Replication via Invalidating the Applicability of the Fixed Point Theorem

Replication via In v alidating the Applicability of the Fixed P oin t Theorem Gen taIto Maruo L ab., 50 0 El Camino R e al #30 2, Burlingame, CA 94 010, Unite d States. Abstract W e p resen t a construction of a certain in finite complete p artial order (CPO) th at differs from the stand ard construction used in Scott’s denotational seman tics. In addition, w e construct sev eral other infin ite CPO’s. F or some of those, w e apply the usu al Fixed Poi nt Theorem (FPT) to yield a fixed p oin t for ev ery con tin uous function µ : 2 → 2 (where 2 denotes the set { 0 , 1 } ), w hile for the other C PO’s w e cannot in v ok e that theorem to yield su ch fixed p oin ts. Ev ery elemen t of eac h of these CPO’s is a b inary string in the monot ypic form and we show that inv a lidation of the applicabilit y of the FPT to the CPO that Scott’ s constru cted yields the concept of replication. Key wor ds: Denotatio nal Seman tics, Fixed P oin t, Ad j unction, Boundary, LR-transformation, Inte rnal Measurement 1 In tro duction One of the most imp ortant differences b etw een ph ysics and biolo gy is the con tr a st betw een indistinguishable particles and replications. The former is a constrain t/ condition on calcu lations in statistical mec hanics, and there a re t w o t yp es of indistinguishable particles in the ph ysical w orld: b osons and fermions. Con vers ely , DNA replication is one example of the latter. It o ccurs in a cell and is follow ed b y a cell division. In the sense tha t DNA and a cell can b e in terpreted as a co de a nd its deco der, resp ectiv ely , there is a measuremen t pro cess b et w een them. In his theory of denotational sem an tics [15], Scott regards a computer program as an elemen t of a complete partial order (CPO). He constructs a CPO X suc h Email add r ess: cxq02 365@gmail .com (Gen taIto). Preprint s ubmitted to Acta Biotheoretica 27 No ve nber 2007 that X ≃ C( X, 2), whe re 2 denotes the set { 0 , 1 } a nd C( X, 2) is the set of all con tin uous functions f : X → 2. (Here, the t erm c ontinuous function is used in t he context of a CPO, whic h is defined later.) On one hand, he uses the isomorphism b etw een X and C( X, 2) to obtain a fixed p oin t for ev ery con tinuous f unction µ : 2 → 2. (F or a set Y , a function τ : Y → Y has a fixed p oin t if there is some y ∈ Y suc h that τ ( y ) = y .) On the other hand, he regar ds the relation X ≃ C( X , 2) as a domain equation, whic h he uses to find a new meaning for the progr a m as a lo op structure (self-similar lo op program). Gunji interprets the isomorphism as an “abstract” b oundary to whic h the Fixed P oin t Theorem (FPT) (Theorem 6) is applicable, and the domain equ a- tion as an inv a lida tion of the b oundary , that is, an inv alidation of the appli- cabilit y of the FPT, in order t o construct a dynamical system in the contex t of theoretical biology [1]. He argues that any solution to a problem will in- evitably b e a pseud o-solution, and that the pseudo-solution at an y given time step “alwa ys” tr iggers a pro blem to b e solved a t the next time step. This leads to a p erp etual ev olutionary pro cess that allow s fo r emergent prop erties. This p erp etual cy cle of problem and pseudo-solution is called a p erp etual e qui- libr ating m e chanism [9], whic h Matsuno prop osed in the field of researc h in theoretical biology kno wn as In ternal Measuremen t (IM) [9,10,1,2 ]. Recen tly , Matsuno and Gunji emphasized the imp ortance of a type of engine in theoret- ical biology b y whic h dynamical sy stems ev olv e and o ccurrences of distinction con tinue t o alternate with occurrences of in v alidation of the distinction [10,2]. W e may express an IM’s mo del o f living thing s in terms of a triad: t w o differen t logical la ye rs suc h as the Exten t (collection of fragmen ts/elemen ts/parts) and the In ten t (prop erty as a whole), and a mediator/in terface to adjust the t wo la y ers in an “inconsisten t” manner [10,2]. IM claims that the relationship b et w een the t w o lay ers is not consisten tly determined , a nd fo r this reason they are p erpetually changing relative to one another. In one of the mo dels in IM prop osed by Gunji et a l. [2 ], first, the Extent and In tent are regarded to hav e the structures of a la t t ice and a quotien t lattice, r espective ly . The op erations b etw een the t w o la y ers, σ : Exten t → Inten t and ρ : In tent → Exten t , can b e defined as a sheaf (that is, a type o f integration op eration) from a lat tice to a quotien t la ttice, and a r evers e sheaf (that is, a ty p e of differen tiatio n op eration) from a quotien t lat t ice to a lattice, resp ectiv ely , only when the Exten t and Inten t are consisten t with eac h other. Ho w eve r, IM claims a fundamen tal incons istency b et w een the Exten t and Inten t. Hence, second, an observ er who cannot lo ok out o v er the whole lattice (i.e., Ex ten t) is in tr o duced on the rev erse sheaf, ρ : In ten t → Exten t . T his rev eals a colla pse of the lattice structure itself in the Exten t. Third, a new mathematical op eration is in tro duced, called a sk eleton (repairing function), whic h is required to repair the broke n structure in the Exten t. By t he op eration of the sk eleton, the la t t ice structure is reco v ered and then a new quotien t lattice is constructe d b y the 2 sheaf, σ : Exte n t → In ten t . This pro cess (consisting of ρ a nd σ ) is going o n p erp etually . The sk eleton that mediates b etw een the t w o lev els is regarded as a particular expre ssion for the material cause and/or a clock as a particular expression for t ime itself. One of sev eral distinguishing features in Gunji’s mo del (in the conte xt of the curren t pap er) is the role of the observ er who cannot lo ok out o v er the whole lattice (i.e., Exten t). On one hand, IM claims that the t w o la y ers a r e incon- sisten t with eac h other a nd are therefore c hanging p erp etually . On the o t her hand, if w e supp ose that the defectiv e observ er can lo ok out o ver the whole lat- tice, t hen the operatio n betw een the t w o lay ers can be defined consisten tly ( a s sheaf a nd rev erse-sheaf ) and the p erp etual pro cess therefore stops. A ques- tion then arises: Are the tw o lay ers consisten t or inconsisten t? If they are consisten t, then the defectiv e observ er exists, unfort unat ely , just to destro y the structure of the Exten t. If they are inconsisten t, that is, if w e assume the consistency b et w een them and w e obtain a logical paradox suc h as Russel’s parado x [6], what the observ er destro ys is the presupp osition of the consis- tency . Whic h is correct in IM? How ev er, it do es not matter whether he is a destro yer or a life-sa ving s up er-prev en ter, b ecause he cannot lo ok out ov er the whole lattice in an y case. This must b e the essen tial treatmen t of the lo gical parado x in IM. Rosen also considers suc h a para dox in his mo del of a living thing as a metab olism-r ep air system ( M-R system ) [1 3]. The mo del consists of tw o sets: X , whic h is a set of raw materials, a nd Y , whic h is a set of b ehavio rs . It also includes t hree functions: f ∈ F = Hom( X , Y ), called a metab oli c function , g ∈ G = Hom( Y , F ), called a r ep air function , and h ∈ H = Hom ( F , G ), called a r eplic ation function , where he claims g ∈ G and h ∈ H are onto f unctions and Y ≃ H holds. One of t w o remark able features in Rosen’s mo del is that there are no suc h on to functions g ∈ G and h ∈ H . If we assume that they exist, then w e obtain a pa rado x suc h as Russel’s para do x [6 ]. A second re- mark able feature is that an y function in the system can b e b oth a function and an output of a different function. He claims that a cen tr al feature of living things is c omplex i ty . A system is called c omplex if its b eha vior cannot b e captured b y mo dels of that system; otherwise, that system is called simple [14]. The term “complex” for a system ma y b e r eplaced with “ inc omputable ” or “ not wel l-forme d ” for its mo dels. Therefore a system is called complex only if its mo dels are incomputable or not well-formed, and so on. A treatmen t o f the pa rado x is to in vok e hy p erset theory [5 ]. A h yp erse t is defined as a graphable set whic h is a digraph ( “ digraph” is short for directed graph) suc h that a no de can b e either a set or an elemen t of a set, and a directed edge → is the set membership ∋ . A ∋ A , whic h leads to R ussel’s 3 parado x [6], is inte rpreted as just a lo op structure in h yp erset theory . There- fore w e should r edefine the complexit y o f a system so that a system is called complex if it cannot b e we ll-formed in Rosen’s o riginal mo del but can b e w ell-fo rmed (as a lo op structure) in h yp erset theory . Let Π b e a collection of concepts/en tities, and let In t( π ) and Ext( π ) b e the in tension and exten sion of the conce pt π ∈ Π. The triad— In t ( π ) , Ext( π ), and π ∈ Π— is a use ful means to o rganize our though ts. There are at least t w o usages. One is when we assume that, giv en a pair of in tension I and ex tension E , w e try to find a concept π ∈ Π suc h that I = Int( π ) and E = Ext ( π ). Another is when, giv en a concept π ∈ Π, w e try to find its In t( π ) and Ext( π ). When Int( π ) is give n as Hom(Ext ( π ) , 2), where Hom( A, B ) denotes the set of all morphisms from A to B , and 2 denotes a set { 0 , 1 } , if w e assume that In t ( π ) ≃ Ext( π ), then we obtain the pa rado x. W e can regard the paradox in an M-R system a s this case. How ev er, w e hav e t o note t ha t there is not the asso ciated concept π ∈ Π in b oth cases of the original M-R system and its in t erpretatio n with h yp erset theory . Although it is still difficult to specify what the asso ciated π ∈ Π in IM is, w e ma y at least state that it is materia l causatic. In this pap er w e will b e consisten t and consider binary strings in the form of: 0 · · · 0 | {z } u 1 · · · 1 | {z } v , (1) where u is the n umber of 0 ’s and v is the num b er of 1’s, a nd u, v ∈ N = { 0 , 1 , 2 , . . . } . W e will regard (1) as the concept π ∈ Π that w e will cons isten tly consider in this pap er. Whe n w e insert a comma at the cen t er of a binary string to divide the string in to tw o parts, w e regard the left part as the Int( π ) and the righ t part as the Ext( π ). W e should note that the notat io n “0 · · · 01 · · · 1” means that the string consists of finitely man y 0’s, follow ed b y finitely many 1’s. Hence, we mus t explicitly define binary strings one-by -one, suc h as • finitely man y 0’s, follo w ed by finitely man y 1’s • infinitely many 0’s, follow ed b y finitely man y 1 ’s • finitely man y 0’s, follo w ed by infinitely man y 1’s · · · etc. W e designate these strings by (1) in this paper. In this sense, b o t h π ∈ Π and (1) are not strict mathematical statemen ts. W e will construct v arious infinite CPO’s. In each of the CPO’s an elemen t (i.e., a binary string) is in the form of ( 1 ). F or some of those CPO’s, w e can apply the FPT to yield a fixed p oint for ev ery con t in uous function µ : 2 → 2, while for the other CPO’s w e cannot in v oke that theorem to yield suc h fixed p oin ts. In section 2, we construct a countably infinite CPO S in the manner 4 of Scott’s denotat ional semantic s, and we define an isomorphism fro m S to C( S, 2) in the usual w ay . In section 3, w e construct a CPO whic h is isomorphic to S , b y using a completely differen t method. Then in section 4 , we in tro duce a transformation (called the LR-transformatio n) b et wee n certain types of in- finite binary strings that a re defined from differen t sp ecifications of certain finite binary strings. In section 5 w e sho w that the concep t of replication is deriv ed from inv a lidating the applicability of the FPT to the CPO that Scott constructed. 2 Construction of infinite CPO 2.1 Pr eliminaries In this pap er, w e construct a n um b er of linear orders, eac h of whic h is a complete partial o rder. Definition 1 A p artial or der ( D , ⊆ ) is a complete partial order (CPO) if D has a ⊆ -le ast element and every c ountable, monotone non-de cr e asing se quenc e d 0 ⊆ d 1 ⊆ d 2 ⊆ · · · of el e m ents o f D ha s a unique le ast upp er b ound ∪ d i in D . Definition 2 L et ( D , ⊆ D ) and ( E , ⊆ E ) b e C PO’s. A function g : D → E is con tinuous if it satisfies the fol lowing c onditions: (1) If d i ⊆ D d j , then g ( d i ) ⊆ E g ( d j ) . (2) If d 0 ⊆ d 1 ⊆ d 2 ⊆ · · · is a c o untable , m o notone non-d e cr e asing s e quenc e of ele m ents of D , then g ( ∪ d i ) = ∪ ( g ( d i )) . Definition 3 The r elation D ⊆ E holds of CPO’s D , E if ther e e xist c on tin- uous functions e : D → E and p : E → D such that e ◦ p ⊆ id E and p ◦ e = id D . The f unction s e and p ar e c al le d em b edding an d pro jection , r esp e ctively. I f e ◦ p = id E , then D is isomorphic to E , which is denote d by D ≃ E . Prop osition 4 [15] L et D and E b e CPO’s, and le t C( D , E ) b e the s et of al l c ontinuous functions fr om D to E . Then (C( D, E ) , ⊆ ) is also a CPO, the le ast element b eing the c onstant function which maps every d ∈ D to the le ast element of E , and the le ast upp er b ound of the c ountable, monotone non- de cr e asing se quenc e g 0 ⊆ g 1 ⊆ g 2 ⊆ · · · b eing the function ∪ g i define d b y ( ∪ g i )( d ) = ∪ ( g i ( d )) . Let 2 b e the set { 0 , 1 } . Clearly , the linear o r der (2 , ⊆ ) defined b y 0 ⊆ 1 is a CPO ( which we will denote b y 2), so the next result follow s immed iately from Prop osition 4. 5 Corollary 5 L et ( S 0 , ⊆ ) b e a CPO. T hen S 1 ≡ C( S 0 , 2) , S 2 ≡ C(C( S 0 , 2) , 2) , . . . ar e als o CPO’s. Theorem 6 (Fixe d Point The or em) [4 ,3] L et µ : 2 → 2 b e c ontinuous, let ( S, ⊆ ) b e a CPO which is isomorphic to C( S, 2) , and let ϕ b e an isom o rphism (a c on tinuous b ije ction) fr om S to C( S, 2) . F urthermor e, let g : S → 2 b e the c ontinuous function define d by g ( x ) = µ ( ˆ ϕ x ( x )) , wher e ˆ ϕ x denotes ϕ ( x ) . Then g ( ϕ − 1 ( g )) is a fix e d p oint of µ . Pro of By the definition of ϕ , w e ha v e tha t, for ev ery f ∈ C( S, 2) and ev ery x ∈ S , f ( x ) = ˆ ϕ ϕ − 1 ( f ) ( x ) Since g ∈ C( S, 2), we ha v e t ha t, for eve ry x ∈ S , g ( x ) = ˆ ϕ ϕ − 1 ( g ) ( x ) Setting x to ϕ − 1 ( g ) yields g ( ϕ − 1 ( g )) = ˆ ϕ ϕ − 1 ( g ) ( ϕ − 1 ( g )) Also, b y the definition of g , we hav e g ( ϕ − 1 ( g )) = µ ( ˆ ϕ ϕ − 1 ( g ) ( ϕ − 1 ( g ))) Th us µ ( ˆ ϕ ϕ − 1 ( g ) ( ϕ − 1 ( g ))) = g ( ϕ − 1 ( g )) = ˆ ϕ ϕ − 1 ( g ) ( ϕ − 1 ( g )) Hence ˆ ϕ ϕ − 1 ( g ) ( ϕ − 1 ( g )) (= g ( ϕ − 1 ( g ))) is a fixed p oin t of µ .  Preparatory to constructing an infinite CPO, w e cons truct an infinite seq uence of finite CPO’s. W e do this in stages, whic h w e inde x with the natural n um b ers n ≥ 1. (Throughout this paper, the v ariable n denotes a natural num b er, i.e., an elemen t o f the set N = { 0 , 1 , 2 , . . . } . Except where n is sp ecifically res tricted in some w ay —suc h as in t he stages of this construction, where n is restricted to v alues greater than 0—it is to b e assumed that n is any nonnegativ e integer.) A t stage n , w e construc t a se t S n with n elemen ts, and w e de fine a linear order ⊆ n on S n . Since S n is finite, ( S n , ⊆ n ) is a CPO. W e represen t the elemen ts of S n as binary strings (strings of 0’s and 1’s) of length n − 1. La ter, w e will use infinite seq uences of elemen ts of S ∞ n =1 S n to construct t w o infinite sets S . In eac h case, w e will define a linear order ⊆ on S in suc h a wa y that ( S, ⊆ ) is a CPO. Stage 1 Let S 1 = { λ } , where λ is the empt y string (the binary string of length 0). Then ( S 1 , ⊆ 1 ) is trivially a CPO (where λ ⊆ 1 λ ). W e will denote ( S 1 , ⊆ 1 ) b y { λ } . 6 Stage 2 Let S 2 = C( S 1 , 2), and let ( S 2 , ⊆ 2 ) b e the CPO obtained f r om Prop o- sition 4 for D = C( S 1 , 2) and E = 2. Since S 2 is the set of con tin uous functions from { λ } to 2 (where 2 = { 0 , 1 } ), S 2 consists o f the functions ( λ → 0) a nd ( λ → 1). W e will denote those functions b y their outputs (0 and 1, resp ectiv ely). Recall that 0 ⊆ 1 (b y our definition of the CPO 2), so w e will denote ( S 2 , ⊆ 2 ) by { 0 ⊆ 1 } . Stage 3 Similarly , let S 3 = C( S 2 , 2), a nd let ( S 3 , ⊆ 3 ) b e the CPO obtained from Prop osition 4 f o r D = C( S 2 , 2) and E = 2. Since S 3 is the set of con- tin uo us functions fro m S 2 to 2 , S 3 consists of the f unctions (0 → 0; 1 → 0) , (0 → 0 ; 1 → 1), a nd (0 → 1; 1 → 1). The function (0 → 1; 1 → 0) is excluded, b ecause it is not con tinuous (the outputs do not preserv e the order of the inputs). W e will denote the three elemen ts of S 3 b y 00 , 01, and 11, respectiv ely (i.e., for each elemen t of S 3 —equiv alen tly , for eac h con tinuous function fro m S 2 in to 2— w e concatenate the o utputs that corresp ond to the t w o inputs, 0 and 1), so w e will denote ( S 3 , ⊆ 3 ) by { 00 ⊆ 01 ⊆ 1 1 } . . . . Stage n Let S n = C( S n − 1 , 2), and let ( S n , ⊆ n ) b e the CPO obtained fro m Prop o- sition 4 for D = C( S n − 1 , 2) and E = 2. ( S 1 , ⊆ 1 ) = { λ } ( S 2 , ⊆ 2 ) = { 0 ⊆ 2 1 } ( S 3 , ⊆ 3 ) = { 00 ⊆ 3 01 ⊆ 3 11 } ( S 4 , ⊆ 4 ) = { 000 ⊆ 4 001 ⊆ 4 011 ⊆ 4 111 } . . . ( S n , ⊆ n ) = { 0 · · · 000 0 | {z } n − 1 ⊆ n 0 · · · 0001 | {z } n − 1 ⊆ n 0 · · · 0011 | {z } n − 1 ⊆ n 0 · · · 0111 | {z } n − 1 ⊆ n · · · ⊆ n 0001 · · · 1 | {z } n − 1 ⊆ n 0011 · · · 1 | {z } n − 1 ⊆ n 0111 · · · 1 | {z } n − 1 ⊆ n 1111 · · · 1 | {z } n − 1 (2) F or fixed n , there ma y be se v era l w a ys to define the em b edding and pro jection functions e : S n → S n +1 and p : S n +1 → S n , and different em b edding and pro jection functions will lead to differen t infinite sets S . W e now review the standard definitions of em b edding and pro jection, as used b y Scott [15] and others [16,4,3]. 7 2.2 Standar d definition s of emb e dding and pr oje c tion Let S 1 , S 2 , S 3 , . . . be the finite sets defined in (2). W e no w presen t t w o differen t definitions of the em b edding a nd pro jection functions. F or each definition, w e use the pro jection functions to generate a coun tably infinite CPO. F or eve ry n ≥ 1 , define the embedding function e n : S n → S n +1 and the pro jection function p n : S n +1 → S n as sho wn in T able 1. ( No te that these functions satisfy D efinition 3.) 11111 ւր 1111 01111 ւր ւր 111 0111 00111 ւր ւր ւ 11 011 0011 ↔ 00 011 ւր ւ ւ 1 01 ↔ 00 1 ↔ 0001 ↔ 00001 ւ ւ λ ↔ 0 ↔ 00 ↔ 000 ↔ 0000 ↔ 00000 S 1 S 2 S 3 S 4 S 5 S 6 T able 1 Definition of embedd ing and pro jectio n for n = 1 , 2 , . . . , 5. F or example, e 2 : S 2 → S 3 is defined as e 2 (0) = 00 , e 2 (1) = 11, and p 2 : S 3 → S 2 is defined as p 2 (00) = p 2 (01) = 0 , p 2 (11) = 1. No w lab el all t he entrie s in T a ble 1 : Lab el en try s with the n um b er of 1’s in s (i.e., lab el λ , 0, 00, 0 0 0, . . . with 0; lab el 1, 01, 00 1, 00 0 1, . . . with 1; lab el 11, 011, 001 1, 00011, . . . with 2; etc.). This yields T able 2 . 5 ւր 4 4 ւր ւր 3 3 3 ւր ւր ւ 2 2 2 ↔ 2 ւր ւ ւ 1 1 ↔ 1 ↔ 1 ↔ 1 ւ ւ 0 ↔ 0 ↔ 0 ↔ 0 ↔ 0 ↔ 0 S 1 S 2 S 3 S 4 S 5 S 6 T able 2 T able 1 after labeling of the en tries . 8 Our coun tably infinite CPO will consist of all the infinite paths in T able 2 that ha v e t he 0 in S 1 as their in v erse (pro jectiv e) limit (i.e., ev ery elemen t o f our CPO is an infinite seque nce ( s 1 , s 2 , s 3 , . . . ) suc h t ha t, for ev ery i ≥ 1 , s i ∈ S i and s i = p i ( s i +1 )). Lab el ev ery suc h infinite path, either with n or n ′ for some n ∈ N , or with ∞ , as follo ws: 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , . . . ↔ 0 0 , 0 , 1 , 1 , 1 , 1 , 1 , 1 , 1 , 1 , . . . ↔ 1 0 , 0 , 1 , 1 , 2 , 2 , 2 , 2 , 2 , 2 , . . . ↔ 2 . . . 0 , 0 |{z} 2 , 1 , 1 |{z} 2 , 2 , 2 |{z} 2 , 3 , 3 |{z} 2 , 4 , 4 |{z} 2 , . . . ↔ ∞ . . . 0 , 0 , 1 , 1 , 2 , 3 , 4 , 5 , 6 , 7 , . . . ↔ 2 ′ 0 , 0 , 1 , 2 , 3 , 4 , 5 , 6 , 7 , 8 , . . . ↔ 1 ′ 0 , 1 , 2 , 3 , 4 , 5 , 6 , 7 , 8 , 9 , . . . ↔ 0 ′ (3) The infinite path whic h is lab eled with ∞ is the least upp er b ound of the infinite paths lab eled with n for some n ∈ N . Th us w e ha v e a countably infinite CPO with the follo wing structure: Φ = { 0 ⊆ 1 ⊆ 2 ⊆ 3 ⊆ · · · ⊆ ∞ ⊆ · · · ⊆ 3 ′ ⊆ 2 ′ ⊆ 1 ′ ⊆ 0 ′ } (4) F or ev ery n , map the elemen t s of Φ which ar e lab eled with n and n ′ to the con tinuous functions ψ n and ψ n ′ , resp ectiv ely (and ma p the elemen t of Φ whic h is lab eled with ∞ to the con tin uous function ψ ∞ ), a s sho wn in T able 3. F rom the table, it is o bvious that the map ˆ ψ : Φ → C(Φ , 2) whic h consists of the union of the maps n 7→ ψ n , n ′ 7→ ψ n ′ , and ∞ 7→ ψ ∞ preserv es the ordering of Φ. Hence ˆ ψ is a con tin uous function and an isomorphism. This is the usual construction of a coun tably infinite CPO Φ (i.e., the con- struction used b y Scott). Moreo ve r, the usual deriv ation of a fixed p oin t for a contin uous function µ : 2 → 2 is that whic h is g iv en in the statemen t of Theorem 6 . W e obtain that fixed p oin t (namely , g ( ˆ ψ − 1 ( g )), where g is a s in the statemen t of Theorem 6) from T able 3: • if µ (0) = 0 = µ (1), then g is ψ 0 , so the fixed p oint is ψ 0 (0) (= 0) • if µ (0) = 1 = µ (1), then g is ψ 0 ′ , so the fixed p oint is ψ 0 ′ (0 ′ ) (= 1) • if µ (0) = 0 and µ ( 1 ) = 1, then g is ψ ∞ , so the fixed p oin t is ψ ∞ ( ∞ ) (= 0) 9 ψ \ x 0 1 · · · n − 1 n · · · ∞ · · · ( n − 1) ′ n ′ · · · 1 ′ 0 ′ ψ 0 0 0 · · · 0 0 · · · 0 · · · 0 0 · · · 0 0 ψ 1 0 0 · · · 0 0 · · · 0 · · · 0 0 · · · 0 1 . . . ψ n 0 0 · · · 0 0 · · · 0 · · · 0 1 · · · 1 1 . . . ψ ∞ 0 0 · · · 0 0 · · · 0 · · · 1 1 · · · 1 1 . . . ψ n ′ 0 0 · · · 0 1 · · · 1 · · · 1 1 · · · 1 1 . . . ψ 1 ′ 0 1 · · · 1 1 · · · 1 · · · 1 1 · · · 1 1 ψ 0 ′ 1 1 · · · 1 1 · · · 1 · · · 1 1 · · · 1 1 T able 3 The functions ψ in C(Φ , 2) and their v alues ( ψ ( x ) for x ∈ Φ) As an alt ernat iv e, we can define the embedding and pro jection functions as in T able 4, and w e can use those functions to define a countably infinite CPO whic h has a structure differen t from that of Φ. 11111 ւ 1111 ↔ 01111 ւ 111 ↔ 0111 ↔ 00111 ւ 11 ↔ 011 ↔ 0011 ↔ 00011 ւ 1 ↔ 01 ↔ 001 ↔ 0001 ↔ 00001 ւ λ ↔ 0 ↔ 00 ↔ 000 ↔ 0000 ↔ 00000 S 1 S 2 S 3 S 4 S 5 S 6 T able 4 Alternativ e definition of em b eddin g and pro jection for n = 1 , 2 , . . . , 5 . Using the same lab eling sc heme in T able 4 as that whic h w as used in T able 1 yields T able 5. No w lab el eve ry infinite path in T able 5 that has the 0 in S 1 as its inv erse 10 5 ւ 4 ↔ 4 ւ 3 ↔ 3 ↔ 3 ւ 2 ↔ 2 ↔ 2 ↔ 2 ւ 1 ↔ 1 ↔ 1 ↔ 1 ↔ 1 ւ 0 ↔ 0 ↔ 0 ↔ 0 ↔ 0 ↔ 0 S 1 S 2 S 3 S 4 S 5 S 6 T able 5 T able 4 after labeling of the en tries (pro jectiv e) limit, either with some n ∈ N or with ∞ , a s follo ws: 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , . . . ↔ 0 0 , 1 , 1 , 1 , 1 , 1 , 1 , 1 , 1 , 1 , . . . ↔ 1 0 , 1 , 2 , 2 , 2 , 2 , 2 , 2 , 2 , 2 , . . . ↔ 2 0 , 1 , 2 , 3 , 3 , 3 , 3 , 3 , 3 , 3 , . . . ↔ 3 . . . 0 , 1 , 2 , · · · , n − 1 | {z } n , n, n, . . . ↔ n . . . 0 , 1 , 2 , 3 , 4 , 5 , 6 , 7 , 8 , 9 , . . . ↔ ∞ This yields a CPO with the fo llowing structure: Θ = { 0 ⊆ 1 ⊆ 2 ⊆ 3 ⊆ · · · ⊆ ∞} (5) Unfortunately , Θ is not isomorphic to C(Θ , 2). T o see this, we will examine the only t w o mappings from Θ t o C(Θ , 2 ) that could p ossibly b e isomorphisms. One p ossibilit y is that, for ev ery n ∈ N , the elemen t of Θ whic h is lab eled by n w ould b e mapp ed to the function ψ ′ n ∈ C( S, 2) defined by ψ ′ n ( x ) = 1 if x is one of the n great est elemen ts of Θ, and ψ ′ n ( x ) = 0 otherwise (and mapping the elemen t of Θ whic h is lab eled b y ∞ to the all-1 function ψ ′ ∞ ). Ho w ev er, the “ n g r eatest elem en ts of Θ” do not exist, so this mapping is imp ossible. Alternativ ely , w e could map the elemen t of Θ whic h is lab eled by n to the function ψ ′′ n ∈ C(Θ , 2) defined b y ψ ′′ n ( x ) = 1 for eve ry x ∈ Θ with n ⊆ x , 11 and ψ ′′ n ( x ) = 0 otherwise ( a nd mapping the elemen t whic h is lab eled b y ∞ to the all-0 f unction ψ ′′ ∞ ), a s sho wn in T able 6. Note, how ev er, that ψ ′′ y ⊆ ψ ′′ x for elemen ts x, y of Θ with x ⊆ y , so t his mapping do es not preserv e t he ordering of Θ; hence it is not an isomorphism. ψ ′′ \ x 0 1 2 · · · ψ ′′ 0 1 1 1 · · · ψ ′′ 1 0 1 1 · · · ψ ′′ 2 0 0 1 · · · . . . ψ ′′ ∞ 0 0 0 · · · T able 6 The functions ψ ′′ in C(Θ , 2) and their v alues ( ψ ′′ ( x ) for x ∈ Θ ) 3 Another represen tation of Φ and Θ , and beyond W e now use a differen t construction for an infinite CPO that has the same structure as Φ, and then w e go on to construct infinite CPO’s whose structure s differ from that of Φ. 3.1 Non-standar d r epr esen tations of Φ and Θ First, w e construct a coun tably infinite CPO Λ whic h is isomorphic to Φ. Eac h elemen t of Λ is a n infinite binar y string. W e do this directly , without first constructing a sequence of finite CPO’s . Denote the order type of the usual linear order on N (i.e., 0 ⊆ 1 ⊆ 2 ⊆ · · · ) b y ω , and denote the rev erse order (i.e., · · · ⊆ 2 ⊆ 1 ⊆ 0) by ω ∗ . T o construct Λ, w e first obtain one infinite set of infinite binary strings, Ω, b y starting with the a ll-0 string that has order t yp e ω ∗ (i.e., the all-0 string · · · 000), and g enerating the remaining strings in succession b y changing the righ tmost 0 to a 1: Ω = {· · · 000 , · · · 001 , · · · 001 1 , · · · 0011 1 , . . . } This giv es us a linear order that has or der t yp e ω : Ω = { · · · 000 ⊆ · · · 001 ⊆ · · · 0011 ⊆ · · · 00111 ⊆ · · · } (6) 12 Next, w e obtain another infinite set of binary strings, Ω opp , b y starting with the all-1 string that has order t yp e ω (i.e., the all-1 string 111 · · · ), and generating the remaining strings in success ion b y c hanging the leftmost 1 to a 0: Ω opp = { 1 11 · · · , 011 · · · , 0011 · · · , 00011 · · · , . . . } This giv es us a linear order that has or der t yp e ω ∗ : Ω opp = { · · · ⊆ 00011 · · · ⊆ 00 11 · · · ⊆ 01 1 · · · ⊆ 111 · · · } (7) Lab el x ∈ Ω with n , where n is the nu m b er of 1’s in x . Similarly , lab el y ∈ Ω opp with n ′ , where n is the n um b er of 0 ’s in y . Then w e hav e a linear order (Ω ∪ Ω opp , ⊆ ) of order ty p e ω + ω ∗ : 0 ⊆ 1 ⊆ 2 ⊆ · · · ⊆ n ⊆ · · · ⊆ · · · ⊆ n ′ ⊆ · · · ⊆ 2 ′ ⊆ 1 ′ ⊆ 0 ′ This is not a CP O, because Ω has no “sup” (suprem um, i.e., least upp er b ound). The most natural infinite binary string w e could use as t he sup of Ω is the a ll-1 string that has o rder ty p e ω ∗ (i.e., the string · · · 111). If w e place that all-1 string b et wee n t he elemen ts of Ω and t he elemen ts of Ω opp (and label it with ∞ ), we obt a in the following CPO: Λ = Ω ′ ∪ Ω opp = { 0 ⊆ 1 ⊆ 2 ⊆ 3 ⊆ · · · ⊆ ∞ ⊆ · · · ⊆ 3 ′ ⊆ 2 ′ ⊆ 1 ′ ⊆ 0 ′ } , (8) where Ω ′ = Ω ∪ {· · · 111 } . This CPO has order type ω + 1 + ω ∗ , as do es Φ in (4), so Λ ≃ Φ. W e can a ssign to x ∈ Λ the function ψ x in T able 3 . Hence the map x 7→ ψ x is a con tin uous function and an isomorphism from Λ to C(Λ , 2). F or ev ery con tinuous function µ : 2 → 2, we obtain the same fixed p oin t as t he one w e found earlier (where w e applied Theorem 6 to the isomorphism ˆ ψ : Φ → C(Φ , 2)). Definition 7 L et x b e an element of a line ar or der ( S, ⊆ ) . T hen x has an imme diate neighb or in ( S, ⊆ ) if x has an imme diate pr e de c esso r and/or an imme diate suc c essor in ( S, ⊆ ) . F or example, b oth · · · 000 a n d 111 · · · have an imm e diate neighb or in Λ ( · · · 000 has an imme diate suc c essor (namely, · · · 001 ), and 111 · · · has an imm e diate pr e de c esso r (namely, 011 · · · )), but · · · 111 has no imme diate neighb or in Λ . Clearly , Ω ′ ≃ Θ. How ev er, Ω ′ 6≃ Ω opp , b ecause Ω ′ has order type ω + 1 and 13 Ω opp has order type ω ∗ . Since Ω ′ ≃ Θ 6≃ C(Θ , 2) ≃ C(Ω ′ , 2) , w e hav e Ω ′ 6≃ C(Ω ′ , 2). Th us, for a contin uous function µ : 2 → 2, w e cannot apply Theorem 6 to yield a fixed p oint of µ . 3.2 CPO of or der typ e ω + 1 + 1 + ω ∗ ; adjunction In Λ, the infinite binary string · · · 1 11 is not only the sup of Ω but also the “inf ” (infim um, i.e., greatest low er bound) of Ω opp . How ev er, the most natural infinite binary string w e could use as t he inf of Ω opp is the all-0 string that has order ty p e ω (i.e., the string 000 · · · ). If w e place that all-0 string betw een the elemen ts of Ω ′ and the eleme n ts of Ω opp (and lab el it with ∞ ′ ), w e obta in the follo wing CPO: Λ ′ = Ω ′ ∪ Ω ′ (opp) = { 0 ⊆ 1 ⊆ 2 ⊆ 3 ⊆ · · · ⊆ ∞ ⊆ ∞ ′ ⊆ · · · ⊆ 3 ′ ⊆ 2 ′ ⊆ 1 ′ ⊆ 0 ′ , } (9) where Ω ′ (opp) = { 000 · · · } ∪ Ω opp . This CPO has or der t yp e ω + 1 + 1 + ω ∗ , so it is not isomorphic t o any of the linear orders discussed earlier in this pap er. W e can a ssign to x ∈ Λ ′ the contin uous function ψ x in T a ble 7 . Hence the map x 7→ ψ x is a con tinuous function and an isomorphism from Λ ′ to C(Λ ′ , 2). F or ev ery con tin uous function µ : 2 → 2, w e obtain a fixed p oin t (namely , g ( ˆ ψ − 1 ( g )), where g is as in the statement o f Theorem 6) from T able 7: • if µ (0) = 0 = µ (1), then g is ψ 0 , so the fixed p oint is ψ 0 (0) (= 0) • if µ (0) = 1 = µ (1), then g is ψ 0 ′ , so the fixed p oint is ψ 0 ′ (0 ′ ) (= 1) • if µ (0) = 0 a nd µ (1) = 1, then g is ψ ∞ ′ , so the fixed po in t is ψ ∞ ′ ( ∞ ′ ) (= 1) Just as in Λ, b oth · · · 000 a nd 111 · · · ha v e an immediate neighbor in Λ ′ . In addition, · · · 111 and 000 · · · are m utual immediate neigh b ors in Λ ′ . Definition 8 L et x b e an infinite binary string of or der typ e ω or ω ∗ . T h en x opp is the infinite binary s tring which i s gen e r ate d fr o m x by first r eplacing every 0 with a 1 and vic e versa, and then r eversing the or der typ e of the r esulting string. F or exampl e , if x = · · · 00011 , and if the 0’s and 1’s in x ar e r eplac e d with 1 ’s and 0 ’ s , r e s p e ctively, the r esulting s tring is · · · 11100 ; r eversing the or der typ e of · · · 11100 gives x opp = 00111 · · · . The relatio nship whic h consists of the follo wing three conditions ho lds b et w een Ω ′ and Ω ′ (opp) in Λ ′ :  ∀ x ∈ Ω ′  h x opp ∈ Ω ′ (opp) i 14 ψ \ x 0 1 · · · n − 1 n · · · ∞ ∞ ′ · · · n ′ ( n − 1) ′ · · · 1 ′ 0 ′ ψ 0 0 0 · · · 0 0 · · · 0 0 · · · 0 0 · · · 0 0 ψ 1 0 0 · · · 0 0 · · · 0 0 · · · 0 0 · · · 0 1 . . . ψ n 0 0 · · · 0 0 · · · 0 0 · · · 0 1 · · · 1 1 . . . ψ ∞ 0 0 · · · 0 0 · · · 0 0 · · · 1 1 · · · 1 1 ψ ∞ ′ 0 0 · · · 0 0 · · · 0 1 · · · 1 1 · · · 1 1 . . . ψ n ′ 0 0 · · · 0 1 · · · 1 1 · · · 1 1 · · · 1 1 . . . ψ 1 ′ 0 1 · · · 1 1 · · · 1 1 · · · 1 1 · · · 1 1 ψ 0 ′ 1 1 · · · 1 1 · · · 1 1 · · · 1 1 · · · 1 1 T able 7 The functions ψ in C(Λ ′ , 2) and their v alues ( ψ ( x ) for x ∈ Λ ′ )  ∀ y ∈ Ω ′ (opp)  [ y opp ∈ Ω ′ ]  ∀ x ∈ Ω ′   ∀ y ∈ Ω ′ (opp)  [ x ⊆ y opp ↔ x opp ⊇ y ] This relationship is an example of what is called adjunction in category the- ory [8]. Adjunction do es not hold b et w een Ω ′ and Ω opp in Λ = Ω ′ ∪ Ω opp . T o see this, let x = · · · 11 1, a nd note tha t x ∈ Ω ′ but x opp = 00 0 · · · 6∈ Ω opp . 3.3 CPO of or der typ e ω + 1 + ω ∗ ; b ounda ry element and adjunction No w w e construct a new CPO ˆ Λ ′ whic h is similar to Λ ′ but where the equiv- alen ts o f the elemen ts lab eled with ∞ and ∞ ′ are iden tical. Prop osition 9 F or a ∈ Ω ′ (opp) , let ˆ Ω ′ b e the line ar or der wi th underlying set { a } × Ω ′ and o r d e ring define d by ( a, x 1 ) ⊆ ( a, x 2 ) ↔ x 1 ⊆ Ω ′ x 2 F or b ∈ Ω ′ , let ˆ Ω ′ (opp) b e the line ar or der with underlying set Ω ′ (opp) × { b } an d or dering define d by ( y 1 , b ) ⊆ ( y 2 , b ) ↔ y 1 ⊆ Ω ′ (opp) y 2 Thus ˆ Ω ′ = { ( a, · · · 000) ⊆ ( a, · · · 00 1) ⊆ ( a, · · · 0011) ⊆ · · · ⊆ ( a, · · · 111) } ˆ Ω ′ (opp) = { ( 0 00 · · · , b ) ⊆ · · · ⊆ (0011 · · · , b ) ⊆ (011 · · · , b ) ⊆ (111 · · · , b ) } 15 Then ˆ Ω ′ ≃ Ω ′ and ˆ Ω ′ (opp) ≃ Ω ′ (opp) . Mor e over, the line ar or der ˆ Ω ′ ∪ ˆ Ω ′ (opp) that extends the or derings in ˆ Ω ′ and ˆ Ω ′ (opp) and s a tisfi e s ( a, · · · 111) ⊆ (000 · · · , b ) is a CPO in which ( a, · · · 111) is the s up of ˆ Ω ′ and (000 · · · , b ) is the inf of ˆ Ω ′ (opp) . Let ˆ Λ ′ b e the CPO ˆ Ω ′ ∪ ˆ Ω ′ (opp) obtained from Prop osition 9 fo r a = 000 · · · and b = · · · 111, and let m = ( a, b ) (= (000 · · · , · · · 1 11)). Then m is a b oundar y elemen t of ˆ Λ ′ in that m ∈ ˆ Ω ′ ∩ ˆ Ω ′ (opp) and m is bot h the sup of ˆ Ω ′ and the inf of ˆ Ω ′ (opp) . Also, m has no immediate neigh b or in ˆ Λ ′ , and ˆ Λ ′ has order t yp e ω + 1 + ω ∗ , so ˆ Λ ′ ≃ Λ = Ω ′ ∪ Ω opp (and ˆ Λ ′ ≃ C( ˆ Λ ′ , 2)) but ˆ Λ ′ 6≃ Λ ′ = Ω ′ ∪ Ω ′ (opp) . Definition 10 L et ( x, y ) b e an or der e d p air of infinite binary strings e ach o f which is of or der typ e ω or ω ∗ (though x is n ot ne c essarily o f the same or der typ e a s y ). Then ( x, y ) opp = ( y opp , x opp ) . F or exam ple, i f x = · · · 0001 1 and y = 1 1111 · · · , then x opp = 00 111 · · · and y opp = · · · 00000 , s o ( x, y ) opp = ( y opp , x opp ) = ( · · · 00 0 00 , 00111 · · · ) Note that m opp = (000 · · · , · · · 111) opp = (0 00 · · · , · · · 111) = m , and that the follo wing conditions are satisfied: ˆ Ω ′ = { ( x, y ) opp : ( x, y ) ∈ ˆ Ω ′ (opp) } ˆ Ω ′ (opp) = { ( x, y ) opp : ( x, y ) ∈ ˆ Ω ′ } Moreo ver, it can easily b e sho wn that adjunction holds b etw een ˆ Ω ′ and ˆ Ω ′ (opp) in ˆ Λ ′ . 3.4 CPO of or der typ e 1 + ω ∗ + ω + 1 No w w e construct a CPO V with order type 1 + ω ∗ + ω + 1, using the follow ing coun terpart of Prop osition 9: Prop osition 11 F or a ∈ Ω ′ , let Ξ b e the CPO with underl ying set { a } × Ω ′ (opp) and or dering define d by ( a, y 1 ) ⊆ ( a, y 2 ) ↔ y 1 ⊆ Ω ′ (opp) y 2 F or b ∈ Ω ′ (opp) , le t Ξ opp b e the CPO with underlying set Ω ′ × { b } and or dering define d by ( x 1 , b ) ⊆ ( x 2 , b ) ↔ x 1 ⊆ Ω ′ x 2 Thus Ξ = { ( a, 000 · · · ) ⊆ · · · ⊆ ( a, 00 11 · · · ) ⊆ ( a, 011 · · · ) ⊆ ( a, 1 11 · · · ) } Ξ opp = { ( · · · 000 , b ) ⊆ ( · · · 001 , b ) ⊆ ( · · · 0011 , b ) ⊆ · · · ⊆ ( · · · 111 , b ) } 16 Then Ξ ≃ Ω ′ (opp) and Ξ opp ≃ Ω ′ . Mor e over, the line ar or der Ξ ∪ Ξ opp that extends the or d e rings in Ξ and Ξ opp and s a tisfi e s ( a, · · · 111) ⊆ ( · · · 000 , b ) is a CPO in which ( a, · · · 111) is the sup of Ξ and ( · · · 000 , b ) is the inf of Ξ opp . Let V b e the CPO Ξ ∪ Ξ opp obtained from Prop osition 11 f o r a = · · · 000 and b = 111 · · · , and let m ′ = ( a, b ) (= ( · · · 000 , 111 · · · ) ) . W e can lab el the elemen ts of V as follo ws: ( · · · 0 00 , 000 · · · ) ↔ −∞ . . . ( · · · 0 00 , 001 · · · ) ↔ − 2 ( · · · 0 00 , 011 · · · ) ↔ − 1 m ′ = ( · · · 000 , 111 · · · ) ↔ 0 ( · · · 0 01 , 111 · · · ) ↔ +1 ( · · · 0 11 , 111 · · · ) ↔ +2 . . . ( · · · 1 11 , 111 · · · ) ↔ + ∞ (10) Then m ′ is a b o undary elemen t of V in that m ′ ∈ Ξ ∩ Ξ opp and m ′ is b o th the sup of Ξ and the inf of Ξ opp . Also, m ′ has immediate neigh b ors in V , and V ha s order t yp e 1 + ω ∗ + ω + 1. Therefore, V is not isomorphic to an y of the linear orders discussed earlier in this pap er. (See Figure 1 for a comparison of ˆ Λ ′ and V .) Note that m ′ (opp) = ( · · · 000 , 111 · · · ) opp = ( · · · 000 , 111 · · · ) = m ′ , hence that adjunction holds b etw een Ξ and Ξ opp in V . Fig. 1. CPO ’s ˆ Λ ′ and V . Th e b o xes indicate th e elemen ts that ha ve no immediate neigh b or. V is not isomorphic to C( V , 2). F or supp ose otherwise, and let f : V → C( V , 2) b e an isomorphism. Then f preserv es the ordering of V , so f maps the greatest elemen t of V (the elemen t lab eled with + ∞ , whic h is ( · · · 11 1 , 111 · · · )) to the greatest ele men t of C( V , 2), whic h is the all-1 function 1 . Now define a function 17 φ : V → 2 by φ ( x ) =      0 , x = ( · · · 000 , 000 · · · ) 1 , x ∈ V − { ( · · · 000 , 000 · · · ) } Note that ( · · · 000 , 0 00 · · · ) is the least ele men t of V (i.e., t he elemen t lab eled with −∞ ), so φ is con tin uous (hence φ ∈ C( V , 2)). Moreo v er, φ is the immedi- ate predecessor of 1 in C( V , 2 ) , so f − 1 ( φ ) mus t b e the immediate predeces sor of f − 1 ( 1 ) in V ; how ev er, f − 1 ( 1 ) = ( · · · 11 1 , 111 · · · ), whic h has no immediate predecess or. 4 The finite binary strings in S n and t heir infinite extensions; the LR-transformation In this section we in tro duce four w ays to specify the finite binary strings t ha t are elemen ts o f the finite CPO’s S n ( n ≥ 2). Then w e explore the extensions of those specifications to certain infinite binary strings that ar e of order t yp e ω o r ω ∗ , and we de fine an opera t io n (the LR-transformation) o n those infinite strings. 4.1 Sp e cific ations of the strings in S n F or ev ery n ≥ 2, there are four wa ys t o sp ecify the ele men ts of the finite linear order ( S n , ⊆ n ) in (2). Let i, j ∈ N suc h that 1 ≤ i ≤ n and 1 ≤ j ≤ n − 1. Sp ecification I s ( n ) i = l 1 · · · l j · · · l n − 1 , where l j =      0 j ≤ n − i 1 j > n − i (11) 18 Then s ( n ) 1 ⊆ n s ( n ) 2 ⊆ n . . . ⊆ n s ( n ) n , where s ( n ) 1 = 0 · · · 0 | {z } n − 1 s ( n ) 2 = 0 · · · 0 | {z } n − 2 1 s ( n ) 3 = 0 · · · 0 | {z } n − 3 11 . . . s ( n ) n − 1 = 0 1 · · · 1 | {z } n − 2 s ( n ) n = 1 · · · 1 | {z } n − 1 (12) Sp ecification I I s ( n ) i = l 1 · · · l j · · · l n − 1 , where l j =      0 j < i 1 j ≥ i (13) Then s ( n ) 1 ⊇ n s ( n ) 2 ⊇ n . . . ⊇ n s ( n ) n , where s ( n ) 1 = 1 · · · 1 | {z } n − 1 s ( n ) 2 = 0 1 · · · 1 | {z } n − 2 s ( n ) 3 = 00 1 · · · 1 | {z } n − 3 . . . s ( n ) n − 1 = 0 · · · 0 | {z } n − 2 1 s ( n ) n = 0 · · · 0 | {z } n − 1 (14) Sp ecification I I I s ( n ) i = r n − 1 · · · r j · · · r 1 , where r j =      1 j < i 0 j ≥ i (15) 19 Then s ( n ) 1 ⊆ n s ( n ) 2 ⊆ n . . . ⊆ n s ( n ) n , where s ( n ) 1 , s ( n ) 2 , . . . , s ( n ) n are as in Sp ecifica- tion I, that is, s ( n ) 1 = 0 · · · 0 | {z } n − 1 s ( n ) 2 = 0 · · · 0 | {z } n − 2 1 s ( n ) 3 = 0 · · · 0 | {z } n − 3 11 . . . s ( n ) n − 1 = 0 1 · · · 1 | {z } n − 2 s ( n ) n = 1 · · · 1 | {z } n − 1 (16) The only difference b etw een (16) and (12 ) is the order o f indexation of the 0’s and 1’s in each elemen t: from right to left in (16), and from left to rig h t in (12). Sp ecification IV s ( n ) i = r n − 1 · · · r j · · · r 1 , where r j =      1 j ≤ n − i 0 j > n − i (17) Then s ( n ) 1 ⊇ n s ( n ) 2 ⊇ n . . . ⊇ n s ( n ) n , where s ( n ) 1 , s ( n ) 2 , . . . , s ( n ) n are as in Sp ecifica- tion I I, tha t is, s ( n ) 1 = 1 · · · 1 | {z } n − 1 s ( n ) 2 = 0 1 · · · 1 | {z } n − 2 s ( n ) 3 = 00 1 · · · 1 | {z } n − 3 . . . s ( n ) n − 1 = 0 · · · 0 | {z } n − 2 1 s ( n ) n = 0 · · · 0 | {z } n − 1 (18) The only difference b etw een (18) and (14 ) is the order o f indexation of the 0’s and 1’s in each elemen t: from right to left in (18), and from left to rig h t in (14). 20 4.2 Sp e cific ations of the c orr esp onding infinite strings F or eve ry n ≥ 2, all four sp ecifications yield the same set of finite bina r y strings, S n . When n → ∞ , the four sp ecifications yield four differen t sets of infinite binary strings, whic h w e can define a s follows: T yp e I Let i ≥ 1, and let t i = ˘ l 1 ˘ l 2 ˘ l 3 · · · , where, for ev ery j ≥ 1, ˘ l j = lim n →∞ l ( n ) j and l ( n ) j is the j th bit (counting from the left) of the finite string s ( n ) i in Sp ecification I. Let j ≥ 1, and recall that l ( n ) j is defined for ev ery n suc h that i ≤ n and j ≤ n − 1, and t ha t l ( n ) j =      0 j ≤ n − i 1 j > n − i (19) The conditions i ≤ n and j ≤ n − 1 are equiv alent to n ≥ i and n ≥ j + 1, resp ectiv ely . Since i, j ≥ 1, w e ha ve j + i ≥ i and j + i ≥ j + 1. Th us f o r ev ery n ≥ j + i, l ( n ) j is defined; moreo v er, the condition j ≤ n − i is satisfied, so l ( n ) j = 0. F ro m this it follo ws that ˘ l j (= lim n →∞ l ( n ) j ) = 0. F or ev ery i ≥ 1 , t i = 000 · · · ; hence ( { t i : i ≥ 1 } , ⊆ ) is the trivial linear order { 000 · · · } . Recall tha t 000 · · · is the inf o f Ω opp in Λ ′ . T yp e I I Let i ≥ 1, a nd let t i = ˘ l 1 ˘ l 2 ˘ l 3 · · · , where, for ev ery j ≥ 1, ˘ l j = lim n →∞ l ( n ) j and l ( n ) j is the j th bit (counting from the left) of the finite string s ( n ) i in Sp ecification I I. Let j ≥ 1. F or ev ery n ≥ j + i, l ( n ) j is defined and l ( n ) j =      0 j < i 1 j ≥ i (20) Th us ˘ l j (= lim n →∞ l ( n ) j ) =      0 j < i 1 j ≥ i F or ev ery i ≥ 1 , t i is the infinite binary string of order t yp e ω that (from left to righ t) consists of ( i − 1) 0’s follo w ed b y infinitely man y 1’s. In the finite case, 21 the elemen ts of S n are ordered as s ( n ) n ⊆ · · · ⊆ s ( n ) 1 . Using the coun t erpar t o f that ordering sc heme in the infinite case, w e obtain t i +1 ⊆ t i for ev ery i ≥ 1, whic h yields the infinite linear order { · · · ⊆ 0011 · · · ⊆ 01 1 1 · · · ⊆ 111 1 · · · } This linear order, whic h has order type ω ∗ , is Ω opp . T yp e I I I Let i ≥ 1, and let t i = · · · ˘ r 3 ˘ r 2 ˘ r 1 , where, f or ev ery j ≥ 1, ˘ r j = lim n →∞ r ( n ) j and r ( n ) j is the j th bit (coun ting fr o m the rig h t) of the finite string s ( n ) i in Sp ecification I I I. Let j ≥ 1. F or ev ery n ≥ j + i, r ( n ) j is defined and r ( n ) j =      1 j < i 0 j ≥ i (21) Th us ˘ r j (= lim n →∞ r ( n ) j ) =      1 j < i 0 j ≥ i F or ev ery i ≥ 1 , t i is the infinite binary string of order t yp e ω ∗ that (from righ t to left) consists of ( i − 1) 1’s follow ed b y infinitely man y 0’s. In the finite case, the eleme n ts of S n are ordered as s ( n ) 1 ⊆ · · · ⊆ s ( n ) n . Using the coun terpart of that ordering sc heme in the infinite case, w e obtain t i ⊆ t i +1 for ev ery i ≥ 1 , whic h yields the infinite linear order { · · · 0000 ⊆ · · · 0001 ⊆ · · · 0011 ⊆ · · · } This linear order, whic h has order type ω , is Ω. T yp e I V Let i ≥ 1, and let t i = · · · ˘ r 3 ˘ r 2 ˘ r 1 , where, f or ev ery j ≥ 1, ˘ r j = lim n →∞ r ( n ) j and r ( n ) j is the j th bit (coun ting fr o m the rig h t) of the finite string s ( n ) i in Sp ecification IV. Let j ≥ 1. F or ev ery n ≥ j + i, r ( n ) j is defined and r ( n ) j =      1 j ≤ n − i 0 j > n − i (22) 22 F or ev ery n ≥ j + i , the condition j ≤ n − i is satisfie d, so r ( n ) j = 1. F rom this it follo ws t ha t ˘ r j (= lim n →∞ r ( n ) j ) = 1. F or ev ery i ≥ 1 , t i = · · · 111; hence ( { t i : i ≥ 1 } , ⊆ ) is the trivial linear order { · · · 1 11 } . Recall tha t · · · 111 is the sup of Ω in b oth Λ a nd Λ ′ . 4.3 LR-tr ansformation There is a natural relationship b etw een ty p es I and IV, in tha t the sole infinite string x of type I (namely , 000 · · · ) is y opp for the sole infinite string y of type IV (namely , · · · 11 1)—and, of course, y = x opp . Similarly , ev ery infinite string x of t yp e I I is y opp for some infinite string y of t yp e I I I, and ev ery infinite string y of t yp e I I I is x opp for some infinite string x of type I I. There is also a natural op eration that transforms an infinite string of t yp e I to an infinite string of t yp e I I I (and vice v ersa), and an infinite string of t yp e I I to an infinite string of type IV (and vice v ersa). F or an infinite string t i of an y of the four t yp es, this op eration consists of replacing j with n − j in the definition of l ( n ) j in (19), l ( n ) j in (20), r ( n ) j in (21), or r ( n ) j in (22), as appropriate, and rev ersing the order ty p e of t i . W e will refer to this op eration as the LR- transformation, b ecause it con v erts an infinite string whose bits are indexed starting from the left (L) to an infinite string whose bits are indexe d starting from the right (R). The LR- and opp-transformations for all fo ur t yp es of infinite strings are depicted in Fig. 2. Fig. 2. LR-transformation and opp-transf orm ation F or an ordered pair ( x, y ) where x and y are infinite binary strings of one of the four t yp es specified in section 4.2 (thoug h x is not ne cessarily o f the same t yp e as y ), define ( x, y ) LR as ( x LR , y LR ). 23 5 Replication T able 8 lists prop erties o f t he CPO’s Λ , Λ ′ , ˆ Λ ′ , and V . CPO Ad j unction FPT Bo undary Order t yp e Λ No Applicable N/A ω + 1 + ω ∗ Λ ′ Y es Applicable N/A ω + 1 + 1 + ω ∗ ˆ Λ ′ ( ≃ Λ) Y es Applicable m ω + 1 + ω ∗ V ( 6≃ Λ ′ ) Y es Not app licable m ′ 1 + ω ∗ + ω + 1 T able 8 CPO’s and their prop erties. FPT sta nds for “Fixed Poin t Theorem.” In contin uum mechanic s and thermo dynamics, the basic notion of a b o dy as a whole (i.e., In tent) and parts a s infinitesimal b o dy elemen ts (i.e., Exten t) has b een discusse d for sev eral decades. There arises the crucial problem of in- tegration, i.e., o f understanding how the bo dy can glue the infinitesimal ther- mo dynamical systems to o btain the global one. Ow en prop oses to appro a c h this problem through the notion of sheaf [12]. The theory of par t s a nd a bo dy natura lly deals with sub-bo dies (whic h mus t form a Bo olean alg ebra) [11]. Law v ere tak es particular note of b oundaries (whic h a re not sub-b o dies). He p oin ts to a cartesian closed part ia lly-ordered set a s a con v enien t algebraic structure whic h includes these features (i.e., sub- b o dies and bo undaries) [7], in whic h → is thought of as ⊇ and hence cartesian pro duct b ecomes ∪ while exponentiation b ecomes a binary op eratio n akin to subtraction, whic h is c haracterized b y A ⊇ C \ B ⇔ A ∪ B ⊇ C. Then w e can define ∼ A = 1 \ A where 1 denotes the whole b o dy; thus ∼ A is the smallest ob ject suc h that ∼ A ∪ A = 1. And w e can define the b oundary of A a s ∂ A = A ∩ ∼ A. Though L awv ere stated that the notion of b oundary is just that o f logical con tr a diction (within the realm of closed sets), how can w e distinguish t he b oundary elemen t m of the CPO ˆ Λ ′ (to whic h the FPT is applicable) from the one m ′ of the CPO V (to whic h the FPT is not a pplicable)? It mig ht b e consisten t for m to b e a logical con tradiction; how ev er it is not for m ′ . ˆ Λ ′ and V are transformed in to eac h other by the LR- t r ansformation. Since the FPT is applicable to ˆ Λ ′ but inapplicable to V , they are differen t en tities. 24 Therefore the LR-transformatio n transforms an entit y in to a different en tit y . F or example, the infinite binary string 0111 · · · is transformed in to a differ- en t infinite binar y string · · · 11 1 b y the L R -transformation. This means that 0111 · · · implies t ha t it is defined with T yp e I I and · · · 111 implies that it is defined with Type IV (see section 4 .2). That is, in this case, w e can state: (i) Ev ery existing infinite binary string remem b ers ho w it w as defined a nd constructed. On the con tr a ry , the LR - transformation can b e interpre ted a s replacing j with n − j in the definition of an“ existing” infinite binary string (and then “turn- ing t he string around”), t ha t is, w e ma y regard t w o differen t infinite binary strings which are transformed into each other by the LR-transformation as t w o differen t sides of the same coin. (Therefore, if there are t w o CPO’s suc h that the FPT is applicable to one but inapplicable to the other, and one of them is con verted to the o ther b y this interpretation of the LR-tra nsformation, then w e ma y in terpret the conv ersion as an in v alidation o f the applicability of the FPT.) That is, we can state that an observ er who stands at the right endpoint observ es 01 11 · · · as · · · 111. Therefore, the string 0111 · · · itself do es not im- ply that it is defined with T yp e I I . The LR-transformatio n is in terpreted as a tr a nsformation of the observ er’s p ositions (i.e., the left endp oint or the righ t endp oin t of an infinite binary string), and therefore the LR- t ransformation acts as if it did nothing to an existing bina r y string. (The coun terpart of the LR-transformation in the finite strings in S n (see section 4) is j ust an iden tit y transformation.) Th us we can in tro duce the concept π ∈ Π, corresp onding to the exis ting infinite binary string. So, in this case, w e state: (ii) No one, b esides the string, know s how it w as defined and constructed. Though ( x, y ) LR for suc h ordered pairs of infinite strings is w ell defined, the classification of suc h ordered pairs o f infinite strings according to the four t yp es is not. Consider ˆ Λ ′ , for example, whic h is of order t yp e ω + 1 + ω ∗ . There is a natural isomorphism ϕ 1 : ˆ Λ ′ → Ω ∪ Ω ′ opp = Ω ∪ { 000 · · · } ∪ Ω opp that pairs elemen t x of Ω with elemen t (000 · · · , x ) of ˆ Λ ′ (for ev ery x ∈ Ω); pairs elemen t y of Ω opp with eleme n t ( y , · · · 111) of ˆ Λ ′ (for ev ery y ∈ Ω opp ); and pairs 0 00 · · · with the b oundary elemen t m = (000 · · · , · · · 111 ) of ˆ Λ ′ . No w Ω is ty p e I I I, 000 · · · is type I, and Ω opp is t yp e I I, so w e could sa y tha t { (000 · · · , x ) : x ∈ Ω } is t yp e I II, m is t yp e I, and { ( y , · · · 111) : y ∈ Ω opp } is t yp e I I. Ho w ev er, there is also a natural isomorphism ϕ 2 : ˆ Λ ′ → Λ = Ω ′ ∪ Ω opp = Ω ∪ { · · · 11 1 } ∪ Ω opp that pair s · · · 111 with the b oundary elemen t m (and is otherwise iden tical 25 to ϕ 1 ). Since · · · 111 is t yp e IV, w e could just as well sa y that m is t yp e IV. No w conside r V , whic h has order t yp e 1 + ω ∗ + ω + 1. Th us V ≃ Ω ′ opp ∪ Ω ′ = { 000 · · · } ∪ Ω opp ∪ Ω ∪ {· · · 111 } , whic h has t he de comp osition I + I I + II I + IV. How ev er, there is no “natural” isomorphism in the se nse of ϕ 1 or ϕ 2 . An y isomorphism that pairs eleme n t x of Ω ′ with eleme n t ( x, 1 11 · · · ) o f V (fo r eve ry x ∈ Ω ′ ) cannot also pair elemen t y of Ω ′ opp with elemen t ( · · · 000 , y ) of V (for ev ery y ∈ Ω ′ opp ): If y = 111 · · · , then ( · · · 000 , y ) = m ′ , the b oundary elemen t of V , whic h w ould hav e b een paired with elemen t x = · · · 000 of Ω ′ b y suc h an isomorphism. Similarly , an y isomorphism that pairs elemen t y of Ω ′ opp with elemen t ( · · · 000 , y ) of V ( f or ev ery y ∈ Ω ′ opp ) cannot also pair elemen t x of Ω ′ with eleme n t ( x, 111 · · · ) of V ( f or ev ery x ∈ Ω ′ ). Th us there is no “natural” w ay t o assign types to the elemen ts of V . Th us ˆ Λ ′ and V a re transformed in to eac h ot her by the LR-transformat io n; ho w ev er the classific ation o f V according to the four t yp es is not w ell defined. ˆ Λ ′ is “ naturally” isomorphic to a linear order that can b e dec omp osed as either I I I + I + I I (corresp onding to ϕ 1 ) or I I I + IV + I I (corresp onding to ϕ 2 ), while V is isomorphic to a linear order that has the decomp o sition I + I I + I I I + IV; ho w ev er there is no “natura l” isomorphism (in the same sen se that ϕ 1 and ϕ 2 are natural). Th us, eve ry infinite binary string of ˆ Λ ′ is in terpreted b y (i) , while ev ery infinite binary string of V is in terpreted by (ii). Here is a crucial a m biguity where w e can debate whether the LR- transformation ˆ Λ ′ to V is a transformatio n b et w een differen t en tities or whether it acts as if it do es nothing to the existing binar y string, and therefore whe ther or not w e can in tro duce the concept π ∈ Π, corresp onding to the existing infinite binary string. Ho w ev er, w e do not stic k to the problem to determin e which is righ t , as IM do es not. R ecall that in G unji’s mo del of IM [2 ], it w as not es sen tia l to in- quire whethe r the defectiv e observ er destroy s the Ex ten t or simply in v alidates the pr esupp osition that leads to the para dox. In the same sense, we stated that bot h (1) and π ∈ Π are not strict mathematical statemen ts in section 1. In order for ˆ Λ ′ and V to b e transformed into eac h other by the interpretation (i) of the LR-tra nsfor ma t io n, the t w o exclus iv e decompositions “I I I + I + I I” and “I I I + IV + I I” hav e to b e com bined in to o ne pseud o-decomp osition “I I I + IV + I + I I”. So, con v ersely , we r a ther consider Λ ′ and V are transformed in to eac h other, where w e insert a comma at the cen ter of a binary string to divide the string in to tw o parts when Λ ′ is transformed in to V , and also w e ignore the comma a nd stand at either the left or rig ht endp oin t of the infinite binary string when V is transformed in to Λ ′ . W e will refer to this op eration (whic h inherits the properties (i) and (ii) of t he LR-transformat io n) as the LCR-transformation. That is, first, w e cop y the b oundary eleme n t m = 26 (000 · · · , · · · 11 1) o f ˆ Λ ′ , and, second, we pro ject it in to 000 · · · and · · · 111, b oth of whic h are elemen ts o f Λ ′ and ma y b e in terpreted as In t( m ) and Ext( m ), resp ectiv ely: m = (000 · · · , · · · 111) − − − → Cop y        m In t( − ) − − − − − → Pro j ection 000 · · · m Ext( − ) − − − − − → Pro j ection · · · 111 (23) Since w e cannot p o ssess t wo iden tical elemen ts in a linear order, the t wo pro cesses that “ c opy and then pr oje ct ” hav e to segue as if they are just o ne pro cess. W e call it a r eplic ation . Th us the f o ur infinite CPO’s (i.e., Λ , ˆ Λ ′ , Λ ′ , and V ) constructed in this pap er are link ed as: Λ ∼ ← − − − − − → Dualization ˆ Λ ′ − − − − − − → Replication Λ ′ ← − → LCR V (24) The first CPO, Λ, (whic h is the one that Scott constructed) is isomorphic to the second one, ˆ Λ ′ . W e call the construction of ˆ Λ ′ from Λ (wh ic h w e discussed in subsection 3.1) a dualization in (24 ). The third one, Λ ′ , is transformed in to the fourth one, V , b y the LCR-tra nsforma t ion. Here is one imp orta n t p oin t. W e could construct a c omplemen tarity b et wee n an isomor phism ψ 1 : V → Ω ′ opp ∪ Ω ′ that pairs elemen t · · · 000 of Ω ′ with m ′ , and an isomorphism ψ 2 : V → Ω ′ opp ∪ Ω ′ that pairs elemen t 111 · · · o f Ω ′ opp with m ′ . Usually , for a b oundary elemen t b ∈ Υ ∩ Υ (opp) , where Υ is a linear o r der, we deem it just to be b ∈ Υ and b ∈ Υ (opp) (25) On the contrary , w e condemned it t o the complimen ta ry situation: either b ∈ Υ or b ∈ Υ (opp) (26) corresp onding to either ψ 1 or ψ 2 (eac h of whic h is a “natural” isomorphism in the same sense that ϕ 1 and ϕ 2 are nat ur a l), and then w e in terpreted (26) as: b 1 ∈ Υ a nd b 2 ∈ Υ (opp) (27) Here is the place where we can find the concept of replication a s the pro- cess from (26) to (2 7) (corresp onding to the pro cess fro m ˆ Λ ′ to Λ ′ ) b y an in v alidat io n of the applicabilit y of the FPT to the CPO, Λ; namely the pro - cess is kic ked b y the b oundary m and the iden tit y and/or indistinguishabilit y b et w een b 1 and b 2 is main tained by the b oundary m ′ . 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