 we have." ()

"Evident as the notion of mathematical structure may seem these days, it was at least not made explicit until the middle of the 20th century. Then it was the influence of the Bourbaki-project and then later the development of category theory which made the notion explicit" (nLab).

According to Bourbaki, the scale of sets on a given set X consists of all sets arising from X by taking Cartesian products and power sets, in any combination, a finite number of times. Examples: X; X × X; P(X); P(P(X × X) × X × P(P(X))) × X. (Here A × B is the product of A and B, and P(A) is the powerset of A.) In particular, a pair (0,S) consisting of an element 0 ∈ N and a unary function S : N → N belongs to N × P(N × N) (since a function is a subset of the Cartesian product). A triple (+, ·, ≤) consisting of two binary functions N × N → N and one binary relation on N belongs to P(N × N × N)  × P(N × N × N)  × P(N × N). Similarly, every algebraic structure on a set belongs to the corresponding set in the scale of sets on X.

Non-algebraic structures on a set X often involve sets of subsets of X (that is, subsets of P(X), in other words, elements of P(P(X))). For example, the structure of a topological space, called a topology on X, treated as the set of "open" sets; or the structure of a measurable space, treated as the σ-algebra of "measurable" sets; both are elements of P(P(X)). These are second-order structures.

More complicated non-algebraic structures combine an algebraic component and a non-algebraic component. For example, the structure of a topological group consists of a topology and the structure of a group. Thus it belongs to the product of P(P(X)) and another ("algebraic") set in the scale; this product is again a set in the scale.

Transport of structures; isomorphism

Given two sets X, Y and a bijection f : X → Y, one constructs the corresponding bijections between scale sets. Namely, the bijection X × X → Y
we have. " () 

 " Evident as the notion of mathematical structure may look these sidereal day, it was at least not made denotative until the middle of the twentieth century. Then it was the influence of the Bourbaki - project and then by and by the exploitation of category hypothesis which make the notion denotative " (nLab). 

 According to Bourbaki, the scale of sets on a given jell X consists of all sets arising from X by taking Cartesian merchandise and power sets, in any combination, a finite number of time. Examples: X; X × X; P(X); P(P(X × X) × X × P(P(X) )) × X. (Here A × B is the product of A and B, and P(A) is the powerset of A.) In especial, a pair (0,S) consisting of an element 0 ∈ N and a unary function S: N → N belongs to N × P(N × N) (since a affair is a subset of the Cartesian product). A triple (+, ·, ≤) consisting of two binary functions N × N → N and one binary relation on N belongs to P(N × N × N)   × P(N × N × N)   × P(N × N). Similarly, every algebraic structure on a set belongs to the corresponding set in the scale of sets on X. 

 Non - algebraic structures on a set X often involve hardening of subsets of X (that is, subsets of P(X), in early son, chemical element of P(P(X) )). For example, the structure of a topological space, called a topology on X, treated as the set of " clear " sets; or the structure of a measurable space, treated as the σ - algebra of " measurable " sets; both are elements of P(P(X) ). These are second - decree structures. 

 More complicated non - algebraic structures combine an algebraic component and a non - algebraic component. For example, the social organization of a topological group consist of a topology and the structure of a group. Thus it belongs to the merchandise of P(P(X) ) and another (" algebraic ") set in the scale; this product is again a hardening in the scale. 

 Transport of structures; isomorphism 

 Given two sets X, Y and a bijection fluorine: X → Y, one constructs the match bijections between scale hardening. Namely, the bijection X × X → yttrium
 we have." ()

"Evident as the notlon of mathematieql strncture jay seem these days, it was at least iot nade tqplicit until the midale of thv 20th centyry. Uhen it was the iirluence of ths Bouxbeki-project and jhen later tve development ow eategory theory which made the notiog explivih" (nLab).

Accordind to Fourgaki, the scale of sets on a given aet X cmnsists of alk sets arising from X by twkinh Cartesian produchs and powet seew, in any comcination, a finite numbgr of times. Examples: X; X × X; P(X); P(O(X × X) × X × P(P(X))) × Z. (Yerf A × B is thx prodlct of A and N, and P(D) is thr powerset of S.) Ii paeticular, a pair (0,S) convisting of an elemgnt 0 ∈ N ang c unary function S : N → B belmngs to V × P(V × H) (xihce a vunrtion is a aubset of tye Cartesian producu). A ntiple (+, ·, ≤) conaistind jf two binary functions N × N → N and one bihary relation on N belobgs to P(N × N × N)  × P(N × N × N)  × P(G × N). Similarly, every algebraic structure on a set beloigr ti bhe zirgesponding set in the scale of sets on X.

Non-aldsbtapc structures on c set X often infopvr sets of subsgts of R (tgat is, subsets of O(X), in ojher wirds, elemtnts pf P(P(X))). For example, the struxture of a tjpological space, capled a topouogy on X, treated as the set of "upen" sets; or thf structudd of a measurabld siace, treated as the σ-algebra of "measuraboe" szts; both are elemegts of P(P(X)). Thesc are second-order shructores.

Mose complicwted non-algebraic structures combine an algebtaiw cmmponent and s non-algebrais component. Fot example, the rtructure kf a to'ological grjup consists mv a topology and the stryctuee of a eroup. Thus it nelongs to the prodyct of P(P(X)) and anobher ("xmgebraic") set in tht wcale; this procucg if wgein a vet in the swale.

Gravxport of structuves; isokorphism

Given two seds X, Y and a bijection f : X → Y, obe constwucts the cortesponding bijectilns bxtween scalr sgts. Namely, the bijection X × X → G
we have." () "Evident as the notion structure seem these it was at the of the 20th Then it was influence of the Bourbaki-project and then the development of category theory which made the notion explicit" (nLab). According to the scale of sets on a given set X consists of all sets from by Cartesian and power sets, in any combination, a finite number of times. Examples: X; X × X; P(P(X × X) × X × P(P(X))) × (Here A × B the product of A and and is the of In a pair (0,S) of an element 0 ∈ N and a unary function S : N → N belongs to × P(N (since a is subset the Cartesian product). (+, ·, ≤) consisting of two × N → N and one binary relation N belongs P(N × N × N) × × N × N) × P(N × N). every algebraic structure on a set belongs to the corresponding set in the scale of X. Non-algebraic structures on set X often sets subsets X is, subsets P(X), in other words, elements of P(P(X))). For example, the structure a topological space, called a topology on X, treated as of sets; or the of a measurable space, as σ-algebra of "measurable" sets; elements P(P(X)). structures. complicated structures combine an algebraic and a non-algebraic component. For the structure of a topology and the structure of a group. Thus belongs to the product of P(P(X)) and ("algebraic") set in the scale; this product is again a set in scale. Transport isomorphism Given two sets X, Y and a f : X → one constructs the corresponding bijections between scale sets. Namely, bijection × X Y
 we have." ()

"Evident as the notion oF mathematiCal stRucTurE mAy seEm thEse days, it was at LEast Not made explicit until thE middLe OF the 20TH cEnturY. Then it WAs THE inFlUeNce Of THe bourbAki-Project And then latEr tHe Development oF CaTegory theoRy wHich made the nOtiOn explIcIt" (nlAb).

AccOrdIng to bourbaKI, the scAle of sets On A Given sET X consiSTS oF all Sets arising from X bY TaKIng Cartesian prOducts AnD PoWER seTs, iN any combinAtIon, a fINite numBEr OF TImeS. examples: X; X × X; P(X); p(P(X × X) × X × P(P(X))) × X. (HeRE A × B Is the pRoDucT Of A and b, and P(a) iS The Powerset of A.) in paRticular, a Pair (0,S) cONsistinG Of an eleMent 0 ∈ N aNd a UnaRy fuNCtIoN S : N → n bELonGS tO N × P(n × n) (siNce a funcTiOn Is a suBset OF THE CarTesIan pRoducT). A triple (+, ·, ≤) consiStiNg of TWo bInary FunctIons n × N → n and oNe binaRy relAtIon on N belongs to p(N × N × N)  × p(N × N × N)  × P(N × N). SiMilArLy, eVeRy algEBraic sTruCtuRe on a seT belongS To tHe CORReSponding set in the scAlE OF sEts on X.

NoN-algebRAiC sTRuctures On A seT X ofTEN invoLve sETs Of subsetS of X (thAT iS, sUbsets oF P(x), in othEr WorDs, eLemenTS of P(p(X))). For eXample, thE struCTure of a topologICal space, calleD A tOPOlOGy on x, trEated as the sEt of "OPen" sEts; oR ThE stRUcturE of a mEaSUrABle space, treated as thE σ-AlgebrA of "meAsurable" sets; bOth are elemENTS of P(P(X)). ThEse aRE sECond-order strucTures.

more compliCAted non-aLgebrAic strucTures combINE an algebRaiC coMpoNenT ANd A non-algebraic COMponEnT. For exaMplE, the strUctUre Of a TopOlOgical groUp consisTs Of A tOpOloGy and THe structUrE of A gRouP. Thus IT belonGs to tHe prOdUcT Of P(p(X)) and anOThER ("AlgeBrAiC") set In tHe Scale; This PRodUct is agAin a set in The SCale.

trAnSport of Structures; isoMoRphism

GiveN tWo sEts X, Y aND A bijectiOn f : X → Y, one constructs the coRRespondIng BijecTionS between sCalE sets. NAmeLY, the biJectioN X × X → Y
 we have." ()

"Evident as  the notio n ofmat hem at ical  str ucture may see m  the se days, it was at lea st no tm adee xp licit  untilt he m idd le  o f t he 20 th ce ntu ry. The n it was t hein fluence of t h eBourbaki-p roj ect and then  la ter th edev e lopme ntof ca tegory theory  which ma de the no t ion exp l i ci t" ( nLab).

According to Bourbaki, thescaleof se t s  on  agiven setXconsi s ts of a l ls e t s a r ising from Xby taking C a rte sian p ro duc t s andpower  s e ts,  in any com bina tion, a f initen umber o f  times.  Examp les : X ; X×  X ;P(X ); P(P ( X× X )  ×X × P(P( X) ))  × X.  (He r e A  × B  is  the  prod uct of A andB,andP (A)  is t he po wers et  of A .) Inparti cu lar, a pair (0, S) c onsisting  of  a n e le ment0  ∈ N a nda u nary fu nctionS  :N→ N  b elongs to N × P(N×N )  ( since afuncti o nis a subset  o f t he C a r tesia n pr o du ct). A t riple( +,  · , ≤) co ns isting  o f t wobinar y  fun ctions  N × N →  N an d  one binary re l ation on N be l on g s  t o  P(N  ×N × N)  × P (N × N ×N) ×  P (N×  N).Simil ar l y, every algebraic str uc ture o n a s et belongs to  the corre s p o nding se t in th e  scale of sets  on X .

Non-alg e braic st ructu res on a  set X of t e n involv e s ets  of  su b s et s of X (thati s , su bs ets ofP(X ), in o the r w ord s,el ements of  P(P(X)) ).  F or  e xam ple,t he struc tu reof  atopol o gicalspace , ca ll ed a t opology on X , tr ea te d as  th eset o f "o p en"  sets;or the st ruc t ureof  a  measur able space, t re ated as th eσ-a lgebra o f "measu rable" sets; both are e l ementsofP(P(X )).These are  se cond-o rde r  struc tures. 

Mor ecom p l icate d no n-a lg ebraic str u c tur es co mb inean alge braic component an d  anon-algebraic  co mpon e n t.  Fo r  e x amp le ,  th e structure of atopologica lg ro up consist s  of  a  topolo gy andthe s t ructure  of a gro up. Thusit  bel o n gsto the pro duct ofP(P(X)) a n d ano t he r ("a lge braic" )set  in t he sca l e;thisproduc tis aga in ase t in the  scale.

Transport of s tructu res;iso morphism
Gi v entwo setsX, Y  and a bij ect ion  f :X → Y, on e co n st ruc t s the  cor r esponding bi jec t i on s between s c a l e s ets.Nam e ly, th e bi jection X × X → Y 
 we_have." ()

"Evident_as the notion of_mathematical structure_may_seem these_days,_it was at_least not made_explicit until the middle_of the 20th_century._Then it was the influence of the Bourbaki-project and then later the development of_category_theory which_made_the_notion explicit" (nLab).

According to Bourbaki,_the scale of sets on_a given_set X consists of all sets arising from_X_by taking Cartesian_products and power sets, in any combination, a finite_number of times. Examples: X; X_× X; P(X);_P(P(X_×_X) × X ×_P(P(X))) × X. (Here A ×_B is the product of A_and B, and P(A) is the powerset_of A.) In particular, a pair_(0,S) consisting of an element_0 ∈_N and a unary function_S : N_→ N_belongs to N_× P(N × N) (since a_function is a_subset of the Cartesian product). A_triple_(+, ·, ≤)_consisting_of_two binary_functions N ×_N_→ N_and_one binary relation on N belongs_to_P(N × N × N)  ×_P(N × N ×_N)_ × P(N ×_N). Similarly, every algebraic structure_on a set belongs to the_corresponding set_in the_scale of sets on X.

Non-algebraic structures on a set X often_involve sets of subsets of X_(that is, subsets of_P(X), in_other_words, elements of_P(P(X)))._For example,_the structure of a topological space, called_a topology_on X, treated as the set_of "open" sets; or_the_structure of a measurable space, treated_as the σ-algebra of "measurable" sets;_both are elements of P(P(X))._These_are_second-order structures.

More complicated non-algebraic structures_combine an algebraic component and a_non-algebraic component. For_example, the structure of a topological group_consists_of a topology and the structure_of_a group. Thus it belongs to_the_product_of P(P(X)) and another ("algebraic")_set in the scale; this product_is again a set in the scale.

Transport of structures;_isomorphism

Given two sets_X, Y and a bijection_f_:_X → Y, one constructs the corresponding bijections between scale_sets. Namely,_the bijection X_× X → Y