 in computer science) and is just the complement of the associated equivalence relation. The notion of a distinction of a partition is the partition logic analogue of an element of a subset in subset logic. For instance, given two partitions π = {B} and σ = {C} on a universe set and two subsets S and T of a universe set, the partition join π ∨ σ combines the distinctions of the partitions, i.e., dit (π ∨ σ) = dit (π) ∪ dit (σ), just as the subset join or union S ∪ T combines the elements of the subsets (see [12] or [16] for further developments). 12 In QM, the extension of concepts on finite dimensional Hilbert space to infinite dimensional ones is well-known. Since our expository purpose is conceptual rather than mathematical, we will stick to finite dimensional spaces. 12 3.2 What is a vector space partition? A partition π = {B} on a set U is a set of subsets whose direct sum (i.e., disjoint union) is the whole set, i.e., a direct sum decomposition of the set. The corresponding vector space concept is a set of subspaces of a vector space whose direct sum is the vector space, i.e., a direct sum decomposition of the vector space. In terms of the basis principle, we could apply the set partition π = {B} of a set U to a basis set {bu}u∈U, then each block B generates a subspace VB and the set of subspaces {VB}B∈π is a direct sum decomposition of the vector space spanned by the basis set. Thus the lift or transport of the concept of a set partition is a direct sum decomposition of a vector space. In particular, it is not a set partition of a vector space that is compatible with the vector space operations, i.e., a quotient space V/W as would be defined by each subspace W ⊆ V with the equivalence relation v ∼ v′ if v−v′ ∈W. While a partition on a set is essentially the same as a quotient set (or equivalence relation on the set), the vector-space lift of a set partition is not a quotient vector space but a direct sum decomposition of a vector space. In lifting or transporting the partitional concepts for sets to vector spaces, we are making the choices guided by the set-to-basis-set connection which yields the mathematical machinery of quantum mechanics. This is not particularly
in computer science) and is just the complement of the associated equality relation back. The notion of a distinction of a partition is the division logic analogue of an element of a subset in subset logic. For instance, given two partition π = { B } and σ = { C } on a universe stage set and two subsets S and T of a universe hardening, the partition join π ∨ σ blend the distinctions of the partitions, i.e., dit (π ∨ σ) = dit (π) ∪ dit (σ), just as the subset join or union S ∪ thymine combines the elements of the subsets (learn [ 12 ] or [ 16 ] for further developments). 12 In QM, the extension of concepts on finite dimensional Hilbert space to infinite dimensional one is well - known. Since our expository purpose is conceptual rather than mathematical, we will stick to finite dimensional space. 12 3.2 What is a vector space partition? A partition π = { B } on a set uracil is a set of subsets whose direct sum (i.e., disjoint union) is the whole set, i.e., a direct sum decomposition of the stage set. The corresponding vector space concept is a hardening of subspaces of a vector space whose direct sum is the vector quad, i.e., a direct sum decomposition of the vector space. In term of the basis principle, we could apply the set partition π = { B } of a hardened U to a basis set { bu}u∈U, then each block b-complex vitamin generate a subspace VB and the set of subspaces { VB}B∈π is a direct sum decomposition of the vector quad spanned by the basis set. therefore the lift or transportation of the concept of a set partition is a lineal sum decomposition of a vector space. In particular, it is not a set division of a vector space that is compatible with the vector space operations, i.e., a quotient space volt / W as would be defined by each subspace W ⊆ five with the equivalence relation v ∼ v′ if v−v′ ∈W. While a division on a set is essentially the same as a quotient set (or equivalence relation on the set), the vector - space lift of a set partition is not a quotient vector space but a lineal sum decomposition of a vector outer space. In lifting or transporting the partitional concepts for sets to vector spaces, we are making the choices guided by the hardening - to - basis - set connection which yields the mathematical machinery of quantum mechanics. This is not particularly
 in computer science) and is just the complgmwnt of the aasociatea equivalence relation. The nltuon od a distinction of a pxrtition ps the paetitmon logic analogns of an elemehb of c wubset in subsgt logic. For instance, givet gwl partitions π = {B} and σ = {C} on a univewse set ajd two subsets S amq T kf a universe set, the partition jojn π ∨ σ cmmbines the dostinctions of the partitilns, l.e., dit (π ∨ σ) = dit (π) ∪ dlt (σ), just as the wubset join ur union S ∪ T combines the elements of the subsets (sed [12] or [16] for furtyee dfeelopments). 12 In QI, the extenslpn of wonceptx on finite dikenviobal Hilbert space to mnfinite dimensional ones is feml-known. Since our ezposijory [urpuwe ks doiceltual gatier than mafhematical, qe will stick to fimiev dimensional spacef. 12 3.2 What is a vector space partition? A pagtitjon π = {B} on a set U is a set of subsets whose direct stm (i.e., disjoint union) is the whole set, i.e., a direct sum vezomkowitiov ov the set. The corresponding vector space concqlt ix a set of subfpaces of a vfcyjr space whosg direcc shm is the vector soace, i.e., a dieect sum qecokposition of the vector spaxe. In terms jd the basis princi'le, we could appky thr set partition π = {B} of c set H to a basid set {bu}u∈H, then each block B nenarates a subspace VB and tre set of subxpaces {XB}B∈π os a dyrect sum fecomimsition of the vechor skace s[anned by hhe basis set. Thus the lift or vcansport of tne cotcept of a seb partition is w direct sum dgcompositnon of a vector apace. Ii particular, it is not a vgt partition mf a veceor wpacw that kr compatible woth the vvcuor space iperations, i.e., a quptignf space V/W as wiule be defined by eazh fuhs'ace R ⊆ V with the equkvaurnce felation v ∼ x′ if v−v′ ∈W. While a partithon kn a set is essentoajly the wame as w quotient sey (or equivalence rtlatioi on tie set), thg vector-space lift of a set parfition is nob a quotient dectir space but a direct sum decomposition of a vector space. In lifting ir transporting the kartitional cpncepes for seds to vector spaces, qe are making the choices guided by ths set-tm-basid-set connection which yields the mathematical machinery of quantum mechanucs. Thms is not parficukarly
in computer science) and is just the the equivalence relation. notion of a the logic analogue of element of a in subset logic. For instance, given partitions π = {B} and σ = {C} on a universe set and subsets S and T of a universe set, the partition join π ∨ combines distinctions the i.e., dit (π ∨ σ) = dit (π) ∪ dit (σ), just as the subset join union S ∪ T combines the elements of subsets (see [12] or for further developments). 12 In the of concepts finite Hilbert to infinite dimensional is well-known. Since our expository purpose is conceptual rather than mathematical, we will stick to finite dimensional 12 3.2 a vector partition? partition = {B} on U is a set of subsets (i.e., disjoint union) is the whole set, i.e., direct sum of the set. The corresponding vector concept is a set of subspaces of a space whose direct sum is the vector space, i.e., a direct sum decomposition of the In terms of the principle, we could the partition = of a U to a basis set {bu}u∈U, then each block B generates subspace VB and the set of subspaces {VB}B∈π is a decomposition the vector space by the basis set. the or transport of the a partition sum of vector space. In particular, is not a set partition a vector space that space operations, i.e., a quotient space V/W as be defined by each subspace W ⊆ with the equivalence relation v ∼ v′ if v−v′ ∈W. While a on a essentially the same as a quotient set (or relation on the set), vector-space lift of a set partition is not a vector but a sum decomposition of vector space. In or transporting the for sets vector we choices guided by the set-to-basis-set connection yields mathematical machinery of quantum mechanics. is
 in computer science) and is jusT the compleMent oF thE asSoCiatEd eqUivalence relatIOn. ThE notion of a distinction oF a parTiTIon iS ThE partItion loGIc ANAloGuE oF an ElEMeNt of a SubSet in suBset logic. FOr iNsTance, given twO PaRtitions π = {B} And Σ = {C} on a universE seT and twO sUbsETs S anD T oF a uniVerse sET, the paRtition joIn Π ∨ Σ combiNEs the diSTInCtioNs of the partitions, I.E., dIT (π ∨ σ) = dit (π) ∪ dit (σ), jusT as the SuBSeT JOin Or uNion S ∪ T combInEs the ELements OF tHE SUbsETs (see [12] or [16] for furTher developMEntS). 12 In QM, tHe ExtENsion oF concEpTS on Finite dimenSionAl Hilbert Space tO InfinitE DimensiOnal onEs iS weLl-knOWn. siNce OuR ExpOSiTorY PurPose is coNcEpTual rAtheR THAN matHemAticAl, we wIll stick to finIte DimeNSioNal spAces. 12 3.2 WHat iS a VectoR space PartiTiOn? A partition π = {B} oN a seT U is a set oF suBsEts WhOse diREct sum (I.e., dIsjOint uniOn) is the WHolE sET, I.E., a Direct sum decomposiTiON Of The set. ThE correSPoNdINg vector SpAce ConcEPT is a sEt of SUbSpaces of A vectoR SpAcE whose dIrEct sum Is The VecTor spACe, i.e., A direcT sum decoMposiTIon of the vector SPace. In terms of THe BASiS PrinCipLe, we could apPly tHE set PartITiOn π = {b} Of a seT U to a BaSIs SEt {bu}u∈U, then each block b gEneratEs a suBspace VB and thE set of subsPACEs {VB}B∈π is A dirECt SUm decompositioN of thE vector spaCE spanned By the Basis set. thus the liFT Or transpOrt Of tHe cOncEPT oF a set partitioN IS a diReCt sum deComPositioN of A veCtoR spAcE. In particUlar, it is NoT a SeT pArtItion OF a vector SpAce ThAt iS compATible wIth thE vecToR sPAce OperatiONs, I.E., A quoTiEnT spaCe V/w aS woulD be dEFinEd by eacH subspace w ⊆ V wITh thE eQuIvalencE relation v ∼ v′ if V−v′ ∈w. While a parTiTioN on a seT IS essentiAlly the same as a quotient sET (or equiValEnce rElatIon on the sEt), tHe vectOr-sPAce lifT of a seT partItIon IS Not a qUOTiEnt VeCtor space bUT A diRect sUm DecoMpositiOn of a vector space. In LIftIng or transporTinG the PARtItiONaL ConCePTs fOR Sets to vector spaCes, we are maKiNG tHe choices gUIdeD bY the set-To-basis-Set coNNection Which yielDs the mathEmAticAL MacHinery of quAntum mecHanics. ThiS Is not PArTiculArlY
 in computer science) andis just th e com ple men tof t he a ssociated equi v alen ce relation. The notio n ofad isti n ct ion o f a par t it i o n i sth e p ar t it ion l ogi c analo gue of anele me nt of a subs e tin subsetlog ic. For inst anc e, giv en  tw o  part iti ons π  = {B} and σ= {C} onau nivers e  set an d tw o su bsets S and T ofa  u n iverse set, th e part it i on j oin  π∨ σ combin es  thed istinct i on s o f t h e partitions,  i.e., dit( π ∨  σ) =di t ( π ) ∪ di t (σ) ,j ust  as the sub setjoin or u nion S ∪ T com b ines th e elem ent s o f th e  s ub set s( see [1 2]o r [ 16] forfu rt her d evel o p m e nts) . 1 2 In  QM,the extension  of  con c ept s onfinit e di me nsion al Hil bertsp ace to infinite  dim ensionalone siswe ll-kn o wn. Si nce  ou r expos itory p u rpo se i s  c onceptual rather t ha n ma thematic al, we wi ll stick to  f ini te d i m ensio nals pa ces. 123.2 Wh a tis  a vect or  space  p art iti on? A part itionπ = {B}on as et U is a seto f subsets who s ed i re c t su m ( i.e., disjo intu nion ) is th e w h ole s et, i .e . ,a  direct sum decompo si tion o f the  set. The cor responding v e ctor spa ce c o nc e pt is a set of  subs paces of a vector s pacewhose di rect sumi s  the vec tor  sp ace , i . e .,  a direct sum d ecom po sitionofthe vec tor  sp ace . I nterms ofthe basi spr in ci ple , wec ould app ly  th eset  part i tion π  = {B } of  a  s e t U  to a b a si s set{b u} u∈U,  th en  each  blo c k B  genera tes a sub spa c e VB  a nd  the se t of subspace s{VB}B∈π is  a  di rect s u m  decompo sition of the vector sp a ce span ned  by t he b asis set.  Th us the  li f t or t ranspo rt of  t hec o ncept o fa s et  partition i s a  dire ct  sum  decomp osition of a vecto r  sp ace. In parti cul ar,i t  i s n o ta  se tp art i t ion of a vector  space tha ti scompatible wit hthe vec tor spa ce op e rations , i.e., a  quotient  s pace V /Was would b e define d by each subsp a ce  W ⊆V w ith th eequ ivale nce re l ati on v∼ v′ i fv−v′ ∈ W. Wh il e a part ition on a set is essen tially  thesam e as a qu oti e ntset (or e quiv alence rel ati onon th e s e t), t he v e ct or- s pacelift of a setp ar tit i o nis not a qu o t i ent  vect ors pace b ut a  direct sum decom p osition of a v ecto r spa ce. In l if ting or transp ort in g the part it ional conce pts forse t s tovector  space s, we a r e  m a king t he c hoi ces guide d b yt he set- to -b a sis-se t co nn ection  which yiel d s  the mathematica l mac h i neryo f q uantu mmechani c s. T his is not  particular ly
 in_computer science)_and is just the_complement of_the_associated equivalence_relation._The notion of_a distinction of_a partition is the_partition logic analogue_of_an element of a subset in subset logic. For instance, given two partitions π_=_{B} and_σ_=_{C} on a universe set_and two subsets S and_T of_a universe set, the partition join π ∨_σ_combines the distinctions_of the partitions, i.e., dit (π ∨ σ) =_dit (π) ∪ dit (σ), just_as the subset_join_or_union S ∪ T_combines the elements of the subsets_(see [12] or [16] for further_developments). 12 In QM, the extension of_concepts on finite dimensional Hilbert space_to infinite dimensional ones is_well-known. Since_our expository purpose is conceptual_rather than mathematical,_we will_stick to finite_dimensional spaces. 12 3.2 What is_a vector space_partition? A partition π = {B}_on_a set U_is_a_set of_subsets whose direct_sum_(i.e., disjoint_union)_is the whole set, i.e., a_direct_sum decomposition of the set. The corresponding_vector space concept is_a_set of subspaces of_a vector space whose direct_sum is the vector space, i.e.,_a direct_sum decomposition_of the vector space. In terms of the basis principle, we_could apply the set partition π_= {B} of a_set U_to_a basis set_{bu}u∈U,_then each_block B generates a subspace VB and_the set_of subspaces {VB}B∈π is a direct_sum decomposition of the_vector_space spanned by the basis set._Thus the lift or transport of_the concept of a set_partition_is_a direct sum decomposition of_a vector space. In particular, it_is not a_set partition of a vector space that_is_compatible with the vector space operations,_i.e.,_a quotient space V/W as would_be_defined_by each subspace W ⊆_V with the equivalence relation v_∼ v′ if v−v′ ∈W. While a partition on_a set is_essentially the same as a_quotient_set_(or equivalence relation on the set), the vector-space lift of_a set_partition is not_a quotient vector space but a direct sum decomposition of_a vector space. In lifting or transporting_the partitional concepts for sets to vector spaces, we are making_the choices guided by the set-to-basis-set connection which_yields the mathematical machinery of quantum mechanics._This is not particularly