JournalofAlgebraicHyperstructures andLogicalAlgebras
Volume3,Number2,(2022),pp.17-24
IntroductiontoSuperHyperAlgebraandNeutrosophic SuperHyperAlgebra
F.Smarandache1
1TheUniversityofNewMexico,Mathematics,Physics,andNaturalScienceDivision,705GurleyAve., Gallup,NM87301,USA smarand@unm.edu
Abstract
Inthispaperwerecallourconceptsof nth-Power SetofaSet,SuperHyperOperation,SuperHyperAxiom, SuperHyperAlgebra,andtheircorrespondingNeutrosophicSuperHyperOperation,NeutrosophicSuperHyperAxiomandNeutrosophicSuperHyperAlgebra.Ingeneral,inanyfieldofknowledge,oneactuallyencountersSuperHyperStructures(ormoreaccurately(m,n)SuperHyperStructures).
ArticleInformation
CorrespondingAuthor: F.Smarandache; Received:March2022; Accepted:Invitedpaper; Papertype:Original.
Keywords: SuperHyperOperation, SuperHyperAlgebra,NeutrosophicSuperHyperAlgebra, SuperHyperStructures.
A classical m-aryHyperOperation ◦∗ m isdefinedas: ◦ ∗ m : Hm → P∗(H), forinteger m ≥ 1.For m =1onegetsa UnaryHyperOperation. The classicalHyperStructures arestructuresendowedwithclassicalHyperOperations. TheclassicalHyperOperationsandclassicalHyperStructureswereintroducedby F .Marty[12]in 1934.
2. Definitionofthe nth-PowerSetofaSet: The nth-PowersetofaSetwasintroducedin[16, 18, 19]inthefollowingway: P n(H),asthe nth-PowersetoftheSet H,forinteger n ≥ 1,isrecursivelydefinedas: P 2(H)= P (P (H)), P 3(H)= P (P 2(H))= P (P (P (H))), , P n(H)= P (P (n 1)(H)),where P ◦(H) def = H,and P 1(H) def = P (H).
The nth-PowersetofaSetbetterreflectsourcomplexreality,sinceaset H (thatmayrepresent agroup,asociety,acountry,acontinent,etc.)ofelements(suchas:people,objects,andingeneral anyitems)isorganizedontosubsets P (H),andthesesubsetsareagainorganizedontosubsetsof subsets P (P (H)),andsoon.That’sourworld.
3. NeutrosophicHyperOperationandNeutrosophicHyperStructures[12]:
IntheclassicalHyperOperationandclassicalHyperStructures,theempty-set ∅ doesnotbelong tothepowerset,or P∗(H)= P (H) \{∅}
However,intherealworldweencountermanysituationswhenaHyperOperation ◦ is indeterminate,forexample a ◦ b = ∅ (unknown,orundefined), or partiallyindeterminate,forexample: c ◦ d = {[0 2, 0 3], ∅}
Inoureverydaylife,therearemanymoreoperationsandlawsthathavesomedegreesofindeterminacy(vagueness,unclearness,unknowingness,contradiction,etc.),thanthosethataretotally determinate.
That’swhyin2016wehaveextendedtheclassicalHyperOperationtotheNeutrosophicHyperOperation,bytakingthewholepower P (H)(thatincludestheempty-set ∅ aswell),insteadof P∗(H)(thatdoesnotincludetheempty-set ∅),asfollows.
3
.
2 NeutrosophicHyperStructures:
ANeutrosophicHyperStructureisastructuredendowedwithNeutrosophicHyperOperations.
4 DefinitionofSuperHyperOperations:
Werecallour2016conceptsofSuperHyperOperation,SuperHyperAxiom,SuperHyperAlgebra, andtheircorrespondingNeutrosophicSuperHyperOperationNeutrosophicSuperHyperAxiomand NeutrosophicSuperHyperAlgebra[16].
Let P n ∗ (H)bethe nth-powersetoftheset H suchthatnoneof P (H), P 2(H), ,P n(H) containtheemptyset ∅
Also,let P n(H)bethe nth-powersetoftheset H suchthatatleastoneofthe P (H), P 2(H),··· , P n(H)containtheemptyset ∅
TheSuperHyperOperationsareoperationswhosecodomainiseither P n ∗ (H)andinthiscase onehas classical-typeSuperHyperOperations,or P n(H)andinthiscaseonehas NeutrosophicSuperHyperOperations,forinteger n ≥ 2.
4 1A classical-typeBinarySuperHyperOperation ◦∗ (2,n) isdefinedasfollows: ◦ ∗ (2,n) : H 2 → P n ∗ (H), where P n ∗ (H)isthe nth-powersetoftheset H,withnoempty-set.
4 2 Examplesofclassical-typeBinarySuperHyperOperation:
1)Let H = {a,b} beafinitediscreteset;thenitspowerset,withouttheempty-set ∅,is: P (H)= {a,b, {a,b}},and: P 2(H)= P (P (H))= P ({a,b, {a,b}})= {a,b, {a,b}, {a, {a,b}}, {b, {a,b}}, {a,b, {a,b}}}, ◦ ∗ (2,2) : H 2 → P 2 ∗ (H). ◦∗ (2,2) a b a {a, {a,b}}{b, {a,b}} b a {a,b, {a,b}}
Table1: Example1ofclassical-typeBinarySuperHyperOperation
2)Let H =[0, 2]beacontinuousset. P (H)= P ([0, 2])= {A | A ⊆ [0, 2],A =subset}, P 2(H)= P (P ([0, 2])) Let c,d ∈ H ◦ ∗ (2,2) : H 2 → P 2 ∗ (H)
◦∗ (2,2) c d c {[0, 0.5], [1, 2]} {0.7, 0.9, 1.8} d {2.5} {(0.3, 0.6), {0.4, 1.9}, 2}
Table2: Example2ofclassical-typeBinarySuperHyperOperation
4 2 Classical-type m-arySuperHyperOperation (oramoreaccuratedenomination(m,n) SuperHyperOperation)
Let U beauniverseofdiscourseandanon-emptyset H, H ⊂ U .Then: ◦ ∗ (m,n) : Hm → P n ∗ (H), wheretheintegers m,n ≥ 1, Hm = H × H ×···× H m times , and P n ∗ (H)isthe nth-powersetoftheset H thatincludestheempty-set.
ThisSuperHyperOperationisan m-aryoperationdefinedfromtheset H tothe nth-powerset oftheset H
4 3 Neutrosophic m-arySuperHyperOperation (ormoreaccuratedenomination Neutrosophic (m,n)-SuperHyperOperation):
Let U beauniverseofdiscourseandanon-emptyset H, H ⊂ U .Then: ◦(m,n) : Hm → P n(H), wheretheintegers m,n ≥ 1, and P n(H)isthe n-thpowersetoftheset H thatincludestheempty-set.
5. SuperHyperAxiom:
A classical-typeSuperHyperAxiom ormoreaccuratelya(m,n)-SuperHyperAxiom is anaxiombasedonclassical-typeSuperHyperOperations.
Similarly,a NeutrosophicSuperHyperAxiom (orNeutrosphic(m,n)-SuperHyperAxiom) isanaxiombasedonNeutrosophicSuperHyperOperations.
Thereare:
• StrongSuperHyperAxioms,whentheleft-handsideisequaltotheright-handsideasin non-hyperaxioms,
• and WeekSuperHyperAxioms,whentheintersectionbetweentheleft-handsideandthe right-handsideisnon-empty.
Forexamples,onehas:
• StrongSuperHyperAssociativity,when(x ◦ y) ◦ z = x ◦ (y ◦ z),forall x,y,z ∈ Hm,wherethe law ◦∗ (m,n) : Hm → P n ∗ (H);
• andWeekSuperHyperAssociativity,when[(x ◦ y) ◦ z] ∩ [x ◦ (y ◦ z)] = ∅,forall x,y,z ∈ Hm
6. SuperHyperAlgebraandSuperHyperStructure:
A SuperHyperAlgebra ormoreaccurately(m n)-SuperHyperAlgebra isanalgebra dealingwithSuperHyperOperationsandSuperHyperAxioms.
Again,a NeutrosophicSuperHyperAlgebra (orNeutrosphic(m,n)-SuperHyperAlgebra) isanalgebradealingwithNeutrosophicSuperHyperOperationsandNeutrosophicSuperHyperAxioms.
Ingeneral,wehave SuperHyperStructures (or(m n)-SuperHyperStructures),andcorresponding NeutrosophicSuperHyperStructures.
Forexample,thereareSuperHyperGrupoid,SuperHyperSemigroup,SuperHyperGroup,SuperHyperRing,SuperHyperVectorSpace,etc.
7 DistinctionbetweenSuperHyperAlgebravs.NeutrosophicSuperHyperAlgebra:
i. Ifnoneofthepowersets P k(H),1 ≤ k ≤ n,donotincludetheemptyset ∅,thenonehasa classical-typeSuperHyperAlgebra;
ii. Ifatleastonepowerset, P k(H),1 ≤ k ≤ n,includestheemptyset ∅,thenonehasa NeutrosophicSuperHyperAlgebra.
8 SuperHyperGraph (or n-SuperHyperGraph):
TheSuperHyperAlgebraresemblesthe n-SuperHyperGraph[17, 18, 19],introducedbySmarandachein2019,definedasfollows:
8.1 Definitionofthen-SuperHyperGraph:
Let V = {v1,v2, ,vm},for1 ≤ m ≤∞,beasetofvertices,thatcontainsSingleVertices (theclassicalones),IndeterminateVertices(unclear,vague,partiallyknown),andNullVertices (totallyunknown,empty).
Let P (V )bethepowerofset V ,thatincludestheemptyset ∅,too. Then P n(V )bethe n-powersetoftheset V ,definedinarecurentway,i.e.:
P (V ),P 2(V )= P (P (V )),P 3(V )= P (P 2(V ))= P (P (P (V ))), ··· , P n(V )= P (P (n 1)(V )),for1 ≤ n ≤∞,wherebydefinition P 0(V ) def = V
Then,the n-SuperHyperGraph(n-SHG) isanorderedpair: n-SHG=(Gn,En), where Gn ⊆ P n(V ),and En ⊆ P n(V ),for1 ≤ n ≤∞. Gn isthesetofvertices,and En isthesetofedges.
The setofvertices Gn containsthefollowingtypesofvertices:
SinglesVertices (theclassicalones);
IndeterminateVertices (unclear,vagues,partiallyunkwnown);
NullVertices (totallyunknown,empty); and:
SuperVertex (orSubsetVertex),i.e.twooremore(single,indeterminate,ornull)verticesput togetherasagroup(organization).
n-SuperVertex thatisacollectionofmanyverticessuchthatatleastoneisa(n 1)SuperVertexandallother r-SuperVerticesintothecollection,ifany,havetheorder r ≤ n 1.
The setofedges En containsthefollowingtypesofedges:
SinglesEdges(theclassicalones);
IndeterminateEdges (unclear,vague,partiallyunknown);
NullEdges (totallyunknown,empty); and:
HyperEdge (connectingthreeormoresinglevertices);
SuperEdge (connectingtwovertices,atleastoneofthembeingaSuperVertex);
n-SuperEdge (connectingtwovertices,atleastonebeingan n-SuperVertex,andtheotherof order r-SuperVertex,with r ≤ n);
SuperHyperEdge (connectingthreeormorevertices,atleastonebeingaSuperVertex);
n-SuperHyperEdge (connectingthreeormorevertices,atleastonebeingan n-SuperVertex, andtheother r-SuperVerticeswith r ≤ n;
MultiEdges (twoormoreedgesconnectingthesametwovertices);
Loop (andedgethatconnectsanelementwithitself). and:
DirectedGraph (classicalone);
UndirectedGraph (classicalone);
NeutrosophicDirectedGraph (partiallydirected,partiallyundirected,partiallyindeterminate direction).
2Conclusion
Werecalledthemostgeneralformofalgebras,calledSuperHyperAlgebra(ormoreaccuratedenomination(m,n)-SuperHyperAlgebra)andtheNeutrososophicSuperHyperAlgebra,andtheirextensionstoSuperHyperStructuresandrespectivelyNeutrosophicSuperHyperAlgebrainanyfield ofknowledge.
Theyarebasedonthe nth-PowersetofaSet,whichbetterreflectsourcomplexreality,sincea set H (thatmayrepresentagroup,asociety,acountry,acontinent,etc.)ofelements(suchas: people,objects,andingeneralanyitems)isorganizedontosubsets P (H),andthesesubsetsare againorganizedontosubsetsofsubsets P (P (H)),andsoon.That’sourworld.
HopingthatthisnewfieldofSuperHyperAlgebrawillinspireresearcherstostudyingseveral interestingparticularcases,suchastheSuperHyperGroupoid,SuperHyperMonoid,SuperHyperSemigroup,SuperHyperGroup,SuperHyperRing,SuperHyperVectorSpace,etc.
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