Elements of Generalized Ultrametric Domain Theory

Generalized ultrametric spaces are a common generalization of preorders and ordinary ultrametric spaces, as was observed by Lawvere (1973). Guided by his enriched-categorical view on (ultra)metric spaces, we generalize the standard notions of Cauchy sequence and limit in an (ultra)metric space, and of adjoint pair between preorders. This leads to a solution method for recursive domain equations that combines and extends the standard order-theoretic (Smyth and Plotkin, 1982) and metric (America and Rutten, 1989) approaches. 1. I n t r o d u c t i o n A generalized ultrametric space is a set X supplied with a distance function X ( , ) : X x X --* [0,1], satisfying for all x , y , z : X ( x , x ) = 0 and X ( x , z ) < ~ m a x { X ( x , y ) , X ( y , z ) } . This notion generalizes ordinary ultrametric spaces in that the distance need not be symmetric, and different elements may have distance 0. Generalized ultrametric spaces provide a common generalization of both preordered spaces and ordinary ultrametric spaces, as has been observed by Lawvere [14]. Therefore they are of some importance to the domain-theoretic approach to programming language semantics, in which preorders and ordinary ultrametric spaces are the most popular structures. A more direct connection with the world o f semantics is provided by the observation that transition systems can be naturally endowed with a generalized ultrametric that captures their operational behaviour in terms of simulations (see Example 2.1 below). The present paper introduces first some basic concepts such as Cauchy sequence and limit, next introduces so-called metric adjoint pairs, and then describes how these can be used to solve recursive domain equations. The latter can be seen as its main contribution. The paper concludes with some miscellaneous observations on the following topics: algebraicity; an ultrametric generalization o f the category o f SFP objects called SFU; the category o f generalized ultrametric spaces seen as a large ultrametric space; * E-mail: [email protected]. 0304-3975/96/$15.00 (~) 1996--Elsevier Science B.V. All rights reserved Pll S0304-3975(95)00013-8 350 J.J.M.M. Ruttenl Theoretical Computer Science 170 (1996) 349-381 and an equivalent, purely enriched-categorical definition of metric limit, using so-called weighted colimits [6]. The present paper does not deal with admittedly fundamental aspects of domain theory such as completion and topology: for these, the reader is referred to [4]. Our main source of inspiration has been the aforementioned paper by Lawvere, in which he applies insights from enriched-category theory [11] to (ultra)metric spaces. One way of summarizing the relevance of this view is the fact that many properties of generalized ultrametric spaces are determined by the (categorical) structure [0, 1]. Notably the definition of limit of a Cauchy sequence in an arbitrary generalized ultrametric space will be phrased in terms of limits in [0, 1], which are introduced first. The above theory for generalized ultrametric spaces is developed, extending [14], along the lines of a combination of [21] and [3], which deal with the solutions of domain equations in categories of ordered and metric spaces, respectively. This it has in common with the work of Flagg and Kopperman [9] on continuity spaces, and of Wagner [22] on abstract preorders, who aim at a reconciliation of ordered and metric domain theory as well. Furthermore it has similarly been inspired by some of Smyth' results on quasimetric spaces [18]. Unlike [9,22], we do not aim at generality. The category of generalized ultrametric spaces seems rather to be the smallest category (of sets with structure) that contains both the categories of preorders and ordinary ultrametrics. What seems to be new, amongst others, is: two fixed point theorems on generalized ultrametric spaces, generalizing the least and unique fixed point theorems of Tarski and Banach, respectively; the definition and characterizations of metric adjoint pairs; two categorical counterparts of the aforementioned fixed point theorems, based on the use of metric adjoint pairs, and generalizing the ones of [21] and [3]; the definition and characterization of the subcategory SFU of bifinite spaces; and the purely enrichedcategorical definition of metric limit in terms of weighted colimits. 2. Generalized ultrametric spaces Generalized ultrametric spaces are introduced and shown to be [0, 1J-categories in the sense of Lawvere. In order to see this, a brief recapitulation of Lawvere's enrichedcategorical view of metric spaces is presented. For us, one of the main benefits of Lawvere's approach is the insight that many properties of generalized ultrarnetric spaces are determined by the unit interval of real numbers [0, I]. The section concludes with a brief discussion of the category of all generalized ultrametric spaces, and a few basic definitions. (The subsection on [0, 1]-categories can be skipped at first reading, except for the very basic Proposition 2.2, which will be used time and again.) A generalized ultrametric space (gum for short) is a set X together with a function x ( , ) : x × x --, [0, l] which satisfies, for all x, y, and z in X, 1. X ( x , x ) = 0, and J. J. M.M. Rutten I Theoretical Computer Science 1 70 (1996) 349-381 351 2. X(x ,z ) <~max{X(x, y) ,X(y , z ) } , where 2 is the so-called strong triangle inequality ("strong" because we have max instead of +) . The real number X(x, y) will be called the distance from x to y. A generalized ultrametric space generally does not satisfy 3. if X(x, y) = 0 and X(y , x ) = 0 then x = y, 4. X(x, y) = X(y ,x ) , which are the additional conditions that hold for an ordinary ultrametric space. Therefore it is sometimes called a pseudo-quasi ultrametric space. A quasi ultrametric space is a gum which satisfies axioms 1, 2, and 3. A gum satisfying 1, 2, and 4 is called a pseudo ultrametric space. Examples 2.1. 1. Pseudo, quasi, and ordinary ultrametric spaces are generalized ultrametric spaces. 2. Any preorder (P, <~) (where ~< is a reflexive and transitive binary relation on P ) can be viewed as a generalized ultrametric space, by defining a distance, for p and q in P, {01 i f p ~ < q ' P(P'q) ---if p ~ q. By a slight abuse of language, any gum stemming from a preorder in this way will itself be called a preorder. 3. The set A °~ of finite and infinite words over some given set A with distance function, for v and w in A °°, 0 if v is a prefix of w, A ~( v 'w ) = 2 -n otherwise, where n is the length of the longest common prefix of v and w. 4. The set [0, 1] with distance, for r and s in [0, 1], [0,1](r ,s) = { 0 i f r> / s , s i f r < s. Note that [0, 1] is a quasi ultrametric space. 5. The set 03 = {0, 1 . . . . } U {o}, with distance, for x and y in 03, S 0 if x ~< y, 03(x, Y) / 2 y if x > y. 6. A transition system is a pair (S, ~ } consisting of a set S of states and a transition relation ~ C_ S x S. Let (~<,) , be a sequence of relations on S inductively defined by ~ < 0 = S x S a n d <<-n+l = {(s,t) E S x S [Vs' E S s.t. s ,s' ~t ' E S s.t. t ~t' and St~ntt}. For s and t in S, S(s,t) = inf {2-" [ s<<.,t} defines a generalized ultrametric on S, which measures the extent to which the transition steps from s can be simulated by steps from t. 352 J.J.M.M. Rutten / Theoretical Computer Science 170 (1996) 349-381 2.1. Generalized ultrametric spaces are [0, 1]-cate#ories We briefly review Lawvere's [14] conception of metric spaces as ~g'-categories [18, 11]. Then we shall follow and further elaborate his approach for the special case of generalized ultrametric spaces, which will be shown to be [0, 1]-categories. The main point is that, in general, many properties of ~/'-categories derive from the structure on the underlying category ~K'. The starting point is a category "K" together with a functor which is symmetric and associative, and has a unit object k (up to isomorphism). This defines a so-called symmetric monoidal structure on ~/'. The category ~ is required to be complete and cocomplete (i.e., all limits and colimits in ~ should exist), and its monoidal structure should be closed: that is, there exists an internal hom functor Horn : ~g "°p x ~/" --+ such that for all a in ~//', the functor Hom(a , ) (mapping b in ~e ~ to Hom(a,b)) is right adjoint to the functor a ® (which maps b in ~¢r to a@b). A ~-category, or a category enriched in U, is any set (more generally, class) X together with the assignment of an object X(x, y) of ~ to every pair of elements (x, y) in X; the assignment of a ~//'-morphism X(x, y) ® X ( y , z ) --+ X(x ,z ) to every triple (x, y,z) of elements in X; and the assignment of a ~//'-morphism k ~ X(x ,x) to every element x in X, satisfying a number of naturality conditions (omitted here since they are trivial in the particular case we are interested in; see [14, 5]. For instance, the category of all sets is a (complete and cocomplete) symmetric monoidal closed category (where ® is given by the Cartesian product, and any one element set is a unit), The corresponding ~f-categories are just ordinary categories: X(x , y ) is given by the homset of all morphisms between two objects x and y in a category X, and the ~-morphisms that are required to exist are just functions defining the composition of morphisms, and giving identity morphisms. Generalized ultrametric spaces can now be seen to be [0, 1]-enriched categories as follows. First of all, [0, 1] is shown to be a complete and cocomplete symmetric monoidal closed category. It is a category because it is a preorder (objects are the real numbers between 0 and 1; and for r and s in [0, 1] there is a morphism from r to s if and only if r>~s). It is complete and cocomplete: equalizers and coequalizers are trivial (because there is at most one arrow between any two elements of [0, 1]), the product r × s of two elements r and s in [0, 1] is given by max {r,s}, and their coproduct r+s by min{r,s}. More generally, products are given by sup, and coproducts J,J.M.M. Ruttenl Theoretical Computer Science 170 (1996) 349-381 353 are given by inf. The monoidal structure on [0, 1] is given by max : [0, 1] × [0, 1] ~ [0, 1], assigning to two real numbers their maximum, which is symmetric and associative, and for which 0 is the unit element. (Note that in this particular case the monoidal product is identical to the categorical product.) Consider the following internal horn functor [0, 1 ] ( , ) : [0, l] °p X [0, 1] ---4 [0, 1], defined (as in Example 2.1) by, for r and s in [0, 1], f 0 if r ~>s, [0, 1](r,s) / s i f r < s. The following fundamental equivalence states that [O, 1 ] ( r , ) is fight adjoint to max { r , } , for any r in [0, 1]: Proposition 2.2. For all r, s, and t in [0, 1], max {r, t} >~s i f and only i f t >~ [0, 1](r,s). As a consequence, [0, 1] is a (complete and cocomplete symmetric) monoidal closed category. (In fact, since the monoidal structure is given by the categorical product on [0, 1], it is even Cartesian closed.) Now [0, 1J-categories are precisely the generalized ultrametric spaces introduced at the beginning of this section: sets X together with a function assigning to x and y in X an object, i.e., a real number X(x, y) in [0, 1]. The existence of a [0, 1]-morphism from X(x, y ) ® X(y , z ) = max {X(x, y ) ,X(y , z ) } to X(x,z) gives the second, and the existence of a morphism from k = 0 to X(x,x) gives the first of the axioms for generalized ultrametric spaces. 2.2. The category o f generalized ultrametric spaces As mentioned above, many constructions and properties of generalized ultrametric spaces are determined by the category [0, 1]. Important examples are the definitions of limit and completeness, presented in Section 3. Also the category of all gum's, which is introduced next, inherits much of the structure of [0, 1]. Let Gum be the category with generalized ultrametric spaces as objects, and nonexpansive functions as arrows: i.e., functions f : X --~ Y such that for all x and x' in X, Y ( f (x ), f (x') ) <.X (x, x'). (Non-expansive fimctions are precisely the [0, l]-functors between [0, 1J-categories.) A function f is isometric if for all x and x ' in X, Y( f (x ) , f ( x ' ) ) = X(x,x') . 354 J.J.M.M. Rutten I Theoretical Computer Science 170 (1996) 349-381 Two spaces X and Y are called isometric (isomorphic) if there exists an isometric bijection between them. The product X x Y of two gum's X and Y is defined as the Cartesian product of their underlying sets, together with distance, for (x, y} and (x', y') in X x Y, X x Y((x,y), (x',y'}) =max{X(x ,x ' ) , Y(y ,y ' )} . Note that this definition uses the product (max) of [0, 1]. The exponent of X and Y is defined by yX = { f : X ---* Y [ f is non-expansive }, with distance, for f and # in yX, Y X ( f ,9) = sup{Y(f(x) ,9(x)) l x 6 X} . The fact that the category [0, 1] is monoidal (Cartesian) closed implies that the category Gum is monoidal (Cartesian) closed as well: i.e., for all gum's X, Y, and Z, zx×r ~ (zr) x. In the category Gum, all limits and colimits exist. Moreover, they are constructed at the level of their underlying sets. Formally: Theorem 2.3. Let U : Gum ---* Set be the functor that maps a 9um to its underlyin9 set ("forgetting" its metric structure). The functor U creates all limits and all colimits. Proof. Limits are easy but colimits are less trivial. They involve the use of the socalled "least-cost" [14] or "shortest-path" [20] distance. For details see [16]. [] 2.3. A few basic definitions This section is concluded by a number of constructions and definitions for generalized ultrametric spaces that will be used in the sequel. The opposite X °p of a gum X is the set X with distance ~t r °P(x ,x / ) = X(Xt,X). With this definition, the distance function X ( , ) can be described as a function X ( , ) : X °p × X ~ [0, 1]. Using Proposition 2.2 one can easily show that X ( , ) is non-expansive, i.e., a morphism in the category Gum. We saw that any preorder P induces a gum. (Note that a partial order induces a quasi ultrametric and that the non-expansive functions between preorders are precisely J.J.M.M. Ruttenl Theoretical Computer Science 170 (1996) 349-381 355 the monotone functions.) Conversely, any gum X gives rise to a preorder (X, ~ O ~tNVn>.N, X(x.,xn+l)<.~. Note that this is equivalent to the more familiar condition: W > 0 3N Vn>>.m>~N, X(xz,Xn)<<.a, 356 J.,I.M.M. Rutten/ Theoretical Computer Science 170 (1996) 349-381 because of the strong triangle inequality. Since our metrics need not be symmetric, the following variation exists: a sequence (xn)~ is backward-Cauchy if W > 0 ~tNVn~>N, X(x.+l,x.)<<.~. If X is an ordinary ultrametric space then forward-Cauchy and backward-Cauchy both mean Cauchy in the usual sense. If X is a preorder then forward-Cauchy sequences are eventually increasing: there exists an N such that for all n ~ N , x, <~xn+l. (Increasing sequences in a preorder are also called ~o-chains.) Similarly backward-Cauchy sequences are eventually decreasing. Cauchy sequences in [0, 1], with the generalized ultrametric of Section 2, are particularly simple: every forward-Cauchy sequence either converges to 0 or is eventually decreasing; dually, every backward-Cauchy sequence either converges to 0 or is eventually increasing. Proposition 3.1. A sequence (r,)n in [0, 1] is forward-Cauchy i f and only i f either: Ve > O ~N Vn>/N, rn <~e, or: ~N Vn>~N, rn>.rn+l. Dually, it is backward-Cauchy i f and only i f either: Ve > O ~ Vn>~N, rn <~e, or: ~V Vn>>.N, rn <~rn+l. Proof. We prove only the first statement, the verge to 0 or that are eventually decreasing Conversely, let (rn)n be forward-Cauchy in [0, VN 3n >~ N, rn > ~. second being dual. Sequences that conare easily seen to be forward-Cauchy. 1]. Suppose there exists e > 0 such that We claim that there exists an N such that for all n>~N, r, > e; for suppose not: VN ~n>~N, rn~e. Because (rn). is forward-Cauchy, there exists M such that for all m>.M, [0,1] (rm, rm+l)<~e. Consider nl ~>M with rn, <<.~, and consider n2>~na with rn2 > ~. Then ,~ < rn2 = [0, 1](rn~,rn2) [definition of the distance on [0,1]] ~< e, a contradiction. Therefore let N be such that for all n >>.N, rn > e. Let M ~>N such that for all m>~M, [0, 1](rm, rm+z)~<~, which is equivalent to rm+~ <~ max{e, rm} by Proposition 2.2. Because r,n > ~, for all m>~M, this implies rm+l <~rm. [] Because Cauchy sequences in [0, 1] are that simple, the following definitions are easy as well: the forward-limit of a forward-Cauchy sequence (rn)n in [0, 1] is given J. J. M.M. Rutten I Theoretical Computer Science 170 (1996) 349-381 357 by limrn = sup inf rk. n k>~n Similarly, the backward-limit of a backward-Cauchy sequence (rn)n in [0, 1] is 1Lmr, = sup inf rk. n k>~n The following proposition shows how forward-limits and backward-limits in [0, 1] are related (cf. [23]. Proposition 3.2. For a forward-Cauchy sequence (r,), in [0, 1], and r in [0, 1], [0, 1](limrn, r) = lira[0, 1](rn, r). For a backward-Cauchy sequence ( rn) n in [0, 1], and r in [0, 1], [0, 1](r, ILm rn) ---lim[0, 1](r, rn). A proof follows easily from the followin9 elementary facts: Lemma 3.3. For all V C[O, 1] and r in [0, 1], 1. [0, 1](inf V,r) = supoev[O, 1](v,r); 2. [0, 1](r, sup V) = sup~v[O, 1](r, v). Forward-limits and backward-limits in an arbitrary generalized ultrametric space X can now be defined in terms of backward-limits in [0, 1]: Definition 3.4. Let X be a generalized ultrametric space. An element x in X is a forward-limit of a forward-Cauchy sequence (x , ) , in X, x = limxn__, iff Vy E X, X (x , y ) = limX(xn, y). Dually, an element x in X is a backward-limit of a backward-Cauchy sequence (Xn)n in X, x = lirnx, iff Vy C X, X (y , x ) = l imX(y, xn). In Section 11, an alternative, equivalent definition of forward-limit and backwardlimit will be given, which is, from an enriched-categorical point of view, more attractive. It will be based on the notions of weighted colimit and weighted limit. Definition 3.4 makes use of the following. Proposition 3.5. Let (xn)n be a sequence in X and y in X. I f (x,)n is forwardCauchy in X then the sequence (X (x , , y ) ) , is backward-Cauchy in [0, 1]. I f (x,)n is backward-Cauchy in X then the sequence (X (y,x~))n is backward-Cauchy in [0, 1]. 358 J.J.M.M. Rutten/ Theoretical Computer Science 170 (1996) 349-381 Note that it follows from Proposition 3.2 that our earlier definitions of forward-limit and backward-limit in [0, 1] are consistent with Definition 3.4. For an ordinary ultrametric space X, the above definitions of forwardand backwardlimit are the same and coincide with the usual notion of limit: x = limxn = limxn if and only if re > 0 3NVn>~N, X(xn,x) < e. ---r ~-The implication from left to right is straightforward. For the converse, note that it follows from Proposition 3.1 that for y in X, the sequence (X(x~,y))n, which is both forwardand backward-Cauchy, either converges to 0 or eventually becomes constant. In both cases, X(x, y) = limX(xn, y) = l imX(y, xn). + I f X is a partial order and (x~)~ is a chain in X then x = limx~ if and only i f V y EX, x<~xy ~=~Vn>>,O, Xn<~Xy, i.e., x = [Ixn, the least upperbound of the chain (Xn)n. Similarly, backward-limits of backward-chains correspond to greatest lowerbounds. Since the rest of this paper mostly deals with forward-Cauchy sequences and forwardlimits, we shall simply write Cauchy for forward-Cauchy, and limxn rather than lira Xn. Note that subsequences of a Cauchy sequence are Cauchy again. I f a Cauchy sequence has a limit x, then all its subsequences have limit x as well. Cauchy sequences may have more than one limit. All limits have distance 0, however. As a consequence, limits are unique in quasi ultrametric spaces. A function f : X ~ Y between gum's X and Y is continuous if it preserves limits: i f x = limx~ in X then f ( x ) = lim f ( x , ) in Y. For ordinary ultrametric spaces, this is the usual definition. For partial orders, it means preservation of least upperbounds of og-chains. In dealing with generalized ultrametric spaces, one should be prepared to reconsider some basic intuitions about ordinary ultrametric spaces. For instance, any non-expansive function between ordinary ultrametric spaces is continuous. But: Remark 3.6. The notions of "non-expansive" and "continuous" function between generalized ultrametric spaces are incomparable. An example of a function that is continuous but not non-expansive is f : 03 ~ 03 defined, for x in 03, by 0 if x = 0, f ( x ) = x 1 i f 0 < x < o g , 09 if x = 09, J. J. M.M. Rutten I Theoretical Computer Science 170 (1996) 349-381 359 where 03 is supplied with the generalized ultrametric as defined in Example 2.1. For instance, 03( f (2) , f (1 ) ) = 03(1,0) = 1 ~ 2 -1 = 03(2, 1). Any function between partial orders that is monotone but not continuous (i.e., least-upperbound preserving) yields an example of the converse. A generalized ultrametric space X is complete if every Cauchy sequence in X has a limit. For instance, [0, 1] is complete. I f X is a partial order completeness means that X is an co-complete partial order, or cpo for short: all co-chains have a least upperbound. For ordinary ultrametric spaces, the above definition of completeness is the usual one. Limits are unique in complete quasi ultrametric spaces, which therefore are well suited for the construction of fixed points. There are at least two ways: Theorem 3.7. Let X be a complete quasi ultrametric space and f : X ~ X nonexpansive. 1. I f f is continuous and if there is x in X with x<~xf(x) (i.e., X ( x , f ( x ) ) = 0), then f has a fixed point, which is the least (with respect to <<.x) fixed point above X. 2. I f f is continuous and contractive: 3e < 1Vx, y EX, X(f(x) , f (y))<<.e .X(x ,y ) , and if, moreover, X is non-empty, then f has a unique fixed point. (Note that contractiveness does not imply continuity; for an example see below.) Proof. 1. Suppose f is continuous and let x be such that X(x,f(x)) = 0. The sequence (x, f ( x ) , f2(x) . . . . ) is trivially Cauchy because f is non-expansive. Since X is complete this sequence has a limit y. By continuity of f , f ( y ) = l i m f ( f n ( x ) ) = l i m F ( x ). In quasi ultrametric spaces, limits are unique, thus y = f ( y ) . I f x <<.xz and f ( z ) = z, for z in X, then it follows that y <<.xz. 2. Suppose that f is both continuous and contractive. Let x be any element in X and consider again the sequence ( x , f ( x ) , f 2 ( x ) . . . . ). Because f is contractive this sequence is Cauchy: for all n>>.O, x(fn(x) , fn+l(x))<~e n "X(x , f (x ) ) . AS in 1, a fixed point y is obtained by completeness of X and continuity of f . Suppose z is another one. Then X ( y , x ) = X ( f ( y ) , f ( z ) ) <<. e . X ( y , z ) whence X ( y , z ) = 0. Similarly X(z, y) = O. Because X is a quasi ultrametric space this implies y = z. [] Part 1 generalizes the theorem of Knaster-Tarski that continuous functions on an cocomplete partial order with a least element, have a least fixed point. Part 2 generalizes Banach's contraction theorem. Both part 1 and part 2 above are special instances of a slightly more general theorem (on quasi metric spaces) in [18]. Consider the set 03 of the natural numbers with infinity with the distance induced by the usual ordering, but for the value of 03(1,0), which is 1 rather than 1. Let 360 J.J.M.M. Rutten I Theoretical Computer Science 170 (1996) 349-381 f : O3 ~ o3 map any n ~> 0 to 0, and co to 1. Then f is contractive but not continuous since lim n = co, whereas lim f ( n ) # f(co). In order to prove that a function f : P ~ Q between partial orders is continuous (that is, preserves least upperbounds), one usually establishes first that f is monotone, from which then half of the proof follows: i f x = IlXn and f is monotone, then Xn <~px implies f(Xn) <~ o f ( x ) whence II f (x~) <~ a f (x ) . Similarly (and more generally), nonexpansiveness of a function between generalized ultrametric spaces implies "half of its continuity"; more precisely: Proposition 3.8. Let X and Y be generalized ultrametric spaces, f : X --~ Y a nonexpansive function, and (Xn) n a Cauchy sequence in X with limxn = x in X. For all y i n Y , lim Y(f(Xn), y) <<. Y( f (x ) , y). Proof. Because f is non-expansive and (Xn) n is Cauchy, the sequence ( f (Xn ) )n is again Cauchy. By Proposition 3.5, the sequence (Y(f(xn), y))n is backward-Cauchy in [0, 1], for any y in Y, and hence has a backward-limit. The inequality follows from [0, 1](Y(f(x), y), lim Y(f(Xn), y)) = lira[O, 1](Y(f(x) ,y) , Y ( f ( x . ) , y ) ) [Proposition 3.2] <~ lim Y( f (Xn) , f (x ) ) [ y ( _ , y ) . yop ~ [0, 1] is non-expansive] <~ lirnX(x.,x) [ f is non-expansive] = X( l imx . , x ) [definition of limit] = X(x ,x) z O, and the definition of the metric on [0, 1] (Example 2.1). [] Proposition 3.8 comes in handy in the following. Proposition 3.9. Let X and Y be generalized ultrametric spaces. 1. I f Y is complete then yX is complete. 2. Let [X ~ Y] = { f : X --* Y I f is both non-expansive and continuous}, with distance as in yX. This defines a generalized ultrametric space, which is complete whenever Y is. Proof. The proof combines, as it were, both the proofs (of the same statements) for partial orders and ordinary ultrametric spaces, and is somewhat more complicated than both proofs individually. We list the main steps: consider a Cauchy sequence (f~)n in X Y. We have to show: there is f in X r with limfn = f ; and if all of the fn are moreover continuous then so is f . ZJ.M.M. Ruttenl Theoretical Computer Science 170 (1996) 349-381 361 1. Definition: for any x in X, the sequence (fn(X))n is Cauchy in Y. It has a limit, to be called f ( x ) , because Y is complete. This defines a function f : X --~ Y. 2. A useful observation: V~ > 0 3N Vn>~N Vx E X, Y( fn (x ) , f (x ) ) < ~. 3. From 2, it follows that f -lim fn. 4. Using 3, one can prove that f is non-expansive. This proves part 1 of the theorem. 5. It remains to be shown that f is continuous if all of the f , are. Let lim x~ = x be a converging sequence in X, and let y be in Y. By Proposition 3.8 and 4, lim Y(f(xn) , y) <~ Y( f (x ) , y). 6. Using 2 and the fact that the functions fn are continuous, one can also prove the converse: Y( f ( x ) , y) <~ lira Y(f(Xn), y). From this and 5, it follows that l imf(xn) = f ( x ) . Thus f is continuous. [] The following fact will be useful later. Lemma 3.10. The composition of functions, viewed as a function o : [Y ---, Z] x [X Y] ~ IX ~ Z], for generalized ultrametric spaces X, Y, and Z, is non-expansive and continuous. 4. Distance and order Generalized ultrametric spaces have been introduced as generalizations of ordinary ultrametric spaces. Their definition has been guided by enriched-categorical motivations. In this subsection, we shall briefly show that, alternatively, generalized ultrametric spaces can be presented as generalized preorders. A strong argument in favour of the original metric definition is the applicability of various insights from enriched category theory (see [4] for more examples). Still the presentation of a generalized ultrametric space as a generalized preorder can be useful, because it allows in certain cases a translation of familiar notions from the theory of ordered spaces into a metric variant thereof. An example will be the notion of e-adjoint pair in Section 5. A generalized ultrametric space X induces a family {~<~c_x x X [ ~ c [o, 1]} of preorders on X defined, for e E [0, 1] and x and y in X, by