Following the pioneering work of Scott, domain of approximation (eentially countably baed continuou partial order) have been ued to tudy computability

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1 Partializing Stone Space uing SFP domain Fabio lei, Paolo Baldan, Furio Honell Dipartimento di Matematica e Informatica, via delle Scienze 208, 33100, Udine (Italy) falei,baldan,honellgdimi.uniud.it btract In thi paper we invetigate the problem of \partializing" Stone pace by \Sequence of Finite Poet" (SFP) domain. More pecically, we introduce a uitable ubcategory SFP m of SFP which i naturally related to the pecial category of Stone pace 2-Stone by the functor MX, which aociate to each object of SFP m the pace of it maximal element. The category SFP m i cloed under limit a well a many domain contructor, uch a lifting, um, product and Plotkin powerdomain. The functor MX preerve limit and commute with thee contructor. Thu, SFP domain which \partialize" olution of a vat cla of domain equation in 2-Stone, can be obtained by olving the correponding equation in SFP m. Furthermore, we compare two claical partialization of the pace of Milner' Synchronization Tree uing SFP domain (ee [3], [15]). Uing the notion of \rigid" embedding projection pair, we how that the two domain are not iomorphic, thu providing a negative anwer to an open problem raied in [15]. Keyword: Denotational Semantic, Domain Theory, Stone Space. Introduction The problem of nding an appropriate \partialization" of a pace of total element, arie in everal area of Mathematic and Computer Science when dealing with computational approximation. point can be taken a total if it can be eparated from all the other point of the pace by an intrinic property. \partialization" of a pace of total element can be viewed a a homeomorphic embedding of the pace onto the maximal element of a domain. Partial element can then be een a the repreentative of poibly intenional propertie of the original pace. Partially upported by EC HCM project Lambda Calcul Type, CHRX-CT

2 Following the pioneering work of Scott, domain of approximation (eentially countably baed continuou partial order) have been ued to tudy computability on real number and on other metric pace (ee e.g. [19, 14, 10, 12]). In thi paper we invetigate the \partialization" of 2-Stone pace by SFP domain, rt conidered by bramky (ee [1, 2, 3]). Both kind of pace play a fundamental r^ole in the denotational emantic of concurrency. The importance of SFP domain i unquetionable (ee [16]). The relevance of 2-Stone pace, i.e. countably baed, totally diconnected compact Haudor pace, arie from the fact that compact ultrametric pace, a category of pace widely ued in metric emantic (ee [8]), are 2-Stone pace. natural partialization of a 2-Stone pace hx; i by a Scott domain can be immediately obtained a the ideal completion of the collection K ne (X) of non-empty compact open ubet of X, ordered by revere incluion D X 1 = Idl(K ne (X); ). Such domain are extenional in the ene that dierent partial element approximate dierent et of maximal element. However, thi cla of domain i not cloed under ignicant domain contructor, uch a lifting and Plotkin powerdomain, in that uch contructor add point that are meaningle w.r.t. the topology of the induced pace. nother extenional partialization can be obtained by aociating to a 2- Stone pace X, the tree D X 2 of cloed ball of a metrization of X, ordered by revere incluion (a in [6, 7, 12]). In the etting of compact ultrametric pace and non-ditance increaing function, domain contructor can be dened on thee tree inducing the correponding metric contructor on the pace of maximal element. Thi olution, however, i not completely atifactory ince the contructor are quite \ad hoc". In thi paper we explore the approach of [1] and conider, even non extenional, SFP domain. We exploit the fact that both 2-Stone pace and SFP domain hare the nitary property of being limit of equence of nite dicrete tructure, namely nite dicrete pace and nite partial order, repectively. In fact, at the level of nite tructure, we have that: i) partial order are cloed under many domain contructor, i.e. lifting (:)?, eparated um +, product and Plotkin powerdomain P P l ; ii) the ubpace of maximal element of a partial order i a dicrete pace, and every dicrete pace can be viewed a uch a ubpace, for uitable partial order; iii) the natural functor MX commute in an obviou way with the domain contructor in i). Thu, at the level of nite tructure one can dene compoitionally natural partialization of dicrete pace. In thi paper we generalize to the!-limit what happen at nite level. In particular we introduce a uitable ubcategory SFP m of SFP ep cloed under limit a well a the above mentioned domain contructor. The ubpace of maximal element of an object in SFP m i a 2-Stone pace, and every 2-Stone pace can be viewed a uch a ubpace, for a uitable object in SFP m. Since the functor MX from SFP m to 2-Stone i 2

3 !-continuou, we can dene SFP domain which \partialize" olution of a vat cla of domain equation in 2-Stone, by olving the correponding equation in SFP m. partialization which ha been extenively tudied in the literature by bramky [3] and Milove, Mo, Ole [15] i that of Milner' Synchronization Tree, or equivalently the cloure of the pace of hereditarily nite hyperet. Thi pace i homeomorphic to the hyperunivere N! of [13] and it appear quite frequently under dierent mathematical perpective, e.g. a the 2-Cantor pace. In [15] the quetion wa raied a to whether the two partialization given in [3] and [15] are iomorphic. n immediate application of our reult how that the olution of the two domain equation in SFP ep introduced by bramky (ee [2]) and Milove et al. (ee [15]) have iomorphic maximal pace. Uing the notion of rigid embeddingprojection pair we give a negative anwer to the open problem raied in [15]. The technique baed on rigid embedding-projection pair i rather promiing in the analyi of the ne tructure of domain. Uing the above reult, we can how furthermore that there i a plethora of non-iomorphic olution of reexive domain equation having the hyperunivere N! a pace of total element. It i a matter of further invetigation which of thee (if any) i the mot appropriate partialization of the univere of hyperet. Throughout the paper we ue tandard notation and baic fact of Domain Theory and Topology (ee [17, 11]). In Section 1 we give the baic denition and we recall ueful fact about SFP domain and Stone pace. In Section 2 we dicu extenional partialization. In Section 3 we introduce the category SFP m and how that it i cloed under variou domain contructor. In Section 4 we relate 2-Stone pace to SFP m domain uing the functor MX. In Section 5 we dicu partialization of hyperunivere. Finally in Section 6 we how that the reult of ection 3-4 cannot be extended to function pace contructor and that the compactne condition i neceary. Thi paper grew out from ome initial reult preented by the author at the 1994 meeting in Renne of the EEC project MSK (Mathematical Structure for Concurrency). The extended abtract of thi paper i publihed in [5]. 1 Stone Space and SFP Domain We tart by recalling denition and baic fact about Stone pace and SFP' domain (ee [17], [11] for more detail). Both kind of object are nitary in the ene that they can be obtained a limit of equence of nite object in the correponding categorie. Denition Stone pace i a compact topological pace with a countable bai of clopen et. 3

4 Propoition 1.2 Let hx; i be a topological pace. The following are equivalent: 1. hx; i i a 2-Stone pace; 2. hx; i = lim hhx n ; n i; f n i (X n nite, n dicrete topology); 3. hx; i i compact and ultrametrizable with d : X X! f0g [ f2?n g n. Let Top be the category of topological pace and continuou function. We denote with 2-Stone the full ubcategory of Top coniting of 2-Stone pace. Given two cpo' D and E, an embedding-projection pair (ep-pair) from D to E i any pair of continuou function i : D! E, j : E! D uch that ij v Id E and j i = Id D. We denote by CPO ep the category of CPO' and embedding-projection pair. Let hd n ; p n i be a equence in CPO ep and let D be it limit. For all n we denote with i n and j n the component of the ep-pair p n and with n = h n ; n i the canonical ep-pair from D n into the limit. For any n; m uch that n < m, i nm : D n! D m denote i m?1 i n. Similarly j mn : D m! D n denote j n j m?1. Denition 1.3 Sequence of Finite Poet (SFP) domain i a domain which i iomorphic to the direct limit of a directed equence of nite CPO' in CPO ep. We denote by SFP ep the full ubcategory of CPO ep coniting of SFP'. Let X be a ubet of the collection K(D) of compact element of D and let U(X) denote the et of minimal upper bound of X. U(X) i aid to be complete if for each upper bound y of X there exit x 2 U(X) uch that x v y. Finally U (X) denote the mallet et containing X and cloed under U. Propoition 1.4 Let (D; v) be a partial order. Then D i an SFP if and only if D i an!-algebraic CPO and whenever X i a nite et of nite element of D, then U(X) i a complete nite et of upper bound of X and U (X) i nite. If D atie only the rt two of the three condition above it i called a 2=3 SFP, or equivalently a coherent!-algebraic domain. The following lemma illutrate an intereting property of maximal element ubpace of 2=3 SFP domain with repect to Scott and Lawon topologie. Lemma 1.5 Let D be a 2=3 SFP domain. Then the topologie L and S over Max(D), induced repectively by the Lawon and the Scott topologie over D, coincide. Proof. bai for (Max(D); S) i given by fmax(" a) j a 2 K(D)g, while a ubbai for (Max(D); L) i given by Max(" a); Max((" a) c ) j a 2 K(D)g, hence Lawon topology i ner than Scott topology.2 We now prove that Max((" a) c ) i an open et in (Max(D); S). Let x 2 Max((" a) c ). Since D i!-algebraic there exit an increaing equence (a n ) n 4

5 of nite element uch that x = F n a n. We tate that Max(" a n ) Max((" a) c ) for ome n. In fact let u uppoe by contradiction that for all n there i y n 2 Max(" a n ) \ Max(" a). Since D i a 2=3 SFP, the Lawon topology i compact and thu there exit a converging ubequence (y nk ) k whoe limit y i in " a, ince thi et i Lawon cloed. Now, (a nk ) k i an increaing equence, hence a nk v y nh for all h k and thu, by propertie of limit in Lawon metric, a nk v y for all k. Thu F k a n k = x v y. By maximality of x we have that x = y, but thi i a contradiction ince y 2" a. Therefore for each x 2 Max((" a) c ) we nd a nite element b v x uch that the correponding Scott-open et Max(" b) i included in Max((" a) c ) and contain x. Thu Max((" a) c ) i open in (Max(D); S).2 conequence of the next eay propoition i that when dealing with a direct limit of SFP domain, propertie of nite element can be teted at a nite level. Propoition 1.6 Let D = lim hd n ; p n i with D n SFP' and p n ep-pair. Then: 1. u fin (K(D)), 9n: 9u n fin K(D n ) u = n (u n ); 2. 8n: 8u n fin K(D n ): U ( n (u n )) = n (Un(u n )): 2 Extenional Partialization Given a 2-Stone pace hx; i we ay that a SFP domain D induce hx; i if (Max(D); S) ' hx; i, where S denote the topology induced by Scott topology on Max(D). In general, one can nd innitely many SFP domain which induce a given 2-Stone pace hx; i; conider, for intance, all SFP' with a top element. The nite element of any uch domain, however, cannot be interpreted, in general, a the open et (propertie) of the original pace. In order to have \partialization" of 2-Stone pace where nite element repreent propertie of the original pace, it i natural to retrict attention to extenional domain. Denition 2.1 n SFP domain D i extenional if for each nite element d 2 D d = V fz j z 2 Max(D)\ "dg. Notice that even if we retrict attention jut to extenional SFP domain, till we cannot nd a unique domain which induce a given 2-Stone pace on it ubpace of maximal element. Conider, for intance, a at domain and the meet-emilattice generated by it. We dicu briey two poible canonical contruction for embedding homeomorphically a 2-Stone pace X into Max(D) for ome domain D. The rt contruction i uggeted by Stone duality [18] and it i obtained by conidering the collection K ne (X) of non-empty compact open ubet of X, ordered by revere incluion (K ne (X); ) and then it ideal completion, 5

6 D X 1 = Idl(K ne (X); ): or equivalently the collection of non-empty compact ubet (K ne (X); ). Clearly D X 1 i an extenional!-algebraic Scott domain and (Max(D X 1 ); S) = X. Moreover D X 1 i \maximal", in the ene that any other extenional SFP domain that induce X can be embedded by a continuou injective function into D X 1. In fact SFP domain are!-algebraic and each clopen i determined by a nite element. However, extenionality i not preerved by important domain contructor uch a P P l. To ee thi it i enough to apply P P l to the extenional nite SFP domain D = fa; bg?. lternative extenional partialization are uggeted by [19, 6, 7, 12]. They are baed on the fact that each 2-Stone pace X i metrizable with an ultrametric d : X X! f0g [ f2?n g n. Hence one can conider D X 2 = Idl(f B(x; 2?n ) j n 2 INg; ). D X 2 i an!-algebraic CPO where incomparable element have no upper bound, i.e. D X 2 i a (nitely branching) tree. Maximal element of D X 2 can be identied with maximal chain in (f B(x; 2?n ) : n 2 INg; ) and the function f : (Max(D X 2 ); S)! (X; (X)) mapping a chain (B n ) n to the ole point in T n B n i a homeomorphim. Thi partialization contain only element correponding to a ytem of dijoint clopen et. In [6, 7] it i hown that uch tree (of formal ball), and level preerving function, can be turned into a category BTree, which i equivalent to the carteian cloed category KUM of compact ultrametric pace and non expanive function. The equivalence i etablihed by a functor that aociate to each tree the pace of maximal element with the induced topology. In BTree we can dene domain contructor, uch a lifting, product, um, function pace and powerdomain, which induce on the pace of maximal element the correponding metric contructor. Thi partialization i not completely atifactory ince it require to retrict oneelf to particular continuou function (i.e. non expanive function) and to conider contructor on tree which are quite \ad hoc". 3 The Category SFP m In view of the reult of the previou ection, in order to have a well behaved cla of partialization, we are led to drop the extenionality condition and to focu on a wider cla of SFP domain. In thi ection we dene a ubcategory SFP m of SFP ep uch that every object in SFP m induce a 2-Stone pace. Contructor over SFP m are dened in the tandard way. We etablih a connection between thee contructor and the correponding one over 2-Stone, uing the functor Max. Then, a domain equation in 2-Stone can be tranlated into a domain equation in SFP m, in uch a way that the olution of the latter i a partialization of the former. The following propoition i a direct conequence of Lemma 1.5. Propoition 3.1 Let (D; v) be a 2=3 SFP. Then (Max(D); S) i a econd countable, totally diconnected pace. 6

7 Not all SFP domain induce a compact pace on the ubpace of maximal element. Conider, for intance, IN?. natural and ucient, but not neceary, condition on D for compactne to hold i that there exit a direct equence with limit D, where projection preerve maximal element. In order to ingle out a uitable category of uch SFP domain (ee Denition 3.6), we need ome preliminary reult. Denition 3.2 Let D and E be SFP'. n ep-pair p = hi; ji : D! E i called maximal preerving pair, or M-pair, if for all x 2 Max(E), j(x) 2 Max(D) (i.e. j(max(e)) Max(D)). Notice that if p = hi; ji : D! E i an M-pair then j(max(e)) = Max(D). In fact, by urjectivity of j, for all x 2 Max(D) there exit y 2 E uch that j(y) = x. Hence if y 0 2 Max("y) we have j(y 0 ) = x. Moreover, compoition of M-pair i an M-pair. We denote by lim m! hd n; p n i the limit of a directed equence of nite CPO' and M-pair. Lemma 3.3 Let D = lim m!hd n ; p n i n. Then given x = (x n ) n 2 D, x i maximal in D if and only if, for all n, j n (x) = x n i maximal in D n. Proof. ()) Let u uppoe that x n0 i not maximal in D n0 for ome n 0. Since for all n > n 0, x n0 = j nn0 (x n0 ) and, by hypothei, p n0n i an M-pair we can conclude that x n i not maximal in D n for all n n 0. Therefore 8n n 0 :9z n 2 Max(D n ):x n v z n ; x n 6= z n : (1) Let u build a tree T level by level in the following way: - level 0 contain the root without any label; - level 1 contain a node for each z n0 2 Max(D n0 ) uch that x n0 v z n0 ; - level k > 1 contain each node of level k? 1, labelled with z n0+k?2 ha a on for every z n0+k?1 2 Max(D n0+k?1 ) uch that j n0+k?2 (z n0+k?1 ) = z n0+k?2. The tree T i nitely branching, ince each D n i nite, and it i innite by 1. Therefore by Koenig Lemma there exit an innite path in T, ay root; z n0 ; z n0+1 ; z n0+2 ; : : : ; uch that 8n n 0 :z n 2 Max(D n ), z n = j n (z n+1 ) and x n v z n ; x n 6= z n. Completing thi equence with the initial element z i = j n0i(z n0 ) for i < n 0, we obtain a equence z = (z n ) n 2 D uch that x v z; x 6= z. Therefore x i not maximal. (() Let x n 2 Max(D n ) for all n. If y 2 D, x v y, by denition of the order in D we have x n v y n for all n. Uing maximality of the x n, we conclude x n = y n for all n, hence x = y. 2 Continuity in Lawon topology i a tronger notion than continuity in Scott topology, but one can eaily check that projection are alo Lawon continuou. Thi imple remark i ueful in proving the following: 7

8 Lemma 3.4 Let D = lim m! hd n; p n i n. Then Max(D) i Lawon cloed, hence compact. Proof. Let (x k ) k be a equence in Max(D) converging to x 2 D. Since projection are Lawon continuou, for all n we have lim k n (x k ) = n (x). Therefore, by nitene of D n, there exit k 0 uch that n (x k ) = n (x) for all k k 0 and thu n (x) 2 Max(D n ) for all n. Hence by Lemma 3.3 the element x i maximal in D. 2 Theorem 3.5 Let D = lim m!hd n ; p n i n. Then (Max(D); S) i a 2-Stone pace. Proof. Since D i an SFP, by Propoition 3.1, (Max(D); S) ha a countable bai of clopen et. Moreover, by Lemma 3.4, Lawon topology on Max(D) i compact. Recalling that on the maximal point of a 2=3 SFP Scott and Lawon topologie coincide, we conclude that (Max(D); S) i compact and thu it i a 2-Stone pace. 2 Finally we can introduce the category of SFP domain we hall work with: Denition 3.6 The category SFP m ha a object SFP domain that are limit of directed equence of nite CPO' and M-pair. Morphim are M-pair, the identity and compoition are tandard. We give an intrinic characterization of SFP m object. Thi will be eential in proving ome intereting propertie of SFP m uch a the cloure with repect to direct limit. Denition 3.7 We ay that an SFP (D; v) atie the M-condition if 8u fin K(D):9v fin K(D) uch that: i) u v, ii) Max(U (v)) v Max(D), where v i Smyth preorder (i.e. u v v i 8y 2 v:9x 2 u:x v y). In order to how that SFP m object are exactly thoe SFP' which atify the M-condition we proceed a follow. Firt we prove that the limit, taken in SFP ep, of a equence hd n ; p n i in SFP m i a limit in SFP m. Then we how that the M-condition i preerved under limit. Uing thee fact and that every nite CPO atie the M-condition, we can eaily prove the deired reult. Lemma 3.8 Let D = lim! hd n ; p n i (the limit i computed in SFP ep ), with hd n ; p n i directed equence in SFP m. Then x = (x n ) n 2 Max(D) if and only if 8n:x n 2 Max(D n ): Proof. (() i obviou. ()) Let x = (x n ) n 2 Max(D) and for all n let y n 2 Max(D n ) uch that x n v y n. Since p n are M-pair we can build, for all k, a equence z (k) 2 n Max(D n ) whoe component (z (k) ) n 2 Max(D n ) are dened a follow: 8

9 8 < j kn (y k ) if n < k (z (k) ) n = y k if k = n : y n+1 2 j n?1 (y n ) if k < n, inductively By Theorem 3.5 each Max(D n ) i compact and thu, by Tychono Theorem, n Max(D n ), with the product topology, i compact. Therefore z (k) admit a ubequence z (km) converging to z 2 n Max(D n ). By denition of z (k) and taking into account that n (x n ) v n+n 0(x n+n 0), it follow that, if k n, n (x n ) v z (k). In particular, for each n, it mut be n (x n ) v z, hence it follow x v z. Since x i maximal, x coincide with z. Since, for each n, z n 2 Max(D n ), we get the thei. 2 Lemma 3.9 Let D = lim! hd n ; p n i, with hd n ; p n i directed equence in SFP m. If each D n atie the M-condition then alo D atie M-condition. Proof. Let u fin K(D). een in Propoition 1.6.1, we can nd n and u n n K(D n ) uch that u = n (u n ). Since D n atie the M-condition, there exit v n n K(D n ) uch that u n v n and Max(U (v n )) v Max(D n ). Let v = n (v n ). Clearly u v. Moreover if x 2 Max(D), by Lemma 3.8, n (x) 2 Max(D n ), thu there i a n 2 Max(U (v n )) uch that a n v n (x). By Propoition 1.6.2, a = n (a n ) 2 Max(U ( n (v n ))) = Max(U (v)) and a v n ( n (x)) v x. Therefore Max(U (v)) v Max(D). 2 Theorem 3.10 Let (D; v) be an SFP. Then D i an SFP m object i D ati- e the M-condition. Proof. ()) Let D be an SFP m object; hence D i the limit of a directed equence hd n ; p n i of nite CPO' and M-pair. Clearly each D n, being nite, atie the M-condition and thu, uing Lemma 3.9, we can conclude that D atie M-condition. (() Let D be an SFP that atie the M-condition and let a 0 =?; a 1 ; a 2 ; : : : be an enumeration of nite element in D. Dene inductively an increaing equence (D n ) n of nite ubpace of D a follow: D 0 = fa 0 g, D n+1 = U (v n ), where v n fin K(D) i uch that D n [ fa n g v n and Max(U (v n )) v Max(D) (uch a v n exit ince D atie M-condition). For all n, let p n = hi n ; j n i : D n! D n+1 dened: 8d n 2 D n :i n (d n ) = d n ; 8d n+1 2 D n+1 :j n (d n+1 ) = F fx 2 D n j x v d n+1 g: One can eaily check that p n i a well dened ep-pair. In particular, from the fact that? 2 D n, uing tandard propertie of U, it follow that for d n+1 2 D n+1, the et fx 2 D n j x v d n+1 g i non-empty and directed. Given d n+1 2 Max(D n+1 ) there i a unique d n 2 Max(D n ) uch that d n v d n+1. Firt we prove the exitence of d n. Suppoe n 6= 0 (otherwie the thei 9

10 i trivial). Let x 2 Max(D)\ " D d n+1. Since Max(D n ) v Max(D), it follow that there exit d n 2 D n uch that d n v x. Since D n D n+1, U(fd n ; d n+1 g) i a ubet of D n+1. Moreover it i non-empty, otherwie " d n [ " d n+1 hould be empty. Since d n+1 i maximal, it follow that d n+1 2 U(fd n ; d n+1 g), hence d n v d n+1. for uniquene, if d 0 n 2 Max(D n), d 0 n v d n+1, then U(fd n ; d 0 n g) D n i non-empty. Since d; d 0 are maximal, they mut coincide. Therefore j n (d n+1 ) = F fx 2 Dn j x v d n+1 g i uch unique d n and thu p n i an M-pair. Finally we dene for all n an ep-pair h n ; n i : D n! D: 8d n 2 D n : n (d n ) = d n ; n (d) = F fx 2 D n j x v dgd 2 D: One can eaily check that hd; h n ; n i n i i a cone for the equence hd n ; p n i, and it i initial ince F n n n (d) = F n F fx 2 Dn : x v dg = d, hence D ' lim! hd n ; p n i. Since all D n are nite CPO' and all p n are M-pair we conclude that D i an SFP m object. 2 Corollary 3.11 The category SFP m i cloed under direct limit. Proof. Let hd n ; p n i be a equence in SFP m. By Theorem 3.10 each D n atie the M-condition and thu by Lemma 3.9 the direct limit D = lim! hd n ; p n i atie the M-condition. Therefore D i a SFP m object. 2 Notice that given a 2-Stone pace X the domain D X 1 and D X 2 dened in ection 1 are both SFP object which atify the the M-condition. we mentioned earlier, however, the category SFP m doe not contain all SFP' that induce 2-Stone pace, i.e. the M-condition i only ucient, but not neceary for the compactne of the induced pace. Conider for intance the functor + over SFP ep dened a follow: D + E = def (f(d; 0) j d 2 Dg [ f(e; 1) j e 2 Eg [ f?; g; v ), where for each x; y 6=, x v y if and only if x v D+E y and (? D ; 0) v, (? E ; 1) v. Given two trict function f : D! D 0, g : E! E 0, f + g coincide with f + g on all the element dierent from and it map D+ E to D0 + E0. The action of + over M-pair i hi; ji + hh; ki = hi + h; j + ki: It i eay to prove that the initial olution of the domain equation X ' X + X (repreented in Figure 1) i not in SFP m but that the pace of it maximal element i 2-Stone. We how now that everal domain contructor over SFP ep, namely lifting (:)?, eparated um +, product and Plotkin powerdomain P P l, are functorial over SFP m. The coaleced um i functorial only on SFP m 0, the ubcategory of SFP m coniting of non-trivial SFP domain. From now on it will be undertood that the application of the functor i conned to (object in) SFP m 0. 10

11 ???????? H H H H H H H H H H H H Figure 1: n SFP which i not in SFP m, but uch that it induce a 2-Stone pace on it maximal element pace The function pace contructor i very problematic, ee Section 6 for a brief dicuion of thi iue. We hall ue the characterization of Plotkin powerdomain P P l (D) a the et fx D j X non-empty, convex and Lawon cloedg, with the Egli-Milner ordering. Con(X) denote the leat convex et that contain X. Cl denote the cloure operator in Lawon topology. If f : D! E i a continuou function then P P l (f) : P P l (D)! P P l (E) i dened a P P l (f)(x) = Con(Cl(f(X))). In particular if f i a projection then P P l (f)(x) = f(x). In fact a projection i Lawon continuou, hence f(x) i cloed. Moreover f(x) i convex. The next lemma give a characterization of Max(P P l (D)) for an SFP m object D. It tate that only maximal element of D play an eential role in forming maximal element of the Plotkin powerdomain. It will be ued to how that P P l i functorial on SFP m and correpond, in a ene formalized in Section 4 to the contructor P nco of 2-Stone. Lemma 3.12 Let D be an SFP m object. Then Max(P P l (D)) = fx 2 P P l (D) j X Max(D)g. Proof. Let X 2 P P l (D). If X Max(D) then obviouly X i maximal. For the convere let u uppoe that in X there i a non-maximal point x. Since " X i Lawon cloed in D, it follow Max(" X) =" X \ Max(D) i Lawon cloed too (ince Max(D) i Lawon cloed by Lemma 3.4), hence Max(" X) i in P P l (D). Since X v em Y, X 6= Y, we have X 62 Max(P P l (D)). 2 Since each ubet of Max(D) i clearly convex we have Max(P P l (D)) = fx Max(D) j X Lawon cloedg: Lemma 3.13 Let D; E; D i ; E i (i = 1; 2) be SFP m object and let p : D! E, p i : D i! E i be M-pair. Then: 1. p? : D?! E?, 2. p 1 + p 2 : D 1 + D 2! E 1 + E 2, 3. p 1 p 2 : D 1 D 2! E 1 E 2 (if jd 1 j; jd 2 j > 1), 11

12 ?? =? D 0 D 1 E 0 E 1 D 0 D 1 E 0 E 1 Figure 2: Coaleced um i not functorial over SFP m (arrow repreent projection). 4. p 1 p 2 : D 1 D 2! E 1 E 2, 5. P P l (p) : P P l (D)! P P l (E), are M-pair. Proof. The thei follow immediately by noticing that the functor Max commute with the contructor: 2 Max(D? ) = Max(D); Max(D 1 + D 2 ) = Max(D 1 ) + Max(D 2 ); Max(D 1 D 2 ) = Max(D 1 ) + Max(D 2 ) (if jd 1 j; jd 2 j > 1); Max(D 1 D 2 ) = Max(E 1 ) Max(E 2 ); Max(P P l (D)) = fx 2 P P l (D) : X Max(D)g (by Lemma 3.12): Notice that, a hown in gure 2 if D 1 or D 2 i a one-point CPO then p 1 p 2 can fail to be an M-pair. Hence, a remarked, i not functorial on SFP m. Cloure of SFP m with repect to all contructor dened above eaily follow from a general reult. Lemma 3.14 Let F : (SFP ep ) n! SFP ep be a locally continuou functor that preerve M-pair. If D 1 ; : : : ; D n are SFP m object, then F (D 1 ; : : : ; D n ) i an SFP m object. Proof. Let D 1 ; : : : ; D n be SFP m object. By denition each D i i the limit of a directed equence of nite CPO' and M-pair, i.e. D i = lim! hd (i) k ; p(i) k i. Therefore F (D 1 ; : : : ; D n ) = = F (lim! hd (1) k ; p(1) k i; : : : ; lim!hd (n) k ; p(n) k i) = lim!k1 : : : lim!kn F (hd (1) k 1 ; p (1) k 1 i; : : : ; hd (n) k n ; p (n) k n i) = lim! F (hd (1) k ; p(1) k i; : : : ; hd(n) k ; p(n) k i) = lim! hf (D (1) k ; : : : ; D(n) k ); F (p(1) k ; : : : ; p(n) k )i: 12

13 Hence F (D 1 ; : : : ; D n ) i obtained a limit of a directed equence of nite CPO' F (D (1) k ; : : : ; D(n) k ) and M-pair F (p(1) k ; : : : ; p(n) k ). Therefore it i a SFPm object. 2 Corollary 3.15 Let D; D 1 ; D 2 be SFP m object. Then D?, D 1 +D 2, D 1 D 2, D 1 D 2 and P P l (D) are SFP m object. Proof. It i a traightforward conequence of Lemma 3.14 and local continuity of contructor. concern coaleced um, notice that: if jd i j > 1 (i = 1; 2) then there exit equence of nite CPO' and M-pair hd n (i) ; p (i) n i uch that D i = lim! hd n (i) ; p (i) n i and jd n (i) j > 1 for all n. If jd 1 j = 1 then D 1 D 2 = D 2. 2 Next corollary i an immediate conequence of Lemma 3.13 and Corollary Corollary 3.16 (:)?, +, and P P l are functorial over SFP m. i functorial over the category SFP m 0. 4 Relation between SFP m and 2-Stone In thi ection we relate the categorie SFP m and 2-Stone. Firt of all we how that it i poible to dene an!-continuou functor MX : SFP m! 2-Stone. SFP m object with the ubpace topology i Then we prove that the functor MX i compoitional with repect to the contructor conidered in the previou ection, in the ene that MX(F (D)) ' F (MX(D)), where F i the functor over 2-Stone correponding to F. In thi way equation in 2-Stone and their olution can be decribed by mean of equation and olution in SFP m. Denition 4.1 The contravariant functor MX : SFP m! 2-Stone i dened a follow. For each SFP m object D, MX(D) = (Max(D); S). For each M-pair p = hi; ji : D! E, MX(p) = j jmax(e) : MX(E)! MX(D). It i traightforward to check that MX i well-dened and!-continuou: Theorem 4.2 Let D = lim! hd n ; p n i (hd n ; p n i a directed equence in SFP m or SFP m 0 ). Then MX(D) ' lim hmx(d n ); MX(j n )i: Proof. Let u note rt that lim exactly the ame point. In fact hmx(d n ); MX(j n )i and MX(D) contain x = (x n ) n 2 MX(D), (8n:x n 2 MX(D n )) ^ (x n = j n (x n+1 )) by Theorem 3.8, (8n:x n 2 MX(D n )) ^ (x n = MX(j n )(x n+1 )), x 2 lim hmx(d n ); MX(j n )i: 13

14 We denote i : lim hmx(d n ); MX(j n )i! MX(D n ) the projection on the i-th component Since a bai for the topology in MX(D) i given by Max(" a), for a 2 K(D), a ubbai for lim hmx(d n ); MX(j n )i i given by the et (Max("a i )) with a i 2 K(D i ), i 2 IN. Now we have:?1 i?1 i (Max("a i )) = = f?1 i (y i ) j y i 2 Max(" a i )g = f(y n ) n 2 lim hmx(d n ); MX(j n )i j a i v y i g = fy 2 MX(D) j i (a i ) v yg = Max(" i (a i )): Recalling the characterization of nite element of the limit given by Propoition 1.6 we immediately conclude that the two topologie coincide. Therefore MX(D) and lim hmx(d n ); MX(j n )i are the ame pace. 2 The correpondence between contructor in SFP m and in 2-Stone i formalized a follow: Denition 4.3 Let F : (2-Stone) n! 2-Stone and G : (SFP m ) n! SFP m be functor. F and G are called aociated functor if there exit a natural iomorphim : F (MX; : : : ; MX)! MX G. We now how that (:)?, +, and P P l in SFP m are aociated to the correponding contructor Id (identity), [ (dijoint union), (product), and P nco (hyperpace of non-empty compact ubet) in 2-Stone. 1 Moreover, in SFP m 0, the contructor i aociated to [. Lemma 4.4 Let D, D 1 and D 2 be SFP m object. Then 1. MX(D? ) ' MX(D); 2. MX(D 1 + D 2 ) ' MX(D 1 )[MX(D 2 ); 3. MX(D 1 D 2 ) ' MX(D 1 ) MX(D 2 ); 4. MX(P P l (D)) ' P nco (MX(D)); 5. MX(D 1 D 2 ) ' MX(D 1 )[MX(D 2 ). Proof. 1. Clearly Max(D? ) and Max(D) contain the ame element and the topologie (induced by Scott topology) coincide. 1 The pace Pnco(X) i dened a the et fk X j K non-empty and compactg endowed with the Vietori topology, i.e. the topology having a ubbai: V = fk 2 Pnco(X) j K g and Z = fk 2 Pnco(X) j K \ 6= ;g for 2 (X). 14

15 2. We have Max(D 1 + D 2 ) = Max(D 1 )[Max(D 2 ), and their topologie coincide. In fact K(D 1 + D 2 ) = (K(D 1 ) + K(D 2 )) [ f?g,hence a bai for MX(D 1 + D 2 ) i fmax(" (i; a)) j (i; a) 2 K(D 1 + D 2 ); i = 1; 2g [ fmax("?)g = = ffig Max(" a) j a 2 K(D i ); i = 1; 2g [ fmax(d 1 + D 2 )g = ffig Max(" a) j a 2 K(D i ); i = 1; 2g [ fmax(d 1 )[Max(D 2 ))g; which i alo a bai for MX(D 1 )[MX(D 2 ). 3. gain, we have Max(D 1 D 2 ) = Max(D 1 ) Max(D 2 ), and the topologie coincide. In fact K(D 1 D 2 ) = K(D 1 ) K(D 2 ) and thu MX(D 1 D 2 ) ha a bai the et Max(" (a 1 ; a 2 )), a i 2 K(D i ); i = 1; 2; while a ubbai for MX(D 1 ) MX(D 2 ) i repreented by the et?1 i (Max(" a i )) a i 2 K(D i ); i = 1; 2: Each element i?1 (Max(" a i )) in the ubbai of MX(D 1 ) MX(D 2 ) i open in MX(D 1 D 2 ), ince it can be written a Max(" a i ) Max("?). Moreover for any element Max(" (a 1 ; a 2 )) in the bai of MX(D 1 D 2 ), we have Max(" (a 1 ; a 2 )) = Max(" a 1 ) Max(" a 2 ) =?1 (Max(" 1 a 1)) \?2 (Max(" 1 a 2)); and thu Max(" (a 1 ; a 2 )) i open in MX(D 1 ) MX(D 2 ). 4. We proceed a above. Firt notice that Max(P P l (D)) = P nco (Max(D)). In fact, by Lemma 3.12, we know that maximal element of P P l (D) are nonempty Lawon-cloed ubet of Max(D). Thee ubet are compact in Max(D) ince Lawon and Scott topologie coincide by Theorem 3.5. Let u conider the topologie. The pace MX(P P l (D)) i equipped with the topology induced by Scott topology and thu it ha a bai the et Max(" X), with X 2 K(P P l (D)), i.e. X ha the form Con(fa 1 ; : : : ; a n g), where fa 1 ; : : : ; a n g i a nite et of nite element in D, and it i eay to how that for any uch X we have: X v em Y, fa 1 ; : : : ; a n g v em Y: Thu a bai for MX(P P l (D)) i given by fmax(" u)g ufin K(D): On the other hand, a bai for MX(D) i fmax(" a) j a 2 K(D)g and thu a ubbai for the Vietori topology of P nco (MX(D)) i repreented by the et V Max("a1)[:::[Max("an) ; Z Max("a)) for a; a 1 ; : : : ; a n 2 K(D): 15

16 Let Max(" u), where u = fa 1 ; : : : ; a n g fin K(D) be an element of the bai of the rt topology. The following hold: Y 2 Max(" u), u v em Y, (8i:9y 2 Y:a i v y) ^ (8y 2 Y:9i:a i v y), (8i:Y 2 Z Max("ai) ) ^ (Y 2 V Max("a1)[:::[Max(" an) ), Y 2 V Max("a1)[:::[Max("an) \ T n i=1 Z Max("a i) ; hence Max(" u) i an open et of the econd topology. to the convere, let u conider a generic element of the ubbai of the econd topology. There are two poibility. The rt one i The econd one i V Max("a1)[:::[Max("an) = = fy j fa 1 ; : : : ; a n g v Y g = fy j 9S fa 1 ; : : : ; a n g:s v em Y g = S fmax(" S) : S fa 1 ; : : : ; a n gg. Z Max("a) = = fy j Y \ Max(" a) 6= ;g = fy j f?; ag v em Y g = Max(" f?; ag) : In both cae we conclude that the et are open in the rt topology. Therefore the two topologie coincide. The proof of 5 i the ame a for 2 and i omitted.2 Theorem 4.5 The following functor on SFP m and 2-Stone are aociated: (:)? with Id, with, + with [ and P P l with P nco. Moreover over SFP m 0 i aociated to [ over 2-Stone. Finally compoition of aociated functor i the functor aociated to the compoition. Proof. natural tranformation : F (MX; : : : ; MX)! MX G we can chooe the identical tranformation (D1;:::;D n) = Id (D1;:::;D n), which, by the previou lemma, i an iomorphim. We have to prove that MX(G(p 1 ; : : : ; p n )) = F (MX(p 1 ); : : : ; MX(p n )) for any choice of morphim p 1 ; : : : ; p n in SFP m. Let u chooe p = hi; ji : D! E, p i = hi i ; j i i : D i! E i (i = 1; 2) morphim in SFP m. Then: 1. 8x 2 Max(E? ) = Max(E) we have MX(p? )(x) = MX(p)(x) ince x 6=?. 16

17 2. 8(i; x i ) 2 Max(E 1 + E 2 ) = Max(E 1 )[Max(E 2 ), we have: (MX(p 1 + p 2 ))(i; x i ) = = (i; (j 1 + j 2 )(x i )) = (i; j i (x i )) = (i; MX(p i )(x i )) = (MX(p 1 )[MX(p 1 ))(i; x i ): 3. 8(x 1 ; x 2 ) 2 Max(E 1 E 2 ) = Max(E 1 ) Max(E 2 ), we have: (MX(p 1 p 2 ))(x 1 ; x 2 ) = = (j 1 j 2 )(x 1 ; x 2 )) = (j 1 (x 1 ); j 2 (x 2 )) = (MX(p 1 )(x 1 ); MX(p 2 )(x 2 )) = (MX(p 1 ) MX(p 1 ))(x 1 ; x 2 ): 4. 8X 2 Max(P P l (D)) = P nco (Max(D)), we have: MX(P P l (p))(x) = = P P l (p)(x) = j(x) = MX(j)(X) = P nco (MX(j))(X): The lat tatement of the theorem i trivial. 2 5 Domain Equation for Hyperet In thi ection we apply the theory developed in the previou ection to the tudy of the initial olution of two important domain equation in SFP m, namely: X ' (2 P P l (X? )) X ' 1 + P P l (X) (Eq1) (Eq2) The initial olution D of (Eq1) wa introduced by bramky in [3] in order to partialize Milner' Synchronization Tree. The initial olution E of (Eq2) wa introduced by Milove Mo and Ole in [15] in order to partialize the cloure of the pace of hereditarily nite hyperet. Thi pace of hyperet i homeomorphic in 2-Stone to Milner' Synchronization Tree, a can be een, for intance, by an immediate application of Theorem 4.5 and Theorem 4.2. In [15] the quetion wa raied a to whether the two initial olution in SFP are iomorphic. 17

18 We give a negative anwer to thi open problem by howing that D and E are non-iomorphic. Our proof i baed on the notion of rigid ep-pair. Uing the above reult, we can how furthermore that there i a plethora of non-iomorphic olution of reexive domain equation having the hyperunivere N! a pace of total element. In general, for any SFP domain D 0 uch that U = MX(D 0 ) i a nite dicrete pace, the initial olution of the equation X ' (D 0 + P P l (X)) and (if D 0 ha at leat two point) X ' (D 0 P P l (X? )) induce the hyperunivere N! (U) ([13]). The proof of the fact that D and E are not iomorphic i done through an analyi of the ne tructure of Plotkin powerdomain contructor. Thi allow to how that D contain ome point in a particular relation with the maximal element of D which do not exit in E. We work in SFP ep. Firt we introduce the notion of rigid ep-pair and lit ome of it mot important propertie: Denition 5.1 Let D and E be SFP'. n ep-pair p = hi; ji : D! E i called rigid if 8x 2 D and y 2 E with y v i(x), there exit x 0 2 D uch that x 0 v x and i(x 0 ) = y. Propoition 5.2 Let D and E be SFP' and let p = hi; ji : D! E be an ep-pair. Then the following tatement are equivalent: 1. p i rigid; 2. for all x 2 D and y 2 E, if y v i(x) then i j(y) = y; 3. for all x; x 0 2 D, y 2 E with i(x) v y v i(x 0 ), there exit x 00 2 D uch that x v x 00 v x 0 and y = i(x 00 ). Proof. (1 ) 2) Let x 2 D; y 2 E with y v i(x). Since p i rigid there exit x 0 2 D; x 0 v x uch that y = i(x 0 ). Hence, uing the fact that j i = id D, we have i(j(y)) = i(j(i(x 0 )) = i(x 0 ) = y. (2 ) 3) Let x; x 0 2 D; y 2 E with i(x) v y v i(x 0 ). If we chooe x 00 = j(y), by hypothei y = i(x 00 ) and by monotonicity of j we can derive x v x 00 v x 0. (3 ) 1) Let x 0 2 D; y 2 E with y v i(x 0 ). If we chooe x =? we are in the hypothei of (3) and we immediately conclude. 2 Lemma 5.3 Compoition of rigid ep-pair i a rigid ep-pair. Lemma 5.4 Let D, D 0, E, E 0 be SFP' and let p = hi; ji : D! E, p 0 = hi 0 ; j 0 i : D 0! E 0 be rigid ep-pair. Then 1. p? : D?! E?, 2. p p 0 : D D 0! E E 0, 3. p + p 0 : D + D 0! E + E 0, 4. p p 0 : D D 0! E E 0, 5. P P l (p) : P P l (D)! P P l (E) are rigid ep-pair. 18

19 Proof. Straightforward, uing denition and the characterization of rigid eppair given by Propoition 5.2. The only non-trivial proof i for cae 5. The baic remark i that, ince p i rigid, the embedding i i Lawon continuou and thu P P l (i) = Con(Cl(i(X))) = i(x): Now, let X 2 P P l (D), Y 2 P P l (E) uch that Y v em P P l (i)(x) = i(x), that i 8y 2 Y:9x 2 X:y v i(x) and 8x 2 X:9y 2 Y:y v i(x): Since i i rigid, by the rt relation we have that i(j(y)) = y for all y 2 Y. Thu P P l (i)(p P l (j)(y )) = i(j(y )) = Y. Therefore P P l (p) = hp P l (i); P P l (j)i i rigid. 2 Lemma 5.5 Let D = lim m!hd n ; p n i. ep-pair h n ; n i : D n! D are rigid. If every p n i rigid then the canonical Proof. Let u uppoe that every p n i rigid and let d n 2 D n, x 2 D uch that x v n (d n ). Then n ( n (x)) = n (x n ) = x: In fact for all m, if m < n then ( n (x n )) m = j nm (x n ) = x m. If m > n then ( n (x n )) m = i nm (x n ) = i nm (j mn (x m )) = x m, ince by hypothei x m v ( n (d n )) m = i nm (d n ) and p nm, compoition of rigid ep-pair, i rigid. Therefore h n ; n i i a rigid ep-pair. 2 Finally we are able to tate the property atied by D but not by E. The two reult below are proved uing eentially the fact that both D and E are limit of equence with rigid ep-pair. Hence the property i hown to hold (fail) in the limit by teting it at each nite level. Lemma 5.6 There exit a; b 2 K(D), with a v b; a 6= b uch that 1. 8x 2 D: a v x v b ) x = a _ x = b, 2. 8x 2 D:? v x v a ) x =? _ x = a, 3. Max(" a) = Max(" b). Proof. Let F 1 denote the functor aociated with the rt equation X ' 2 P P l (X? ). If D 0 i the ingle-point CPO and p 0 = hi 0 ; j 0 i i the unique eppair from D 0 to F 1 (D 0 ) then D i the limit of the equence hf n 1 (D 0 ); F n 1 (p 0 )i n. Let n = h n, n i be the canonical ep-pair from F n 1 (D 0 ) to D. Figure 3 and Figure 4 introduce ome notation for element of D n = F n 1 (D 0 ) (n = 0; 1) and give a picture of the tructure of D. Clearly the ep-pair p 0 i rigid and it i an M-pair. Hence uing the cloure reult given by Lemmata 3.13 and 5.4 we can tate that all ep-pair F n 1 (p 0 ) in the directed equence are rigid M-pair. Therefore uing Lemmata 3.3 and 5.5 we conclude that each n i a rigid M-pair. 19

20 f? 0 g p 0 f? 0 ;? 1 g????? 1 f? 0 g Figure 3: The rt two approximation of D: D 0 and D P P l (D).. b 1 = f? 0 g.. p 0 a 1 = f? 1 ;? 0 g....????.? Figure 4: Structure of D ' 2 P P l (D). 20

21 Element in D we are looking for are: a = 1 (a 1 ); b = 1 (b 1 ): In fact a and b are compact element ince a 1 ; b 1 2 K(D 1 ) and clearly a v b; a 6= b ince a 1 v b 1 ; a 1 6= b 1 and 1 i monotone and injective. Moreover condition 1, 2 and 3 are atied: 1. If x 2 D, with a v x v b then 1 (a 1 ) v x v 1 (b 1 ). Since 1 i a rigid ep-pair, by Propoition 5.2 there exit x 1 2 D 1, with a 1 v x 1 v b 1 uch that x = 1 (x 1 ). It follow immediately that x 1 = a 1 or x 1 = b 1 and thu x = a or x = b. 2. If x 2 D,? v x v a then with the ame technique ued for 1 it i eay to how that x =? or x = a. 3. Let x = (x n ) n 2 Max(D) be a maximal element in D. Clearly if b v x then a v x. Let u uppoe a v x, that i a n v x n for all n. In particular a 1 v x 1 and by Lemma 3.3, x 1 i a maximal element in D 1. Looking at the tructure of D 1 we conclude that b 1 = x 1 and thu 2 b = 1 (b 1 ) = 1 (x 1 ) = 1 ( 1 (x)) v x: Lemma 5.7 There are no element a; b 2 K(E), with a v b; a 6= b uch that 1. 8x 2 E: a v x v b ) x = a _ x = b; 2. 8x 2 E:? v x v a ) x =? _ x = a; 3. Max(" a) = Max(" b). Proof. Let F 2 denote the functor aociated with equation X ' 1 + P P l (X). If E 0 i the ingle-point CPO and p 0 = hi 0 ; j 0 i i the unique ep-pair from E 0 to F 2 (E 0 ) then E i the limit of the directed equence hf n 2 (E 0 ); F n 2 (p 0 )i n. Let n = h n, n i be the canonical ep-pair from F n 2 (E 0 ) to E. We tart with a remark on the tructure of E. Since E ' 1 + P P l (E), the domain E i of the hape outlined in Figure 6. Figure 5 introduce ome notation for element of E n = F n 2 (E 0 ) (n = 0; 1; 2). Recalling the denition of P P l we conclude that e = f? 1 g ha exactly two immediate ucceor, correponding to f? 1 ; p 0 g and f? 1 ; f? 0 gg. Now let u uppoe that in E there are two compact element a and b which atify propertie 1-3, hence a = e and b = e 0 or b = e 00. We how that in thi hypothei compact element that atie the propertie hould be found alo in the nite CPO E 2 and thi i clearly a contradiction. in Lemma 5.6 we can eaily ee that each n i a rigid M-pair. By rigidity of 2 it follow that a and b are in the range of 2, i.e. a = 2 (a 2 ); b = 2 (b 2 ) a 2 ; b 2 2 E 2 : 21

22 fp 0 ; f? 0 gg p 0 f? 0 g??? f? 1 ; f? 0 g; p 0 g fp 0 g ff?0 gg f? 1 ; p 0 g?????? f?1 ; f? 0 gg p 0??? f? 1g???? 0? 1? 2 Figure 5: The rt three approximation of E: E 0, E 1 and E p 0??? e ?. P P l (E)??? ? Figure 6: Structure of E ' 1 + P P l (E). 1 + P P l (E) 22

23 In fact recalling that 1 i a monotone injection and reaoning on the tructure of E 2 and E, we can conclude that there i x 2 E 2 uch that a v 2 (x). Thu by rigidity of 2 we have a = 2 ( 2 (a)). imilar argument allow to conclude that b = 2 ( 2 (b)) Therefore a 2 = 2 ( 2 (a 2 )) v 2 ( 2 (b 2 )) = b 2 and a 2 6= b 2. Moreover a 2 and b 2 atie condition 1-3: 1. Let x 2 2 E 2 with a 2 v x 2 v b 2. pplying 2 we obtain a v 2 (x 2 ) v b and thu 2 (x 2 ) = a or 2 (x 2 ) = b. Therefore applying 2 we have x 2 = a 2 or x 2 = b If x 2 2 E 2 with? v x 2 v a 2 then with the ame technique ued for 1 it i eay to how that x 2 =? or x 2 = a Let x 2 2 Max(E 2 ) be a maximal element in E 2. Clearly if b 2 v x 2 then a 2 v x 2. Let u uppoe a 2 v x 2. We can dene y 2 Max(E) a follow: y n = j n2 (x 2 ) for n < 2, y 2 = x 2 and inductively y n+1 2 j n?1 (y n ) for n 2. Thi i indeed a maximal element in E by Propoition 3.3 and a = 2 (a 2 ) v 2 (x 2 ) v x: Therefore b v x an thu b 2 = 2 (a) v 2 (x) = x 2. 2 The following theorem i immediate conequence of Lemma 5.6 and Lemma 5.7. Theorem 5.8 The initial olution of (Eq1) and (Eq2) are not iomorphic. 6 Final remark 1. Given an SFP domain D, the pace MX(D) i a pace with a countable bai of clopen et. One can ak whether Theorem 4.5 can be extended to SFP ep and QStone, the category of totally diconnected eparable Haudor pace and continuou function. The anwer i negative, ince there i no aociated functor to Plotkin powerdomain contructor when we drop the compactne condition. Let D 1 = IN?, D 2 = IN? + IN?. Both Max(D 1 ) and Max(D 2 ) coincide with IN endowed with the dicrete topology. But Max(P P l (D 1 )) i not homeomorphic to Max(P P l (D 2 )) ince the former ha only one limit point, while the latter ha more than one. In fact, in Max(P P l (D 1 )) there i a unique innite et, namely D 1 itelf, while Max(P P l (D 2 )) contain more than one innite element. 2. It would be intereting to extend the reult of Section 4 o a to comprie alo the function pace contructor. Unfortunately 2-Stone i not carteian cloed, in that the pace of continuou function between two 2-Stone pace endowed with the compact open topology i not compact, in general. One could then try to look at leat for the exitence of ome functor over 2-Stone aociated to the function pace contructor over SFP. But even thi i hopele. 23

24 Firt of all maximal function between SFP object do not map maximal element into maximal element, and thu they do not induce in a natural way function between the pace of maximal point. Conider, for intance, D = N lazy, Bool = ftt; g? and take the continuou function parity : D! Bool (dened in the obviou way). It i a maximal element in [D! Bool], but it doe not map the maximal point! 2 D in a maximal element of Bool. But furthermore, function pace of SFP object, with the ame pace of maximal element, can be non-homeomorphic. Conider, for intance, E = fa; b;?g [ fc i j i 2 Ng; ordered a follow: for all i, c i v a; b, and for all x,? v x. Then Max(Bool) and Max(E) are the ame dicrete pace, but the maximal element of the function pace Max([Bool! Bool]) and Max([Bool! E]) are dierent. In fact Max([Bool! Bool]) i a nite dicrete pace containing only four function, while Max([Bool! E]) contain innitely many function. Namely, the function f i (tt) = a, f i () = b, f i (?) = c i, for i 2 IN, and the contant function. ll thee function are iolated point in a topological ene (ince they are nite element in the SFP) and thu Max([Bool! E]) i a innite dicrete pace and hence it i not compact. Thi latter example how alo that SFP m i not cloed w.r.t the function pace contructor. cknowledgment The author are grateful to S. bramky, P. Di Gianantonio, M. Lenia and to all MSK member for ueful comment. Reference [1] Samon bramky. Total v. partial object and xed point of functor. Unpublihed Manucript, [2] Samon bramky. Cook' tour of the nitary non-well-founded et. Talk delivered at BTCS Colloquium, [3] Samon bramky. domain equation for biimulation. Information and Computation, 92(2):161{218, [4] Samon bramky. Domain theory in logical form. nnal of Pure and pplied Logic, 51:1{77, [5] F. lei, P. Baldan and F. Honell. Partializing Stone Space uing SFP domain (Extended btract). LNCS, CP '97. To appear (10 page). [6] F. lei and M. Lenia. Stone duality for tree of ball. Talk delivered at MSK workhop, Koblenz,

25 [7] P. Baldan. xed point theorem for the olution of domain equation in a category of tree. Tei di Laurea, Udine [8] J.W. de Bakker and E. de Vink. Control Flow Semantic. MIT Pre, [9] J.W. de Bakker and J.I. Zucker. Procee and the denotational emantic of concurrency. Information and Control, 54(1/2):70{120, [10] P. Di Gianantonio. Real number computability and domain theory. Information and Computation, 127(1):11{25, [11] J. Dugundji. Topology. llyn and Bacon, [12]. Edalat and R. Heckmann. computational model for metric pace. 1996, to appear. [13] M. Forti, F. Honell, and M. Lenia. Procee and hyperunivere. MFCS '93, LNCS 841:352{367, [14] M.E. Majter-Cederbaum and F. Zetzche. Toward a foundation for emantic in complete metric pace. Information and Computation, 90:217{ 243, [15] M.W. Milove, L.S. Mo, and F.J. Ole. Non-well-founded et modeled a ideal xed point. Information and Computation, 93(1):16{54, [16] Gordon D. Plotkin. powerdomain contruction. SIM Journal on Computing, 5(3):452{487, [17] Gordon D. Plotkin. Domain. Unpublihed Coure Note. Univerity of Edinburgh, [18] Marhall H. Stone. The theory of repreentation for Boolean algebra. Tranaction of the merican Mathematical Society, 40:37{111, [19] K Weihrauch and U. Shreiber. Embedding metric pace into cpo'. Theoretical Computer Science, 16(1):5{24,