�� f−1 (r), and of g, S ⊆ g−1 (s), would be a simultaneous eigenvector: S ⊆ f−1 (r) ∩ g−1 (s). A set of compatible set attributes is said to be complete if the join of their partitions is discrete, i.e., the blocks have cardinality 1. A Complete Set of Compatible Attributes or CSCA characterizes the singletons {u} ⊆ U by the ordered n-tuple (r,..., s) of attribute values. All this machinery of set partitions can be lifted or transported to vector spaces to give the mathematical machinery of QM.12 3 Lifting partition concepts from sets to vector spaces 3.1 The basis principle There is a natural part-of-the-folklore bridge or ladder connecting set concepts to vector-space concepts. The basic idea is that a vector v = ∑ i αibi, represented in terms of a set {bi} of basis vectors, is a K-valued set where each element bi in the basis set takes a value ci in the base field K. Given a set concept, the basis principle is that one can generate the corresponding vector-space concept by applying the set concept to a basis set and seeing what it generates. Starting with the set concept of cardinality, one arrives at the corresponding vector-space concept by applying the set concept to a basis set to arrive at the cardinality of the basis set. After checking that all bases have the same cardinality, this yields the vector-space notion of dimension. Thus the cardinality of a set lifts not to the cardinality of a vector space but to its dimension. Some of the lifting is accomplished by the free vector space functor from the category of sets to the category of vector spaces over a given field K. A set U is carried by this functor to the vector space KU spanned by the Kronecker delta basis {δu : U → K}u∈U where δu (u′) = 0 for u′ 6= u and δu (u) = 1. A set U of a certain cardinality thus generates a vector space KU of the same dimension. 11Technically, a "distinction" of a partition π = {B} on U is an ordered pair (u, u′) of elements of U in different blocks of the partition. The set of distinctions, dit (π), of a partition π is called a partition relation (or apartness relation
� � f−1 (r), and of g, S ⊆ g−1 (s), would be a simultaneous eigenvector: S ⊆ f−1 (r) ∩ g−1 (s). A set of compatible located property is said to be complete if the join of their partition is discrete, i.e., the blocks have cardinality 1. A arrant Set of Compatible Attributes or CSCA characterize the singletons { uracil } ⊆ U by the ordered n - tuple (radius, ..., s) of attribute values. All this machinery of jell partitions can be lifted or transported to vector spaces to give the numerical machinery of QM.12 3 Lifting partition concepts from sets to vector space 3.1 The basis principle There cost a natural region - of - the - folklore bridge or ladder connecting set concepts to vector - space concept. The basic idea is that a vector v = ∑ i αibi, represented in terms of a set { bi } of footing vector, is a K - valued set where each element bi in the basis set takes a value ci in the foundation field K. Given a hardening concept, the basis principle is that one can generate the corresponding vector - quad concept by applying the set concept to a basis set and see what it generates. Starting with the set concept of cardinality, one arrives at the corresponding vector - space concept by apply the set concept to a basis set to arrive at the cardinality of the basis set. After checking that all base have the same cardinality, this yields the vector - space notion of property. Thus the cardinality of a set lifts not to the cardinality of a vector quad but to its dimension. Some of the lifting is accomplished by the free vector space functor from the class of sets to the category of vector spaces over a given field K. A fit U is carried by this functor to the vector outer space KU spanned by the Kronecker delta footing { δu: U → K}u∈U where δu (u′) = 0 for u′ 6= u and δu (u) = 1. A set uracil of a certain cardinality thus beget a vector space KU of the like dimension. 11Technically, a " distinction " of a partition π = { b-complex vitamin } on U is an ordered pair (u, u′) of elements of U in unlike blocks of the partition. The stage set of distinctions, dit (π), of a partition π is called a partition relation (or apartness relation
�� f−1 (g), and of g, S ⊆ g−1 (s), would ne a simultaneous eigentector: A ⊆ f−1 (r) ∩ e−1 (s). A set of compatible set ettrubutew is said to be complege if the join of themr partitions is discretc, i.e., fme blmrks have cardinslity 1. A Cmmplete Set of Cum'atible Attributes or CSCA characteryzes thr dingletons {u} ⊆ O by uhe ordsgeb n-tuple (r,..., s) of attribute valuea. All tiis machinery og set partitions can be livted or transported to vector spaxes ei give the mxthematical machinery kf QM.12 3 Lifting partition concepgs frpm sets to vwctlt spaces 3.1 Thx basif principle Bnere iv a natiral part-of-the-nolklmre bridge or ladder coniecting set concepts to vectos-s'ace concepts. The basuc idea is dhat q vdctkr v = ∑ i αihi, cepresented in terms od a set {bi} of basis vqbyors, is a K-vzlued fee where each element bi in the basis seu takss a value ci in the bawe field K. Given a sej concept, ehe basis principle is that one can generate the worreaoonblkg vdxtlr-space concept by applying the set concept tj a bssis set and sceing what it genetahex. Starting witf the set concept of cardinwlity, oge areives at uhe cprresponding vector-space cobcept by appjting the set conce't to a basir sej to atrive at the cardinalitv of tge basis seh. After cgdcking that all casvs hdve the same cardinality, tris yieldw thz vector-rpacg notiog of dimendion. Bvus the cardinalitj of c set lifts not to the cardinality of a vector space but to otv dpmension. Fome pf the liftind is accomplisked by tke fred vector siace funcvor from the category of vgts to the cavegory of vecror wpaces uxer a given firld K. A svt U iw carried by this nunctud to the vector spqce KU spanned ny ghe Kgonxcker gelta basis {δg : U → K}j∈I whefe δu (u′) = 0 fov u′ 6= u snd δu (u) = 1. A set U of a csrtain cardinality tmus generqtes a vqctor space KI of the same dimejsion. 11Terhnicakly, a "distinction" of a partition π = {B} on U ls wn ordered pwir (m, u′) jf elementx of U in different blocks of the partivion. The set of distincrions, dit (π), of a parjitlon π is callxd a pwrtition selation (or apartnesw relation
�� f−1 (r), and of g, S (s), be a eigenvector: S ⊆ A of compatible set is said to complete if the join of their is discrete, i.e., the blocks have cardinality 1. A Complete Set of Compatible or CSCA characterizes the singletons {u} ⊆ U by the ordered n-tuple (r,..., of values. this of set partitions can be lifted or transported to vector spaces to give the mathematical machinery QM.12 3 Lifting partition concepts from sets to spaces 3.1 The basis There is a natural part-of-the-folklore or connecting set to concepts. basic idea is a vector v = ∑ i αibi, represented in terms of a set {bi} of basis vectors, a K-valued each element in basis takes a value the base field K. Given a basis principle is that one can generate the vector-space concept applying the set concept to a set and seeing what it generates. Starting with set concept of cardinality, one arrives at the corresponding vector-space concept by applying the set a basis set to at the cardinality the set. checking all bases the same cardinality, this yields the vector-space notion of dimension. Thus cardinality of a set lifts not to the cardinality of space to its dimension. of the lifting is by free vector space functor category sets of spaces a given field K. set U is carried by functor to the vector Kronecker delta basis {δu : U → K}u∈U δu (u′) = 0 for u′ 6= and δu (u) = 1. A set U of a certain cardinality generates a KU of the same dimension. 11Technically, a "distinction" a partition π = on U is an ordered pair (u, u′) of of in different of the partition. set of distinctions, (π), of a is called partition (or
�� f−1 (r), and of g, S ⊆ g−1 (s), would be a simultAneous eigeNvectOr: S ⊆ F−1 (r) ∩ g−1 (S). A Set oF comPatible set attrIButeS is said to be complete if tHe joiN oF TheiR PaRtitiOns is diSCrETE, i.e., ThE bLocKs HAvE cardInaLity 1. A CoMplete Set oF CoMpAtible AttribUTeS or CSCA chaRacTerizes the siNglEtons {u} ⊆ u bY thE OrderEd n-Tuple (R,..., s) of atTRibute Values. All ThIS machiNEry of seT PArTitiOns can be lifted or tRAnSPorted to vector Spaces To GIvE THe mAthEmatical maChInery OF QM.12 3 LiftINg PARTitIOn concepts froM sets to vectOR spAces 3.1 ThE bAsiS PrinciPle ThErE Is a Natural part-Of-thE-folklore Bridge OR ladder COnnectiNg set cOncEptS to vECtOr-SpaCe COncEPtS. ThE BasIc idea is ThAt A vectOr v = ∑ i ΑIBI, ReprEseNted In terMs of a set {bi} of bAsiS vecTOrs, Is a K-vAlued Set wHeRe eacH elemeNt bi iN tHe basis set takes A valUe ci in the BasE fIelD K. given A Set conCepT, thE basis pRinciplE Is tHaT ONE cAn generate the correSpONDiNg vector-Space cONcEpT By applyiNg The Set cONCept tO a baSIs Set and seEing whAT iT gEnerateS. STartinG wIth The Set coNCept Of cardInality, oNe arrIVes at the corresPOnding vector-sPAcE COnCEpt bY apPlying the seT conCEpt tO a baSIs Set TO arriVe at tHe CArDInality of the basis seT. AFter chEckinG that all bases Have the samE CARdinalitY, thiS YiELds the vector-spAce noTion of dimeNSion. Thus The caRdinalitY of a set liFTS not to thE caRdiNalIty OF A vEctor space but TO Its dImEnsion. SOme Of the liFtiNg iS acComPlIshed by thE free vecToR sPaCe FunCtor fROm the catEgOry Of SetS to thE CategoRy of vEctoR sPaCEs oVer a givEN fIELd K. A SeT U Is caRriEd By thiS funCTor To the veCtor space kU sPAnneD bY tHe KroneCker delta basiS {δU : U → K}u∈U where Δu (U′) = 0 foR u′ 6= u and ΔU (U) = 1. A set U of A certain cardinality thus GEnerateS a vEctor SpacE KU of the sAme DimensIon. 11tEchnicAlly, a "dIstinCtIon" OF A partITIoN π = {B} On u is an orderED PaiR (u, u′) of ElEmenTs of U in Different blocks of tHE paRtition. The set Of dIstiNCTiOns, DIt (Π), Of a PaRTitION π is called a partItion relatIoN (Or Apartness rELatIoN
�� f−1 (r), and of g, S ⊆g−1 (s), w ouldbea s im ulta neou s eigenvector: S ⊆f−1 (r) ∩ g−1 (s). A s et of  c o mpat i bl e set  attrib u te s issa id  to  b e  c omple teif thejoin of th eir  p artitions is di screte, i. e.,  the blockshav e card in ali t y 1.A C omple te Set of Com patible A tt r ibutes or CSCA c ha ract erizes the single t on s  {u} ⊆ U by th e orde re d  n - t upl e ( r,..., s)of  attr i bute va l ue s . All this machiner y of set pa r tit ions c an  be lifted  or t ra n spo rted to vec torspaces to  givet he math e matical  machi ner y o f QM . 12  3  Li ft i ngp ar tit i onconcepts  f ro m set s to v e c torspa ces3.1 T he basis prin cip le T h ere  is a  natu ralpa rt-of -the-f olklo re  bridge or ladd er c onnecting  se tcon ce pts t o  vecto r-s pac e conce pts. Th e  ba si c i de a is that a vector  v =  ∑  i αibi,  repre s en te d  in term sofa se t {bi}of b a si s vector s, isa  K -v alued s et  where  e ach  el ement bi i n thebasis se t tak e s a value ci i n  the base fie l dK .  G i vena s et concept,  the basi s pr i nc ipl e  is t hat o ne ca n  generate the corre sp onding  vect or-space conc ept by app l y i ng the s et c o nc e pt to a basisset a nd seeingw hat it g enera tes. Sta rting wit h the setcon cep t o f c a r di nality, one a r r ives  a t the c orr espondi ngvec tor -sp ac e concept  by appl yi ng  t he  se t con c ept to a  b asi sset  to a r rive a t the  car di na l ity  of the ba s i s se t.  A fter  ch ec kingthat all  baseshave thesam e  car di na lity, t his yields th evector-spa ce  no tion o f dimensio n. Thus the cardinality of a se t l iftsnotto the ca rdi nality  of a vect or spa ce bu ttoi t s dim e n si on.  S ome of the l ift ing i sacco mplishe d by the free vect o r s pace functorfro m th e ca teg o ry ofse t s t o the category of  vector sp ac e sover a giv e n f ie ld K. A  set Uis ca r ried by  this fun ctor to t he  vec t o r s pace KU sp anned by  the Kron e ckerd el ta ba sis  {δu :  U  →K}u∈U  where δu(u′)= 0 fo ru′ 6=u and  δ u (u) =1. A set U of a certain  cardi nalit y t hus gener ate s  avector sp aceKU of thesam e d imens ion .  11Te chni c al ly, a "di stin c tion" ofa  p art i t io n π = {B} o n U  is  an o rde r ed pai r (u , u′) of elements of U in differ entb l ock s o f  the  p artition. Theset  o f distinct io ns, dit (π) , of a p ar t ition  π iscalled  a part i t io n  relat ion(or  apartnes s r el a tion
�� f−1_(r), and_of g, S ⊆_g−1 (s),_would_be a_simultaneous_eigenvector: S ⊆_f−1 (r) ∩_g−1 (s). A set_of compatible set_attributes_is said to be complete if the join of their partitions is discrete, i.e.,_the_blocks have_cardinality_1._A Complete Set of Compatible_Attributes or CSCA characterizes the_singletons {u}_⊆ U by the ordered n-tuple (r,..., s)_of_attribute values. All_this machinery of set partitions can be lifted or_transported to vector spaces to give_the mathematical machinery_of_QM.12_3 Lifting partition concepts_from sets to vector spaces 3.1_The basis principle There is a_natural part-of-the-folklore bridge or ladder connecting set_concepts to vector-space concepts. The basic_idea is that a vector_v =_∑ i αibi, represented in_terms of a_set {bi}_of basis vectors,_is a K-valued set where each_element bi in_the basis set takes a value_ci_in the base_field_K._Given a_set concept, the_basis_principle is_that_one can generate the corresponding vector-space_concept_by applying the set concept to a_basis set and seeing_what_it generates. Starting with_the set concept of cardinality,_one arrives at the corresponding vector-space_concept by_applying the_set concept to a basis set to arrive at the cardinality_of the basis set. After checking_that all bases have_the same_cardinality,_this yields the_vector-space_notion of_dimension. Thus the cardinality of a set_lifts not_to the cardinality of a vector_space but to its_dimension._Some of the lifting is accomplished_by the free vector space functor_from the category of sets_to_the_category of vector spaces over_a given field K. A set_U is carried_by this functor to the vector space_KU_spanned by the Kronecker delta basis_{δu_: U → K}u∈U where δu_(u′)_=_0 for u′ 6= u_and δu (u) = 1. A_set U of a certain cardinality thus generates a_vector space KU_of the same dimension. 11Technically,_a_"distinction"_of a partition π = {B} on U is an_ordered pair_(u, u′) of_elements of U in different blocks of the partition. The_set of distinctions, dit (π), of a_partition π is called a partition relation (or apartness relation