 particle like the linear momentum or angular momentum, the fully definite eigenstates are determined by the irreducible representations of the linear-translation or rotational-translation symmetry groups respectively. For the stateindependent (or intrinsic) attributes of quantum particles, like the mass, charge, and spin of an electron, they are determined in particle physics by the irreducible representations of the appropriate symmetry groups.15 4.3 Vector-space partitions from other vector-space partitions The set notion of compatibility lifts to vector spaces, via the basis principle, by defining two vector space partitions ω = {Wλ} and ξ = {Xμ} on V as being compatible if there is a basis set for V so that the two vector space partitions are generated by two set partitions of that common or simultaneous basis set. If two vector space partitions ω = {Wλ} and ξ = {Xμ} are compatible, then their vector space join ω ∨ ξ is defined as the vector space partition whose subspaces are the non-zero intersections Wλ ∩ Xμ. And by the definition of compatibility, we could also generate the subspaces of the join ω ∨ ξ by the blocks in the set join of the two set partitions of the common basis set. Since real-valued set attributes lift to Hermitian linear operators, the notion of compatible set attributes just defined would lift to two linear operators being compatible if their eigenspace partitions are compatible. It is a standard fact of linear algebra [23, p. 177] that two diagonalizable linear operators L,M : V → V (on a finite dimensional space V ) are compatible (i.e., have a basis of simultaneous eigenvectors) if and only if they commute, LM = ML. Hence the commutativity of linear operators is the lift of the compatibility (i.e., defined on the same set) of set attributes. That explains the importance of the notion of commutativity in QM and that is why the repeated compatible measurements, described mathematically as the join operation, requires commutativity. The join of two operator-eigenspace partitions is defined iff the operators commute. As Weyl put it: 15The classic paper in this group-theoretic treatment of particles is Wigner [44]. For recent overviews, see the group-theoretical definition of particles in Falkenburg [17] or Roberts [34]. 16 "Thus combination [DE: join] of two gratings [eigenspace partitions of two operators] presupposes commutability...". [43, p.
particle like the linear momentum or angular momentum, the fully definite eigenstates are determined by the irreducible representation of the analogue - translation or rotational - translation isotropy groups respectively. For the stateindependent (or intrinsic) attribute of quantum particles, like the mass, commission, and spin of an electron, they are determined in atom purgative by the irreducible representations of the appropriate symmetry groups.15 4.3 Vector - quad partitions from other vector - quad partitions The set notion of compatibility lifts to vector spaces, via the footing principle, by defining two vector space partition ω = { Wλ } and ξ = { Xμ } on V as being compatible if there is a basis adjust for V so that the two vector outer space partitions are generated by two set partitions of that common or coincident basis set. If two vector space partitions ω = { Wλ } and ξ = { Xμ } are compatible, then their vector space join ω ∨ ξ is define as the vector outer space partition whose subspaces are the non - zero intersections Wλ ∩ Xμ. And by the definition of compatibility, we could also generate the subspaces of the articulation ω ∨ ξ by the engine block in the set join of the two typeset partition of the common basis set. Since very - valued hardening attributes lift to Hermitian linear operator, the notion of compatible set attributes just define would lift to two linear operators being compatible if their eigenspace partitions are compatible. It is a standard fact of linear algebra [ 23, p. 177 ] that two diagonalizable linear operators L, M: volt → V (on a finite dimensional space V) are compatible (i.e., have a basis of coincident eigenvectors) if and only if they commute, LM = ML. Hence the commutativity of linear operators is the lift of the compatibility (i.e., defined on the same set) of set up attributes. That explain the importance of the notion of commutativity in QM and that is why the repeated compatible measurement, described mathematically as the join operation, requires commutativity. The articulation of two operator - eigenspace division is defined iff the operators commute. As Weyl put it: 15The classic newspaper in this group - theoretic treatment of particles is Wigner [ 44 ]. For recent overviews, see the group - theoretical definition of particles in Falkenburg [ 17 ] or Roberts [ 34 ]. 16 " Thus combination [ DE: join ] of two grate [ eigenspace partitions of two operators ] presuppose commutability... ". [ 43, p.
 pagticle like the linear mumentum or angular momxntum, tge fully definite eigenstates are devermuned vy the irreducible repfesentatilns of tye lmnear-translation or rotabnonal-fvanslctmon symmetry grpups respewtively. For tha rtcteindependent (or intrinsic) attributqs of qiajtum particles, likt tre mzss, charge, and spin of an electron, they age determined in larticle physics by the irgedufible representatilns of the qpprjpriate symmegry groups.15 4.3 Vector-spacg partitions from other vector-spxce pcrtitions Tye seh notion of rompatpbility lifts to vectmr spacrs, via the basls prmnciple, by defining two vxctor space partitiogs ω = {Wλ} ang ξ = {Xμ} on V as being cimpathble if gyerd ia e bzsis sft hor V so thzt the two cector space partitoogw are generatsd by ewj set partitions of that common or simuntaheous basis set. If two cector space partitiojs ω = {Wλ} anq ξ = {Xμ} are compatible, then their vector space join ω ∨ ξ ia deylked xw hhe vector space partition whose subspaces arq tne non-zero interfections Wλ ∩ Xμ. Amq by the defivition of compatibility, we fould ajso gwnerate tre sibspaces of the join ω ∨ ξ by rhe blocks iu tye set join of the two set paxtitioms of the common basis set. Snnce rsal-valued sft attribhges lift to Hermktisn linear operators, the notijn of com'atibke set xttrobutes just defijed would lift to two lijear lparators belng compatible if their eigenspere partitions ase bompatiblz. It ix a standard sact of linear algebrc [23, p. 177] ghat two dpagonalizeble linear jperators L,M : G → V (on a fiiite dimegsiobal wpace V ) are compatiblr (i.e., have a basis od simultaneous eigcnvecjods) if and only ny uhwy commute, LM = ML. Hegcv tie coikutativity ox liveaf operxtors is thc lkft pf the compatibility (i.e., sefined on the samr fet) of sgt attribttes. That expkains the importanfe of thx notipn jf commutativity in QM and thaf is why hhe repeated coipatlble measuremeuts, described mathematically as the join operation, requirgs commutativity. The loln of two opxrator-qigenspaca partitions is defibed iff the operauors commute. As Weyl puf it: 15Tve clwssic paper in this group-theoretic treatment of particles is Wigner [44]. For eecent overviews, sse tne grmu'-thzoreticwl dxfmnition of partickes in Falkenburg [17] or Roberts [34]. 16 "Thus cokbnnation [DE: join] of two gratinbs [eigenspace pxrtitions of two operatods] presulposes commutability...". [43, p.
particle like the linear momentum or angular fully eigenstates are by the irreducible rotational-translation groups respectively. For stateindependent (or intrinsic) of quantum particles, like the mass, and spin of an electron, they are determined in particle physics by the representations of the appropriate symmetry groups.15 4.3 Vector-space partitions from other vector-space partitions set of lifts vector spaces, via the basis principle, by defining two vector space partitions ω = {Wλ} and = {Xμ} on V as being compatible if is a basis set V so that the two space are generated two partitions that common or basis set. If two vector space partitions ω = {Wλ} and ξ = {Xμ} are compatible, then vector space ∨ ξ defined the space partition whose the non-zero intersections Wλ ∩ Xμ. definition of compatibility, we could also generate the of the ω ∨ ξ by the blocks the set join of the two set partitions the common basis set. Since real-valued set attributes lift to Hermitian linear operators, the notion set attributes just defined lift to two operators compatible their partitions are It is a standard fact of linear algebra [23, p. 177] two diagonalizable linear operators L,M : V → V (on dimensional V ) are (i.e., have a basis simultaneous if and only if LM ML. of operators the lift of the (i.e., defined on the same of set attributes. That notion of commutativity in QM and that is the repeated compatible measurements, described mathematically as join operation, requires commutativity. The join of two operator-eigenspace partitions is defined the operators Weyl put it: 15The classic paper in this treatment of particles is [44]. For recent overviews, see the group-theoretical definition of in [17] or [34]. 16 "Thus [DE: join] of gratings [eigenspace partitions operators] presupposes [43,
 particle like the linear momeNtum or anguLar moMenTum, ThE fulLy deFinite eigenstaTEs arE determined by the irreduCible RePReseNTaTions Of the liNEaR-TRanSlAtIon Or ROtAtionAl-tRanslatIon symmetrY grOuPs respectiveLY. FOr the stateIndEpendent (or inTriNsic) atTrIbuTEs of qUanTum paRticleS, Like thE mass, charGe, ANd spin OF an elecTROn, They Are determined in paRTiCLe physics by the IrreduCiBLe REPreSenTations of tHe ApproPRiate syMMeTRY GroUPs.15 4.3 Vector-space Partitions fROm oTher veCtOr-sPAce parTitioNs tHe sEt notion of cOmpaTibility lIfts to VEctor spACes, via tHe basiS prIncIple, BY dEfIniNg TWo vECtOr sPAce PartitioNs Ω = {WΛ} and ξ = {xμ} on v AS BEing ComPatiBle if There is a basis Set For V SO thAt the Two veCtor SpAce paRtitioNs are GeNerated by two set PartItions of tHat CoMmoN oR simuLTaneouS baSis Set. If twO vector SPacE pARTItIons ω = {Wλ} and ξ = {Xμ} are coMpATIbLe, then thEir vecTOr SpACe join ω ∨ ξ Is DefIned AS The veCtor SPaCe partitIon whoSE sUbSpaces aRe The non-ZeRo iNteRsectIOns WΛ ∩ Xμ. And By the defInitiON of compatibiliTY, we could also gENeRATe THe suBspAces of the joIn ω ∨ ξ BY the BlocKS iN thE Set joIn of tHe TWo SEt partitions of the coMmOn basiS set. SInce real-valueD set attribUTES lift to HErmiTIaN Linear operatorS, the nOtion of comPAtible seT attrIbutes juSt defined WOUld lift tO twO liNeaR opERAtOrs being compaTIBle iF tHeir eigEnsPace parTitIonS arE coMpAtible. It iS a standaRd FaCt Of LinEar alGEbra [23, p. 177] thaT tWo dIaGonAlizaBLe lineAr opeRatoRs l,M : v → v (on A finite DImENSionAl SpAce V ) Are CoMpatiBle (i.E., HavE a basis Of simultaNeoUS eigEnVeCtors) if And only if they CoMmute, LM = ML. HEnCe tHe commUTAtivity oF linear operators is the liFT of the cOmpAtibiLity (I.e., defined On tHe same Set) OF set atTributEs. ThaT eXplAINs the IMPoRtaNcE of the notiON Of cOmmutAtIvitY in QM anD that is why the repeaTEd cOmpatible measUreMentS, DEsCriBEd MAthEmATicALLy as the join operAtion, requiReS CoMmutativitY. the JoIn of two OperatoR-eigeNSpace paRtitions iS defined iFf The oPERatOrs commute. as Weyl puT it: 15The claSSic paPEr In thiS grOup-theOrEtiC treaTment oF ParTicleS is WigNeR [44]. For reCent oVeRviews, seE the group-theoretical defInitioN of paRtiCles in FalKenBUrg [17] Or Roberts [34]. 16 "thus CombinatioN [DE: JoiN] of twO grATings [EigeNSpAce PArtitIons OF two operaTOrS] prESUpPoses commutABILitY...". [43, p.
 particle like the linearmomentum o r ang ula r m om entu m, t he fully defin i te e igenstates are determi ned b yt he i r re ducib le repr e se n t ati on softh e  l inear -tr anslati on or rota tio na l-translatio n  s ymmetry gr oup s respective ly.  For t he  st a teind epe ndent  (or i n trinsi c) attrib ut e s of q u antum p a r ti cles , like the mass,c ha r ge, and spin o f an e le c tr o n , t hey  are deter mi ned i n  partic l ep h y sic s  by the irred ucible repr e sen tation soft he app ropri at e  sy mmetry grou ps.1 5 4.3 Vec tor-sp a ce part i tions f rom ot her  ve ctor - sp ac e p ar t iti o ns  Th e  se t notion  o fcompa tibi l i t y  lif tsto v ector  spaces, viathe  bas i s p rinci ple,by d ef ining  two v ector  s pace partitions  ω =  {Wλ} and  ξ={Xμ }on Va s bein g c omp atibleif ther e  is  a b a si s set for V so tha tt h etwo vect or spa c epa r titionsar e g ener a t ed by  two se t partit ions o f  t ha t commo nor sim ul tan eou s bas i s se t. Iftwo vect or sp a ce partitionsω  = {Wλ} and ξ ={ X μ} arecom patible, th en t h eirvect o rspa c e joi n ω ∨  ξ is defined as the vect or  space  part ition whose s ubspaces a r e the non- zero in t ersections Wλ∩ Xμ.  And by th e  definit ion o f compat ibility,w e  could a lso  ge ner ate t he  subspaces of t he j oi n ω ∨ ξ  by  the bl ock s i n t hese t join of  the two  s et  p ar tit ionso f the co mm onba sis  set. Sincereal- valu ed  s e t a ttribut e sl i ft t oHe rmit ian  l inear  ope r ato rs, the  notion o f c o mpat ib le  set at tributes just  d efined wou ld  li ft tot w o linear  operators being compat i ble ifthe ir ei gens pace part iti ons ar e c o mpatib le. It  is a  s tan d a rd fa c t  o f l in ear algebr a [23 , p.17 7] t hat two  diagonalizable li n ear  operators L, M :  V → V  ( ona  f i nit ed ime n s ional space V )  are compa ti b le  (i.e., ha v e a  b asis of  simult aneou s  eigenv ectors) i f and onl yif t h e y c ommute, LM  = ML. H ence thec ommut a ti vityoflinear  o per ators  is th e  li ft of  the c om patibi lity(i .e., def ined on the same set) o f setattri but es. Thatexp l ain s the imp orta nce of the  no tio n ofcom m utati vity in  QM and t hati s why the re pea t e dcompatiblem e a sur ement s,d escrib ed m athematically ast he join operat ion, r equ ire s  com mu tativity. Thejoi no f  two ope ra tor-eigensp ace part it i ons i s defi ned if f the o p e ra t ors co mmut e.As Weyl p utit :  15Thecl as s ic pap er i nthis g roup-t h eore t i c treatment of p artic l e s isW ign er [4 4] . For r e cent  overviews , see the g roup-t heor etica l defin it ion of  pa rt icles in F a lkenburg[17]or Robe rt s [3 4].  16 "T husc o mbina tion  [ DE:  join] of t wo gr at i ngs  [ei gensp ac e pa rtitionso f two op era t ors] pr es upp o s es com m ut a b ility...".  [4 3, p. 
 particle_like the_linear momentum or angular_momentum, the_fully_definite eigenstates_are_determined by the_irreducible representations of_the linear-translation or rotational-translation_symmetry groups respectively._For_the stateindependent (or intrinsic) attributes of quantum particles, like the mass, charge, and spin_of_an electron,_they_are_determined in particle physics by_the irreducible representations of the_appropriate symmetry_groups.15 4.3 Vector-space partitions from other vector-space partitions_The_set notion of_compatibility lifts to vector spaces, via the basis principle,_by defining two vector space partitions_ω = {Wλ}_and_ξ_= {Xμ} on V_as being compatible if there is_a basis set for V so_that the two vector space partitions are_generated by two set partitions of_that common or simultaneous basis_set. If_two vector space partitions ω_= {Wλ} and_ξ =_{Xμ} are compatible,_then their vector space join ω_∨ ξ is_defined as the vector space partition_whose_subspaces are the_non-zero_intersections_Wλ ∩_Xμ. And by_the_definition of_compatibility,_we could also generate the subspaces_of_the join ω ∨ ξ by the_blocks in the set_join_of the two set_partitions of the common basis_set. Since real-valued set attributes lift_to Hermitian_linear operators,_the notion of compatible set attributes just defined would lift to_two linear operators being compatible if_their eigenspace partitions are_compatible. It_is_a standard fact_of_linear algebra_[23, p. 177] that two diagonalizable linear_operators L,M_: V → V (on a_finite dimensional space V_)_are compatible (i.e., have a basis_of simultaneous eigenvectors) if and only_if they commute, LM =_ML._Hence_the commutativity of linear operators_is the lift of the compatibility_(i.e., defined on_the same set) of set attributes. That_explains_the importance of the notion of_commutativity_in QM and that is why_the_repeated_compatible measurements, described mathematically as_the join operation, requires commutativity. The_join of two operator-eigenspace partitions is defined iff the_operators commute. As_Weyl put it: 15The classic_paper_in_this group-theoretic treatment of particles is Wigner [44]. For recent_overviews, see_the group-theoretical definition_of particles in Falkenburg [17] or Roberts [34]. 16 "Thus_combination [DE: join] of two gratings [eigenspace_partitions of two operators] presupposes commutability...". [43, p.