid:63)”bedefined
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on H as x(cid:63)y = y◦x for all x,y ∈ H. Then (H,(cid:63)) is a NeutroSemihypergroup (NeutroH -
v
Semigroup).
Proof. Theproofisstraightforward.
Example13. LetM = {m,a,d}and(M,•)betheNeutroSemihypergroupdefinedinExample11.
ByapplyingTheorem1,wegetthat(M,(cid:126))definedinthefollowingtableisaNeutroSemihypergroup
andaNeutroH -Semigroup.
v
(cid:126)
m a d
m a {m,a} d
a a m d
d d d m
Definition10. Let(H,◦)beaNeutroSemihypergroup(NeutroH -Semigroup)andS (cid:54)= ∅ ⊆ H.
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ThenSisaNeutroSubsemihypergroup(NeutroH -Subsemigroup)of Hif(S,◦)isaNeutroSemi-
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hypergroup(NeutroH -Semigroup).
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Remark 6. Let (H,◦) be a NeutroSemihypergroup (NeutroH -Semigroup) and S (cid:54)= ∅ ⊆ H.
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Unlikethecaseinalgebraichyperstructures(Remark2),provingthata◦b ⊆ Sforalla,b ∈ Sdoes
notimplythatSisaNeutroSubsemihypergroup(NeutroH -Subsemigroup)ofH.
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As an illustration of Remark 6, 0(cid:63)0 = {0} ⊆ {0} in Example 5 but {0} is not a
R
NeutroSubsemihypergroupof .
Definition11. Let(H,◦)beaNeutroSemihypergroup(NeutroH -Semigroup)andS (cid:54)= ∅ ⊆ H
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beaNeutroSubsemihypergroup(NeutroH -Subsemigroup). Then
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(1) SisaNeutroLeftHyperidealofHifthereexistsx ∈ Ssuchthatr◦x ⊆ Sforallr ∈ H.
(2) SisaNeutroRightHyperidealofSifthereexistsx ∈ Ssuchthatx◦r ⊆ Sforallr ∈ H.
(3) S isaNeutroHyperidealof H ifthereexists x ∈ S suchthatr◦x ⊆ S and x◦r ⊆ S for
allr ∈ H.
ANeutroSemihypergroup(NeutroH -Semigroup)iscalledsimpleifithasnoproper
v
NeutroSubsemihypergroups(NeutroH -Subsemigroups).
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Symmetry2021,13,535 7of12
Example14. Let(A,+)betheNeutroSemihypergroupdefinedinExample4. Then Aissimple.
Thisisclearas{0}and{1}aretheonlyoptionsforanypossibleproperNeutroSubsemihypergroup
and({0},+)and({1},+)areAntiHypergroupoids.
Example15. Let(M,•)betheNeutroSemihypergroupdefinedinExample11. Then{m,a}isa
NeutroSubsemihypergroupof M.
Example16. Let(Z2,⊕)betheNeutroSemihypergroupdefinedinExample8, M = {(x,0) :
1
x ∈Z},and M = {(0,x) : x ∈Z}. Then M ,M areNeutroSubsemihypergroupsofZ2.
2 1 2
Remark7. TheintersectionofNeutroSubsemihypergroupsmayfailtobeaNeutroSubsemihypergroup.
ThisisclearfromExample16as{(0,0)} = M ∩M isnotaNeutroSubsemihypergroupofZ2.
1 2
Lemma 1. Let (H,◦) be a NeutroSemihypergroup (NeutroH -Semigroup) and A,B be hyper-
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groupoids. If A,BareNeutroSubsemihypergroups(NeutroH -Subsemigroups)ofHthen A∪Bis
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aNeutroSubsemihypergroup(NeutroH -Subsemigroup)ofH.
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Proof. Let A,B be NeutroSubsemihypergroups. Since A and B are hypergroupoids, it
followsthat“◦”isNeutroAssociativeonbothofAandB. Thelatterimpliesthatthereexist
x,y,z,a,b,c,e, f,g ∈ A ⊆ A∪Bsatisfyingsome(orall)ofthefollowingconditionsina
waythat(T,I,F) ∈/ {(1,0,0),(0,0,1)}.
1. T: x◦(y◦z) = (x◦y)◦z;
2. F: a◦(b◦c) (cid:54)= (a◦b)◦c;
3. I: e◦(f ◦g)isindeterminateor(e◦ f)◦gisindeterminateorwecannotfindife◦(f ◦g)
and(e◦ f)◦gareequal.
Therefore, A∪B is a NeutroSubsemihypergroup of H. The proof of (NeutroH -
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Subsemigroupisdonesimilarly.
Example17. Let(Z2,(cid:12))betheNeutroSemihypergroupdefinedinExample9, N = {(x,y) ∈
1
Z2 : x,y ≥ 1}∪{(0,0)}, and N = {(x,y) ∈ Z2 : x,y ≤ 1}∪{(0,0)}. Then N ,N are
2 1 2
NeutroHyperideals of Z2. We show that N is a NeutroHyperideal of Z2 and N may be done
1 2
similarly. Since
[(1,2)(cid:12)(1,3)](cid:12)(1,4) = (1,24) = (1,2)(cid:12)[(1,3)(cid:12)(1,4)]
and
(1,1)(cid:12)[(2,2)(cid:12)(3,3)] = {(1,1),(6,6)} (cid:54)= {(1,1),(3,3),(6,6)} = [(1,1)(cid:12)(2,2)](cid:12)(3,3),
itfollowsthatN isaNeutroSubsemihypergroupofZ2. Having(0,0) ∈ N andforall(r,s) ∈Z2,
1 1
(cid:40)
(0,0) if(r,s) (cid:54)= (1,1);
(r,s)(cid:12)(0,0) = (0,0)(cid:12)(r,s) = ⊆ N
1
{(0,0),(1,1)} otherwise.
impliesthatN isaNeutroHyperidealofZ2.
1
Remark8. TheintersectionofNeutroHyperidealsmayfailtobeaNeutroHyperideal. Thisisclear
fromExample17as{(0,0),(1,1)} = N ∩N isnotaNeutroHyperidealofZ2.
1 2
Lemma 2. Let (H,◦) be a NeutroSemihypergroup (NeutroH -Semigroup) and A,B be hyper-
v
groupoids. If A,BareNeutroLeftHyperideals(NeutroRightHyperidealsorNeutroHyperideals)of
H. Then A∪BisaNeutroLeftHyperideal(NeutroRightHyperidealorNeutroHyperideal)ofH.
Proof. Let A,BbeNeutroLeftHyperidealsofH. Lemma1assertsthat A∪BisaNeutro-
Subsemihypergroup(NeutroH -Subsemigroup)ofH. Since AisaNeutroLeftHyperideal
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Symmetry2021,13,535 8of12
of H, it follows that there exists a ∈ A such that r◦a ⊆ A for all r ∈ H. The latter
impliesthatthereexistsa ∈ A∪Bsuchthatr◦a ⊆ A∪Bforallr ∈ H. Thus, A∪Bisa
NeutroLeftHyperidealofH.
Definition12. Let(H,◦),(H(cid:48),(cid:63))beNeutroSemihypergroups(NeutroH -Semigroups)andφ :
v
H → H(cid:48) beafunction. Then
(1) φiscalledNeutroHomomorphismifφ(x◦y) = φ(x)(cid:63)φ(y)forsomex,y ∈ A.
(2) φiscalledNeutroIsomomorphismifφisabijectiveNeutroHomomorphism.
(3) φiscalledNeutroStrongHomomorphismifforallx,y ∈ A,φ(x◦y) = φ(x)(cid:63)φ(y)when
x◦y ⊆ H, φ(x)(cid:63)φ(y) (cid:42) H(cid:48) when x◦y (cid:42) H, and φ(x)(cid:63)φ(y) is indeterminate when
x◦yisindeterminate.
(4) φiscalledNeutroStrongIsomomorphismifφisabijectiveNeutroOrderedStrongHomomor-
phism. Inthiscasewesaythat(H,◦) ∼= (H(cid:48),(cid:63)).
SI
Example 18. Let (M,•) and (M,(cid:126)) be the NeutroSemihypergroups defined in
Examples 11 and 13, respectively. Then (M,•) ∼= (M,(cid:126)) as φ : (M,•) → (M,(cid:126)) is a
SI
NeutroStongIsomorphism. Here,
φ(m) = a,φ(a) = m, andφ(d) = d.
∼
Theorem2. Therelation“= ”isanequivalencerelationonthesetofNeutroSemihypergroups
SI
(NeutroH -Semigroups).
v
∼
Proof. Bytakingtheidentitymap,wecaneasilyprovethat“= ”isareflexiverelation.
SI
Let A ∼= B. Then there exists a NeutroStrongIsomorphism φ : (A,(cid:63)) → (B,(cid:126)). We
SI
provethattheinversefunctionφ−1 : B → AofφisaNeutroStrongIsomorphism. Forall
b ,b ∈ B,thereexista ,a ∈ Awithφ(a ) = b andφ(a ) = b . Wehave
1 2 1 2 1 1 2 2
φ−1(b (cid:126)b ) = φ−1(φ(a )(cid:126)φ(a ))
1 2 1 2
Weconsiderthefollowingcasesforφ(a )(cid:126)φ(a ).
1 2
Caseφ(a )(cid:126)φ(a ) ⊆ B. HavingφaNeutroStrongIsomorphismandφ(a )(cid:126)φ(a ) ⊆
1 2 1 2
Bimplythata (cid:63)a ⊆ Aandhence,
1 2
φ−1(b (cid:126)b ) = φ−1(φ(a )(cid:126)φ(a )) = φ−1(φ(a (cid:63)a )) = a (cid:63)a = φ−1(b )(cid:63)φ−1(b ).
1 2 1 2 1 2 1 2 1 2
Caseφ(a )(cid:126)φ(a )(cid:42) B.Suppose,togetcontradiction,thatφ−1(φ(a ))(cid:63)φ−1(φ(a )) =
1 2 1 2
a (cid:63)a ⊆ Aorindeterminate. ThenbyusingourhypothesisthatφisNeutroStrongIsomor-
1 2
phism,wegetthatφ(a )(cid:126)φ(a ) ⊆ Borindeterminate.
1 2
Caseφ(a )(cid:126)φ(a )isindeterminate. Suppose,togetcontradiction,thatφ−1(φ(a ))(cid:63)
1 2 1
φ−1(φ(a )) = a (cid:63)a ⊆ A or a (cid:63)a (cid:42) A. Then by using our hypothesis that φ is Neu-
2 1 2 1 2
troStrongIsomorphism,wegetthatφ(a )(cid:126)φ(a ) ⊆ Borφ(a )(cid:126)φ(a )(cid:42) B.
1 2 1 2
∼ ∼ ∼ ∼
Thus, B = Aandhence,“= ”isasymmetricrelation. Let A = BandB = C.
SI SI SI SI
ThenthereexistNeutroStrongIsomorphismsφ : A → Bandψ : B →C. Onecaneasilysee
thatthecompositionfunctionψ◦φ : A → CofψandφisaNeutroStrongIsomorphism.
∼ ∼
Thus, A = Candhence,“= ”isatransitiverelation.
SI SI
Lemma 3. Let (H,◦), (H(cid:48),(cid:63)) be NeutroSemihypergroups (NeutroH -Semigroups) and φ :
v
H → H(cid:48) be an injective NeutroStrongHomomorphism. If M ⊂ H is a NeutroSubsemihyper-
group (NeutroH -Subsemigroup) of H then φ(M) is a NeutroSubsemihypergroup (NeutroH -
v v
Subsemigroup)ofH(cid:48).
Proof. Let MbeaNeutroSubsemihypergroupof H. If“◦”isNeutroHyperoperationon
Mthenitisclearthat“(cid:63)”isNeutroHyperoperationonφ(M). If“◦”isNeutroAssociative
Symmetry2021,13,535 9of12
thenthereexistx,y,z,a,b,c,d,e, f ∈ Msatisfyingsome(orall)ofthefollowingconditions
inawaythat(T,I,F) ∈/ {(1,0,0),(0,0,1)}.
1. T: x◦(y◦z) = (x◦y)◦z;
2. F: a◦(b◦c) (cid:54)= (a◦b)◦c;
3. I: e◦(f ◦g)isindeterminateor(e◦ f)◦gisindeterminateorwecannotfindife◦(f ◦g)
and(e◦ f)◦gareequal.
ThelatterandhavingφaninjectiveNeutroStrongHomomorphismimplythatsome
(orall)ofthefollowingconditionsaresatisfiedinawaythat(T,I,F) ∈/ {(1,0,0),(0,0,1)}.
1. T: φ(x)(cid:63)(φ(y)(cid:63)φ(z)) = (φ(x)(cid:63)φ(y))(cid:63)φ(z);
2. F: φ(a)(cid:63)(φ(b)(cid:63)φ(c)) (cid:54)= (φ(a)(cid:63)φ(b))(cid:63)φ(c);
3. I: φ(e)(cid:63)(φ(f)(cid:63)φ(g))isindeterminateor(φ(e)(cid:63)φ(f))(cid:63)φ(g)isindeterminateorwe
cannotfindifφ(e)(cid:63)(φ(f)(cid:63)φ(g))and(φ(e)(cid:63)φ(f))(cid:63)φ(g)areequal.
Thus, φ(M) is a NeutroSubsemihypergroup. The proof that φ(M) is a NeutroH -
v
SubsemigroupofH(cid:48) isdonesimilarly.
Example 19. Let (M,•) and (M,(cid:126)) be the NeutroSemihypergroups defined in
Examples 11 and 13, respectively. Example 15 asserts that {m,a} is a NeutroSubsemihyper-
group of (M,•). Using Example 18 and Lemma 3, we get that {a,m} = {φ(m),φ(a)} is a
NeutroSubsemihypergroupof(M,(cid:126)).
Lemma4. Let(H,◦),(H(cid:48),(cid:63))beNeutroSemihypergroups(NeutroH -Semigroups)andφ : H →
v
H(cid:48) be a NeutroStrongIsomomorphism. If N ⊆ H(cid:48) is a NeutroSubsemihypergroup (NeutroH -
v
Subsemigroup) of H(cid:48) then φ−1(N) is a NeutroSubsemihypergroup (NeutroH -Subsemigroup)
v
ofH.
Proof. Let N ⊂ H(cid:48) be a NeutroSubsemihypergroup of H(cid:48). If “(cid:63)” is NeutroHyperop-
eration on N then it is clear that “◦” is NeutroHyperoperation on φ−1(N). Let “(cid:63)” be
NeutroAssociative. HavingφisanontoNeutroStrongHomomorphismimpliesthatthere
exist φ(x),φ(y),φ(z),φ(a),φ(b),φ(c),φ(d),φ(e),φ(f) ∈ N satisfyingsome(orall)ofthe
followingconditionsinawaythat(T,I,F) ∈/ {(1,0,0),(0,0,1)}.
1. T: φ(x)(cid:63)(φ(y)(cid:63)φ(z)) = (φ(x)(cid:63)φ(y))(cid:63)φ(z);
2. F: φ(a)(cid:63)(φ(b)(cid:63)φ(c)) (cid:54)= (φ(a)(cid:63)φ(b))(cid:63)φ(c);
3. I: φ(e)(cid:63)(φ(f)(cid:63)φ(g))isindeterminateor(φ(e)(cid:63)φ(f))(cid:63)φ(g)isindeterminateorwe
cannotfindifφ(e)(cid:63)(φ(f)(cid:63)φ(g))and(φ(e)(cid:63)φ(f))(cid:63)φ(g)areequal.
HavingφbeaninjectiveNeutroStrongHomomorphismimpliesthatthereexistx,y,z,a,
b,c,d,e,f ∈ φ−1(N) satisfying some (or all) of the following conditions in a way that
(T,I,F) ∈/ {(1,0,0),(0,0,1)}.
1. T: x◦(y◦z) = (x◦y)◦z;
2. F: a◦(b◦c) (cid:54)= (a◦b)◦c;
3. I: e◦(f ◦g)isindeterminateor(e◦ f)◦gisindeterminateorwecannotfindife◦(f ◦g)
and(e◦ f)◦gareequal.
Thus, φ−1(N) is a NeutroSubsemihypergroup of H. The proof that φ−1(N) is a
NeutroH -SubsemigroupofHmaybedonesimilarly.
v
Theorem3. Let(H,◦),(H(cid:48),(cid:63))beNeutroSemihypergroups(NeutroH -Semigroups)andφ : H →
v
H(cid:48) be a NeutroStrongIsomorphism. Then M ⊆ H is a NeutroSubsemihypergroup (NeutroH -
v
Subsemigroup)ofHifandonlyifφ(M)isaNeutroSubsemihypergroup(NeutroH -Subsemigroup)
v
ofH(cid:48).
Proof. TheprooffollowsfromTheorem2andLemmas3and 4.
Corollary 1. Let (H,◦), (H(cid:48),(cid:63)) be NeutroSemihypergroups (NeutroH -Semigroups) and φ :
v
H → H(cid:48) beaNeutroStrongIsomorphism. ThenHissimpleifandonlyifH(cid:48) issimple.
Symmetry2021,13,535 10of12
Proof. TheprooffollowsfromTheorem3.
Lemma5. Let(H,◦),(H(cid:48),(cid:63))beNeutroSemihypergroups(NeutroH -Semigroups)andφ : H →
v
H(cid:48) beaNeutroStrongIsomorphism. If M ⊆ HisaNeutroLeftHyperideal(NeutroRightHyperideal
or NeutroHyperideal) of H then φ(M) is a NeutroLeftHyperideal (NeutroRightHyperideal or
NeutroHyperideal)ofH(cid:48).
Proof. Let M ⊆ H be a NeutroLeftHyperideal of H. Lemma 3 asserts that φ(M) is a
NeutroSubsemihypergroup(NeutroH -Subsemigroup)of H(cid:48). Having MaNeutroLeftHy-
v
peridealofHimpliesthatthereexistsx ∈ Msuchthatr◦x ⊆ Mforallr ∈ H. Havingφan
ontoNeutroStrongHomomorphismimpliesthatφ(r)(cid:63)φ(x) ⊆ φ(M)foralls = φ(r) ∈ H(cid:48).
Thus,φ(M)isaNeutroLeftHyperidealof H(cid:48). TheproofsofNeutroRightHyperidealand
NeutroHyperidealaredonesimilarly.
Lemma6. Let(H,◦),(H(cid:48),(cid:63))beNeutroSemihypergroups(NeutroH -Semigroups)andφ : H →
v
H(cid:48) beaNeutroStrongIsomorphism. IfN ⊆ H(cid:48) isaNeutroLeftHyperideal(NeutroRightHyperideal
orNeutroHyperideal)ofH(cid:48) thenφ−1(N)isaNeutroLeftHyperideal(NeutroRightHyperidealor
NeutroHyperideal)ofH.
Proof. Let N ⊆ H(cid:48) beaNeutroLeftHyperidealof H. Lemma3assertsthat φ−1(N) isa
NeutroSubsemihypergroup(NeutroH -Subsemigroup)ofH. HavingNaNeutroLeftHy-
v
perideal of H(cid:48) implies that there exists y ∈ N such that s(cid:63)y ⊆ N for all s ∈ H(cid:48). Since
φ is an NeutroStrongHomomorphism, it follows that φ(r◦x) ⊆ N for all r ∈ H where
y = φ(x). Thelatterimpliesthatthereexistsx ∈ φ−1(N)withr◦x ⊆ φ−1(N)forallr ∈ H.
Thus,φ−1(N)isaNeutroLeftHyperidealofH. TheproofsofNeutroRightHyperidealand
NeutroHyperidealaredonesimilarly.
Theorem 4. Let (H,◦), (H(cid:48),(cid:63)) be NeutroSemihypergroups (NeutroH -Semigroups) and φ :
v
H → H(cid:48) be a NeutroStrongIsomorphism. Then M ⊆ H is a NeutroLeftHyperideal (Neu-
troRightHyperideal or NeutroHyperideal) of H if and only if φ(M) is a NeutroLeftHyperideal
(NeutroRightHyperidealorNeutroHyperideal)ofH(cid:48).
Proof. TheprooffollowsfromTheorem2,Lemmas5and 6.
Let H beanynon-emptysetforallα ∈ Γand“· ”beahyperoperationon H . We
α α α
define“◦”on∏α∈ΓHα asfollows: Forall(xα),(yα) ∈ ∏α∈ΓHα,(xα)◦(yα) = {(tα) : tα ∈
x · y }.
α α α
Theorem5. Let(H ,◦ )and(H ,◦ )behypergroupoids. Then(H ×H ,◦)isaNeutroSemi-
1 1 2 2 1 2
hypergroup (NeutroH -Semigroup) if and only if either (H ,◦ ) is a NeutroSemihypergroup
v 1 1
(NeutroH -Semigroup)or(H ,◦ )isaNeutroSemihypergroup(NeutroH -Semigroup)orbothare
v 2 2 v
NeutroSemihypergroups(NeutroH -Semigroups).
v
Proof. Theproofisstraightforward.
Example20. Let(R,(cid:62))bethesemihypergroupdefinedas: x(cid:62)y = {x,y}forallx,y ∈ Rand
(M,·)betheNeutroSemihypergroupdefinedinExample6. Thenthefollowingaretrue.
1. (R×M,◦)isaNeutroSemihypergroup,
2. (M×R,◦)isaNeutroSemihypergroup,and
3. (M×M,◦)isaNeutroSemihypergroup.
Inwhatfollows,wepresentawaytoconstructanewNeutroSemihypergroup(NeutroH -
v
Semigroup)fromanexistingone. Thistoolisofgreatimportancetoprovethatforanypos-
itiveintegern ≥2,thereexistsatleastoneNeutroSemihypergroup(NeutroH -Semigroup)
v
ofordern.