Electronic Notes in Theoretical Computer Science 74 (2003)

URL: http://www.elsevier.nl/locate/entcs/volumeT4.html 17 pages

Norm-weightable Riesz Spaces and the Dual Complexity Space

M. O' Keeffe S. Romaguera2 and M. Schellekens3

August 19, 2003

Abstract

The theory of complexity spaces has been introduced in [Sch95], where the applicability to the complexity analysis of Divide & Conquer algorithms has been discussed. This analysis has been based on the Banach Fixed Point Theorem, which has led to the study of biBanach spaces in [RS98]. In [RS96] we have introduced the dual complexity space as a convenient tool to carry out a mathematical analysis of complexity spaces (cf. also [RS98]). We recall that the complexity space as well as its dual are weightable quasi-metric spaces or, equivalently, partial metric spaces (cf. [Sch95], [RS96] as well as [Kiin93],[KV94] and [Mat94]. Recently it has been shown in [Sch02a] that partial metric spaces correspond dually, in the context of Domain Theory, to semivaluation spaces.

Here, we show that the dual complexity space is the negative cone of a biBanach norm-weightable Riesz space (e.g. [BOU52] and [RS98]) and characterize the class of norm-weightable Riesz spaces in terms of semivaluation spaces. In particular, we show that the quasi-norm of an element of such a Riesz space is the quasi-norm of its projection on the negative cone. Hence, quasi-norms are completely determined by partial metrics, justifying, in this context, O' Neill's analogy between these notions. In [Sch02a], it is shown that quasi-uniform semilattices arise naturally in Domain Theory, which motivates a generalization of our characterization to the context of norm-weightable quasi-uniform Riesz spaces.

1 Background

Throughout this paper the letters R, R+ and u will denote the set of all real numbers, of all nonnegative real numbers and of all nonnegative integer numbers respectively. A function d: X x X ^ R+ is a quasi-pseudo-metric on X iff

1 Supported by Science Foundation Ireland, Enterprise Ireland research grant BR/2001/141 and a PhD studentship under the Boole Centre for Research in Informatics, UCC

2 The second author acknowledges the support of the Spanish Ministry of Science and Technology, grant BFM2000-1111

3 The third author acknowledges the support of Science Foundation Ireland

4 AMS (2000) Subject classification: 54E15, 54E35, 54C35, 46B20,22A26

5 Key words and phrases: Riesz space, quasi-norm, quasi-metric space, norm-weightable, negative cone, dual complexity space, quasi-uniform space, join semilattice.

©2003 Published by Elsevier Science B. V.

1) Vx G X. d(x,x) = 0.

2) Vx, y,z G X. d(x, y) + d(y, z) > d(x, z)

If d is a quasi-pseudo-metric on X, then the function d-1 defined on X x X by d-1 (x,y) = d(y,x), is also a quasi-pseudo-metric on X called the conjugate of d.

A quasi-pseudo-metric space is a pair (X, d) consisting of a set X together with a quasi-pseudo-metric d on X.

In case a quasi-pseudo-metric space is required to satisfy the T0-separation axiom, we refer to such a space as a quasi-metric space.

In that case, condition 1) and the T0-separation axiom can be replaced by the following condition:

1') Vx, y G X.d(x, y) = d(y, x) = 0 ^ x = y.

If d is a quasi-(pseudo)-metric on X, then ds is a (pseudo)metric on X, where Vx, y G X.ds(x, y) = max{d(x, y), d(y, x)}.

A quasi-pseudo-metric space (X,d) is called order-convex if d(x,z) = d(x,y) + d(y, z) whenever z <d< y <d x.

A quasi-(pseudo)-metric d on X is said to be bicomplete if ds is a complete (pseudo)metric on X [FL82].

Examples: The function d1 defined on R x R by d(x, y) = max{y — x, 0} is a quasi-metric on R such that (d1)s is the usual metric on R.

The function d2 defined on (0, to] x (0, to] by d2(x,y) = max{ 1 — X, 0} is also a quasi-metric on (0, to], where we have adopt the convention that — = 0.

The complexity (quasi-metric) space has been introduced in [Sch95] as a part of the development of a topological foundation for the complexity analysis of algorithms. In [RS96] we have introduced the dual complexity (quasi-metric) space as an appropriate tool to carry out a mathematical analysis of complexity spaces (see [RS98]).

We recall that the complexity space (with range (0, to]) is the pair (C,dC), where C = {f : u ^ (0, to] | 2-nj(n) < to} and dC is the quasi-metric defined on

C by dc(f,g) = Er=o2-nmax{gk — jn,0}, whenever f,g G C. dcis called in

[Sch95] "the complexity distance", and intuitively measures relative improvements in the complexity of programs.

The dual complexity space (with range R+) is introduced in [RS96] as a pair (C*,dC*), where C* = {f : u ^ R+ | E^=0 2-nf (n) < to} and dC* is the quasi-metric defined on C * by dC* (f,g) = 2-n max{g(n) — f (n), 0}, whenever f,g G C*.

(C, dC) is isometric to (C*,dC*) by the isometry ^ : C* ^C, defined by ^(f) = 1/f (see [RS96]).

A quasi-metric space (X, d) is vjeightable iff there exists a function w: X ^ R+ such that Wx,y E X.d(x,y) + w(x) = d(y,x) + w(y). The function w is called a vjeighting function, w(x) is the vjeight of x and the quasi-metric d is vjeightable by the function w. A vjeighted space is a triple (X, d, w) where (X, d) is a quasi-metric space weightable by the function w. A weighting function of a weighted quasi-metric space is fading iff the space has points of arbitrary small weight.

We recall that the weighting functions of a weightable quasi-metric space are generated by a unique fading weighting f (e.g. [KV94] or [Sch02a]) in the sense that each weighting is of the form f + c for some real number c > 0.

Examples: The quasi-metric space (R+,d1) is weightable by the identity function, w1(x) = x. The quasi-metric space ((0, ro],d2) is weightable by the function w2(x) = X. The complexity space (C ,dC) is weightable by the function wC where Wf E C.wC(f) = 2nj. The dual complexity space (C*,dC*) is weightable by the function wC* where Wf E C*.wC* (f) = 2-nf (n).

We recall the following definition from [Sch96].

Definition 1.1 If (X,d) is a quasi-metric space then (X,d) is upper weightable iff there exists a weighting function w for (X, d) such that Wx,y E X. d(x, y) < w(y). We refer to such a function w as an upper weighting function. A weighted space (X, d, w) is upper weighted iff w is an upper weighting function.

Examples: The quasi-metric space (R+,d1) is upper weightable by the function w1, the quasi-metric space ((0, ro],d2) is upper weightable by the function w2, the complexity space (C ,dC) is upper weightable by the function wC, while the dual complexity space (C*,dC*) is upper weightable by the function wC*.

A quasi-uniform space is a pair (X, U) consisting of a set X with a filter U on X x X such that

1) WU eu. a c U

2) WU eubV eu.V o V c U.

In that case, U is called a quasi-uniformity on X and its elements are referred to as entourages. The preorder associated with a quasi-uniform space (X, U) is the relation <U defined to be the intersection of all the entourages of U.

A subfamily B of a quasi-uniformity U is a base for U if each entourage contains a member of B.

The quasi-uniformity Ud generated by a quasi-pseudo-metric d on a set X is the filter generated on X x X by the set of relations (Be>0)e, where We > 0.Be = {(x,y)\ d(x,y) < e}. Two quasi-pseudo-metrics are equivalent iff they generate the same quasi-uniformity. Two quasi-pseudo-metric spaces are equivalent iff their quasi-

pseudo-metrics are equivalent.

The topology T(U) associated to a quasi-uniformity U on a set X is the topology generated by the neighbourhood filter base U[x] = {U[x] | U G U}, where Vx G X VU G U. U[x] = {yl (x,y) G U}.

If U is a quasi-uniformity on a set X then the trace quasi-uniformity UlA of U on a subset A of X is defined by: UlA = {U n (A x A)| U gU}.

If (X, U) and (Y, V) are quasi-uniform spaces, then the product quasi-uniformity UxV is the set of all binary relations B on X x Y, such that there is a U gU and a V G V such that for each (x,y) in X x Y, B[(x,y)] = U[x] x V[y]. The topology induced by the product quasi-uniformity is the product topology.

A function f: (X, U) ^ (Y, V) is quasi-uniformly continuous iff VV G V3U G U.f2(U) C V, where f2(U) = {(f (x),f(y))| (x,y) G U}. A quasi-unimorphism f: (X, U) ^ (Y, V) is a bijection such that both f and f-1 are quasi-uniformly continuous.

In case the associated preorder of a quasi-pseudo-metric (quasi-uniform) space is a linear preorder we refer to the space as a linear quasi-pseudo-metric (quasi-uniform) space.

A uniform space is a quasi-uniform space (X, U) which is such that VU G U. U-1 G U. Given a quasi-uniform space (X, U) then the uniform space associated to (X, U) is defined to be the space (X, Us) where Us = {V | V C X x X and 3U gU such that V D U n U-1}.

A vjeak quasi-pseudo-metric (weak quasi-uniform) join semilattice is a quasi-pseudo-metric (quasi-uniform) space which is a join semilattice for its associated preorder. We say that a quasi-pseudo-metric space (X, d) has a maximum x0 G X if x <d x0 for all x G X, where <d is the associated preorder of (X, d).

The terminology of quasi-pseudo-metric (quasi-uniform) (semi)lattice is reserved for quasi-pseudo-metric (quasi-uniform) spaces which are (semi)lattices for which the operations are quasi-uniformly continuous with respect to the product quasi-uniformity Ud xUd (UxU). This is in accordance with the terminology used for the theory of uniform lattices (e.g. [Web91] and [Web]).

As discussed in [Sch02a], quasi-uniform (semi)lattices arise naturally in Domain Theory and include in particular the class of totally bounded Scott domains discussed in [Smy91], the Baire quasi-metric spaces of [Mat95] as well as the complexity spaces of [Sch95].

Each of these structures turns out to satisfy an "optimality condition", which is tightly related to compactness (cf. [Sch02a]).

An optimal weak quasi-pseudo-metric join semilattice is a weak quasi-pseudo-metric join semilattice (X, d) such that d(x U y, y) = d(x, y) for all x,y G X.

We recall that a quasi-pseudo-metric join semilattice (X, d) is optimal if and only

if for all x,y,z E X.d(x U z,y U z) < d(x,y) (cf. [Sch97]). We remark that this equivalent condition to optimality is exactly the more familiar notion of U-invariance as discussed in [Gie80]. Hence we obtain that any optimal weak quasi-pseudo-metric join semilattice is a quasi-pseudo-metric join semilattice and we will simply refer to such structures in the following as "optimal quasi-metric join semilattices".

We recall the following useful generalizations of valuations (e.g. [Bir84]) to the context of semilattices, introduced in [Sch02a]:

If (X, is a join semilattice then a function f: (X, ^ R+ is join-modular iff Wx,y,z E X.f (y U z) - f (x U z) > f (y) - f (x U y) and f is co-join-modular iff

Wx, y,z E X. f (y U z) - f (x U z) < f (y) - f (x U y).

A join valuation on a join semilattice is a join-modular increasing function on this semilattice. A join co-valuation on a join semilattice is a co-join-modular decreasing function on this semilattice.

A real-valued function f on a join semilattice (X, □) is called positive (negative) if Wx,y E X.x n y ^ f (x) < f (y)(f (x) > f (y)).

In the following section we recall the main definitions and results of [RS98] on norm-weightable biBanach spaces.

We provide a brief motivation for the study of biBanach spaces in connection to complexity spaces.

We recall that the complexity analysis of Divide & Conquer algorithms involves functionals on complexity spaces of the following type:

&s(f) = An. if n =1 then c else af (—) + h(n).

Since these functionals are defined in terms of the pointwise operations of addition and of scalar multiplication, which intuitively reflect operations carried out by the algorithm on the given datastructures, it is natural to equip complexity spaces with corresponding operations. This approach directly leads to the study of (semi)linear spaces.

Also, in [Sch95] the complexity analysis of Divide & Conquer algorithms has been carried out via the Banach Fixed Point Theorem. The version of the Banach Fixed Point Theorem used in [Sch95] however is formulated in terms of bicomplete quasi-metric spaces, rather than in terms of say biBanach spaces, as one might expect.

In the following section, we provide the necessary definitions in order to formulate the new approach via biBanach spaces (cf. [RS98]). We also recall the useful notion of norm-weightedness from [RS98], which will allow us to show that the weight of the dual complexity space is the restriction of a quasi-norm of a biBanach space.

2 Norm-weightable biBanach spaces

An ordered linear space is a quadruple (E, Q, +, •) such that (E, +, •) is a linear space, say with neutral element 0 and (E, Q) is an order such that

(1) Wx, y, z E X.x Q y ^ x + z Q y + z

(2) Wx E E WA E R+.x □ 0 ^ Ax □ 0.

Remark: In any ordered linear space, conditions (1) and (2) in fact imply conditions (1') and (2') obtained from (1) and (2) by replacing the implication by an equivalence (cf. [BOU52]).

An element x of an ordered linear space (E, Q, +, •) is positive (negative) iff x □ 0 (x Q 0), where 0 is the neutral element of the linear space.

In our context a semilinear space on R+ is an ordered triple (E, +, •), such that (E, +) is an Abelian semigroup containing the neutral element 0, and • is a function from R+ x E to E such that for all x,y E E and a,b E R+: a • (b • x) = (ab) • x, (a + b) • x = (a • x) + (b • x), a • (x + y) = (a • x) + (a • y), and 1 • x = x.

We recall that every semilinear space is a cone in the sense of Keimel and Roth [KR92]. In the context of this paper we use this terminology rather than the one of semilinear spaces (as used in [RS98]). The motivation for this is that in the context of Riesz spaces, the terminology of cones is traditionally used (e.g. [BOU52]).

We remark that for a linear space E (on R) the traditional definition of a cone with top 0 is a subset of E which is closed under addition and under positive scalar multiplication. It is easy to verify that in the context of linear spaces the two notions of a cone coincide.

Example: The set of all positive elements of an ordered linear space forms a cone, which we refer to as the positive cone of the space. Similarly, the set of all negative elements of an ordered linear space forms a cone, which we refer to as the negative cone of the space.

Let (E, +, •) be a linear space on R. A quasi-norm on E is a nonnegative real-valued function ||.|| on E such that for all x,y E E and a E R+:

W ||x|| = ||-x|| =0 ^ x = 0 (where 0 denotes the neutral element of (E, +));

(ii) ||ax|| = a ||x|| ;

(iii) ||x + yH < ||x|| + ||y|| .

Note that the function ||.||s defined on E by ||x||s = max{||x|| , ||-x||}, for all

x G E, is a norm on E.

If a quasi-norm ||-|| exists on a linear space E, we say that the linear space is quasi-normalizable and refer to the pair (E, ||.||) as a quasi-normed linear space. The quasi-norm ||.|| induces, in a natural way, a quasi-metric d\\.\\ on E, defined

d\\.\\(x, y) = Hy — x|| for all x,y G E.

For a given quasi-normed linear space (E, ||.||), we refer to the order associated to the quasi-metric d\ .\ as the order associated to the quasi-norm.

According to [RS98] a biBanach space is a quasi-normed linear space (E, ||.||) such that the induced quasi-metric d\ .\ is bicomplete.

Example: Let (R, +, •) be the usual Euclidean linear space. For each x G R define ||x|| = max{x, 0}. Then ||.|| is a quasi-norm on R such that ||.||s is the Euclidean norm on R. Hence, (R, ||.||) is a biBanach space. Note that the quasi-metric induced by ||.|| is exactly the quasi-metric d1, defined above.

As in [RS98], we define BR = {f : u ^ R | E^=0 2-n Hf (n)|s < to}. We de-

fine addition and scalar multiplication pointwise on this set and we let

Er=0 2-n Uf (n)|| . Then (BR, M B* ) is a biBanach space (cf. [RS98]). It is an ordered linear space and the dual complexity space (C*,dC*), defined above, is its negative cone, when we consider the order associated to the quasi-norm | .| * .

We remark that condition (ii) of the definition of a quasi-norm is restricted to nonnegative scalars. The reason for this is that the generalization of (ii) to arbitrary scalars would for instance not respect the definition of the quasi-norm ||.|| defined in the preceding example, which induces the quasi-metric d1 on the reals. In fact, put a = —1, x = —1. Then ||ax|| = ||1|| = 1, but |a| ||x|| = 0.

Definition 2.1 A quasi-normed cone is a pair (F, ||.||F) such that F is a nonempty subset of a quasi-normed linear space (E, ||.||), ||.||F denotes the restriction of the quasi-norm ||.|| to F and (F, + , • |i?) is a cone (on R+).

If (F, ||.||F) is a quasi-normed cone, then the restriction to F of the quasi-metric d\\.\\, induced on E by the quasi-norm ||.|| , will be denoted by d\\.\\F.

If in addition the following condition is satisfied, (i) (F,d\\.\\F) is an order-convex optimal quasi-metric join semilattice having a maximum, then (F, ||.||F) is called a norm-weightable cone.

The terminology "norm-weightable" is justified by Corollary 2.6 below.

Definition 2.2 A biBanach cone is a quasi-normed cone (F, ||.||F) such that F is a nonempty subset of the biBanach space (E, ||.||) and F is closed in the Banach space (E, ||.||s). If in addition, the condition (i), in Definition 2.1, is satisfied then, (F, ||.||F) is called a biBanach norm-weightable cone.

Remark: Note that if (F, ||.||F) is a biBanach cone, then d\\.\\F is a bicomplete quasi-metric on F.

It is shown in [RS98] that the (dual complexity) space (C*, ||.||c*) is a biBanach norm-weightable cone, where ||.||c* is the restriction to C* of the quasi-norm ||.||ß* defined above.

The following results have been obtained in [RS98] for the case of quasi-normed semilinear spaces. They have been reformulated here in the context of quasi-normed cones.

Lemma 2.3 Let (F, ||.||F) be a quasi-normed cone such that (F,d\\.\\F) has a maximum. Then the neutral element 0 is the (unique) maximum of (F,d\\.\\F).

Corollary 2.4 Let (F, ||.||F) be a quasi-normed cone of the quasi-normed linear space (E, ||.||) such that (F, dy.yF) has a maximum. Then, ||x - yH < ||x||F , for all x,y E F.

Lemma 2.5 Let (X,d) be an order-convex optimal quasi-metric join semilattice having a maximum element xq. Then (X, d) is upper vjeightable by the vjeighting function w : X ^ R+ defined by w(x) = d(x0, x) for all x E X.

From Lemmas 2.3 and 2.5, we deduce the following result.

Corollary 2.6 Let (F, ||.||F) be a norm-weightable cone. Then the quasi-metric space (F, d||.||F) is upper weightable by the vjeighting function w : F ^ R+ defined by w(x) = ||x||F for all x E F.

3 Norm-weightable Riesz spaces

We recall the definition of a Riesz space (e.g. [BOU52]).

Definition 3.1 A Riesz space is an ordered linear space (E, Q, +, •) such that the order (E, Q) is a lattice.

Example: Let A be a set, then the function space RA, equipped with the pointwise order, pointwise scalar multiplication and pointwise addition, is a Riesz space. The lattice operations are again defined pointwisely.

We introduce some notation which differs somewhat from the traditional notation used in the context of the theory of Riesz spaces (e.g. [BOU52]). Since the dual complexity space is the negative cone of a biBanach space (cf. the example following the notion of a biBanach space), we will focus on a formulation of the theory in terms of negative elements rather than in terms of positive elements as is customary in traditional Riesz space theory. The discrepancy arises because the dual complexity space is equipped with the order <dl which is the converse order of the traditional pointwise order on real valued functions.

As a result we will define the negative part of an element x, that is x-, to be the element x n 0 (rather than the traditional definition as (-x) U 0) and the positive part of x as the element x+ = (-x) n 0 (rather than the traditional definition as x U 0).

If we let |x| = x n (—x) then we have that

(a) x = x- — x+ and (b) |x| = x- + x+. The following results translate dually from [BOU52] to our context:

We have that (a) and (b) are equivalent to:

(c) x- = — (|x| + x) and (d) x+ = — (|x| — x).

We also have

(e) (x + z) n (y + z) = z + (x n y) and

(f) x n y = x + (y — x)- = 1 (x + y + |x — y|).

We remark that a Riesz space is generated by the negative cone, in the sense that each element of the space can be expressed as a difference of two negative elements in a unique way (cf. [BOU52]).

Lemma 3.2 In any Riesz space (E, +, •), the following relations hold:

1) Vx G E. |x| < x-,x+.

2) Vx,y G E.x □ y ^ (—x) □ (—y).

3) a) Vx, y G E. — (x U y) = (—x) n (—y).

3) b) Vx, y G E. — (x n y) = (—x) U (—y).

4) Vx, y G E. (x + y)- □ x- + y-.

Proof. To show 1), we remark that Vx G E.x-,x+ □ 0, where 0 is the neutral element of E. By the fact that (E, +, •) is an ordered linear space we obtain that Vx G E.x- + x+ □ x+ and x+ + x- □ x-.

To show 2), we note that, again since (E, +, •) is an ordered linear space, Vx, y G E.x □ y ^ x + (—x — y) □ y + (—x — y) ^ —y □ —x.

To show 3) a), we first show that Vx,y G E. — (x U y) □ (—x) n (—y). Indeed, if x,y G E then —(x U y) □ (—x) n (—y) (x U y) □ (—x), (—y) ^ x, y □ (x U y), using (2). The result follows since the last inequality holds trivially.

Next we verify that Vx, y G E. — (x U y) □ (—x) n (—y). Indeed, if x,y G E then

— (x U y) □ (—x) n (—y) ^ (x U y) □ —((—x) n (—y)) ^ x □ —((—x) n (—y)) and y □

— ((—x) n (—y)) ^ —x □ ((—x) n (—y)) and — y □ ((—x) n (—y)). The result follows since the last two inequalities are trivial.

We remark that 3) b) follows from a straightforward combination of 2) and 3) a). To show 4), we remark that for any two elements x,y G E, we have that x- □ x and y- □ y. Thus, since we work in an ordered linear space, we have that x- + y- □ x + y- □ x + y. Hence in particular we obtain that (x- + y-) n 0 □ (x + y) n 0. However, since x-,y- □ 0, we obtain that x- + y- □ x- + 0 = x- □ 0.

Hence (x + y)- = (x + y) n 0 □ (x- + y-) n 0 = (x- + y-). So we have that

Wx,y E E. (x + y) □ x + y .

Definition 3.3 A quasi-normed Riesz space is a quasi-normed linear space which is a Riesz space with respect to the order associated to the quasi-norm. A norm-weightable Riesz space is a quasi-normed Riesz space such that the quasi-normed cone of its negative elements is a norm-weightable cone. A biBanach norm-weightable Riesz space is a norm-weightable Riesz space which is a biBanach space.

Proposition 3.4 The biBanach space (BR, ||-||B*) which has the complexity space as negative cone, is a biBanach norm-vjeightable Riesz space.

Proof. We recall from the example of Section 2, following the notion a biBanach space (cf. also [RS98]), that (BR, ||-||B*) is a biBanach space which has the dual complexity space as negative cone and which is norm-weightable.

So it suffices to verify that it is a Riesz space. By the example following the definition of a Riesz space, it suffices to check that the space (BR, ||-||B*) is closed under the pointwisely defined lattice operations.

We verify the case of the join operation and leave the similar verifications for the meet operation to the reader.

We need to verify that if f,g E BR, then f UB* g E BR, where UB* is the supremum operation of the lattice BR, equipped with the order associated to d^.^*. For this we need to verify that 2-n ||(f UB* g)(n)||s < to.

We let U denote the usual supremum operation on R. Then, for f,g eBR and n E u, we have that:

||(f Ub* g)(n)|r = ||(f Ub* g)(n)|| U ||(-(f Ub* g)(n))||

= ((f Ub* g)(n) U 0) U ((-(f Ub* g)(n)) U 0) = ((f (n) n g(n)) U 0) U ((-(f (n) n g(n))) U 0) = ((f (n) n g(n)) U 0) U ((-f (n) U -g(n)) U 0) = ((f (n) U 0) n (g(n) U 0)) U ((-f (n) U 0) U (-g(n) U 0)) < ((f (n) U 0) + (g(n) U 0)) + ((-f (n) U 0) + (-g(n) U 0)) = Hf(n)|| + ||g(n)|| + H-f(n)|| + ||-g(n)|| . Since f,g E BR, the desired result follows.

4 The characterization

We introduce the following notation regarding the theory of U-invariant quasi-metric join semilattices.

Let I denote the class of all U-invariant quasi-metric join semilattices which posses a maximum and let W denote the class of order convex spaces of I.

By Theorem 10 of [Sch02a], which we discuss below, the class W can also be characterized as the class of all weightable spaces of I.

We let W * = {f: (X, ^ (R+, <)| (X, is a join semilattice with a maximum x0, < is the usual order on the reals and f is a negative join co-valuation such that

f (x0) = 0}.

We recall the following useful theorem of [Sch02a] (Theorem 10), adapted to our context of quasi-metric spaces with a maximum.

Theorem 4.1 W* is the dual of W in the following sense: there exists a bijection W* ^ W, defined to be the function which associates to each function f G W* the quasi-metric space (X,df) G W, where X is the domain of f, f is a weighting function for (X, df) and where Vx, y G X. df (x, y) = f (y) — f (x U y). The inverse of ^ is the function which to each space in W associates its unique fading weighting.

Now, we define the class NW to be the class of all norm-weightable Riesz spaces whose negative cone, equipped with the quasi-metric associated to the quasi-norm, belongs to W and the class NW* to be the functions f whose domain is a Riesz space (E, Q, +, •) and whose restriction to the negative cone C- of this Riesz space belongs to W* and satisfies the following conditions:

(1) Vx, y G C-.f(y — x) = f (y) — f (x U y)

(2) Vx,y G C-.f(x + y) < f (x) + f (y)

(3) Vx G C- Va G R+. f (ax) = af (x)

(4) Vx G C-. f(x) = f (—x) = 0 ^ x = 0.

In other words, the function f is a quasi-norm on C- which satisfies (1). Remark: It follows from Proposition 3.4 that the space (BR, ||.||B*) is an element of

Before stating the main results, we introduce a few technical lemmas.

Lemma 4.2 Under hypothesis (2), condition (1) is equivalent to the following two conditions: (1') Vx,y G C-.y Q x ^ f (y — x) < f (y) — f (x) and (1") Vx,y G C-.f(y — x) = f (y — (x U y)).

Proof. Clearly (1) implies (1'). Since, by (1), f (y — x) = f (y) — f (x U y) and

also f (y — (x U y)) = f (y) — f ((x U y) U y) = f (y) — f (x U У), we obtain that

f (y — x) = f (y — (x U y)) and hence (1'').

To show the converse, we remark that, under hypothesis (2), we obtain that f (y) = f ((y — x) + x) < f (y — x) + f (x) and hence f (y — x) > f (y) — f (x). Combined with hypothesis (1') we obtain that (1''') Vx,y G C-.y Q x. f (y — x) = f (y) — f (x).

By (1'') we have that f (y — x) = f (y — (x U y)) and hence, using (1'''), we obtain that f (y — x) = f (y — (x U y)) = f (y) — f (x U y); that is we obtain (1).

Lemma 4.3 Conditions (1'), (2) and (3) imply (5) Vx,y G C-.y Q x ^ f (^^) +

f (^) = f (y)

Proof. Note that by adding (1') and (2), we obtain that Vx,y G C-.y Q x. f (y — x) + f (x + y) < 2f (y). By (2) we also have that f (y — x) + f (x + y) > f ((y — x) + (y + x)) =

f (2y) = 2f (y), where the last equality follows by (3).

So we obtain that Vx,y G C-.y □ x. f (y — x) + f (x + y) = 2f (y). Finally, by dividing both sides by 2 and again using (3), we obtain that Vx,y G C- y □ x ^

f (-) + f ( ^ ) = f (y).

Proposition 4.4 Let E be a Riesz .space and let C- be its negative cone. Then, each quasi-norm f : C- — R+ satisfying (1) can be extended to a quasi-norm f : E — R+, defined by:

(*) Vx G E.f (x) = f (x-). Also, any function f : E — R+ satisfying (1") is entirely determined by its restriction to the negative cone, in the sense that Vx G E. f (x) = f (x-).

Proof. Let E be a Riesz space with negative cone C- and let f : C- — R+ be a quasi-norm satisfying (1).

Since f satisfies (1), we obtain in particular that f is decreasing, since Vx,y G C-.x □ y ^ f (y — x) =_f (y) — f (x) and hence f (y) ^f (x).

We define f by: Vx G E. f (x) = f (x-) and verify that f satisfies (2) — (4). To show (2), we remark that, since f is decreasing, Vx, y G E. f (x + y) = f ((x + y)-) < f (x- + y-) < f (x-) + f (y-) = f (x) + f (y), where the first inequality follows from Lemma 3.2).

We remark that Vx G E Va G R+. (ax)- = ax n 0 = a(x n 0) = a(x-).

To show (3), we remark that Vx G E Va G R+. f (ax) = f ((ax)-) = f (ax-) =

af (x-) = of(x).

To verify (4), we remark that if x = 0 then clearly x- = 0 and x+ = 0 and thus f (x) = f (x-) = f (0) = 0. _ _

Conversely, we assume that f (x) = f (—x) = 0 for some x G E. Then we obtain that f (x-) = f ((—x)-) = 0 and thus f (x-) = f (x+) = 0. Hence f (|x|) < f (x+) + f (x-) = 0. If we assume by contradiction that x = 0 then in particular x- = 0 or x+ = 0 (since x = x- — x+). Say w.l.o.g. that x- = 0 and thus x- \Z 0. Then, since |x| □ x-, we have |x| C 0 and thus f (|x|) > f (0) = 0, since f is negative. Thus we obtain a contradiction.

To show the converse, we remark that that Vx G E. f(x) = f (x- — x+) = f (x- — (x+ U x-)) by (1"). Hence Vx G E. f (x) = f (x- + ((—x+) n (—x-))) by Lemma 3.2). So by (f), we obtain that Vx G X. f (x) = f (x- + ((—x+) + ((—x-) — (—x+))-)) = f (x- + ((—x+) + (—x)-)) = f (x- + (—x+ + x+)) = f (x-).

Theorem 4.5 NW* is the dual of NW in the following sense: there exists a bijection NW* — NW, defined to be the function which associates to each function f G NW* a norm-weightable Riesz space (E, +, - ,f) G NW, where (E, +, •) is the domain of f and where Vx G E. f (x) = f (x-) and f (x-) + f (x+) = f (|x|). The inverse of ^ is the function which to each space in NW associates its quasi-norm.

Proof. Let f G NW*, where say the domain of f is the Riesz space (E, +, •), and let C- be the negative cone of this Riesz space. By Theorem 4.1, we then obtain a

weightable quasi-metric space (C,df), where Vx,y G C.df (x,y) = f (y) — f (x U y), where f is a weighting function of (C,df). Note that C = C-.

Again by Theorem 4.1 and the definition of the class W, we also have that the order of the Riesz space restricted to the negative cone C- , coincides with the associated order of df.

We remark that the weighted space (C-,df,f) is U-invariant and order-convex. By conditions (2), (3) and (4), we know that f is a quasi-norm on C-, where this cone has a maximum 0. Moreover, by Lemma 4.3 we have that

(5) Vx,y G C-.y Q x ^ f(^) + f (^) = f (y).

We remark that, by Lemma 3.2, we know that Vx G E. |x| < x- and thus |x| Q x. So, from (5) and by (c) x- = 1 (|x| + x) and (d) x+ = 1 (|x| — x), we obtain that f (x-) + f (x+) = f (|x|).

From Lemma 4.2 and Proposition 4.4 we also obtain that Vx G E. f(x) = f (x-). It remains to be shown that f is quasi-norm on E. This follows immediately from Proposition 4.4.

To show the converse, let (E, +, •, ||-||) g NW, with a negative cone C-. By Corollary 2.6 and the definition of NW, we obtain that the restriction of the quasi-norm to the negative cone C- of the given norm-weightable Riesz-space is a weighting function of this cone.

Since (C-, d\\^G_) is an optimal quasi-pseudo-metric space, we obtain that Vx, y G C-. Uy — x|| = dHc_ (x,y) = dHc_ (x U y, y) = HyH — ||x U y||, by the weighting equality.

Hence we obtain (1) for ||.|| as well as (2), (3) and (4) since ||.|| is a quasi-norm. Finally, it is straightforward to see that ^ is a bijection.

5 Quasi-uniform Riesz spaces

Definition 5.1 A quasi-uniform Riesz space is a Riesz space equipped with a quasi-uniformity such that the associated order of the quasi-uniformity coincides with the order of the Riesz space. A quasi-uniform Riesz space has a countable base iff its quasi-uniformity has a countable base. A quasi-uniform Riesz space with a countable base is quasi-normalizable iff there exists a quasi-norm on the space such that the quasi-metric induced by this quasi-norm induces the quasi-uniformity on the space. A quasi-uniform Riesz space is norm-weightable iff it is a quasi-normalizable Riesz space such that the Riesz space equipped with the quasi-norm is a norm-weightable Riesz space. In that case we say that the Riesz space is norm-weightable by this quasi-norm.

Remark: A quasi-uniform Riesz space is T0 since the associated preorder of its quasi-uniformity is an order.

We recall some definitions and results from [Sch02a].

Definition 5.2 If (X, U) and (Y, V) are quasi-uniform spaces then a function f: (X, U) ^ (Y, V) is an order quasi-unimorphism iff

1) f is surjective

2) f is strictly increasing with respect to the associated orders

3) VV EV3U EUVx,y E X.x >U y ^ (xUy ^ (fx)V(fy))

4) VV EU3U e VVx,y E X.x >U y ^ (f(x)Uf(y) ^ xVy).

Clearly, every quasi-unimorphism f: (X, U) ^ (Y, V) is an order quasi-unimorphism. We remark that for the case where the domain (X, U) is linear, the notions of an order quasi-unimorphism and that of a quasi-unimorphism are equivalent.

We will focus in the following on order quasi-unimorphisms with range space (R+, Udl) and (R+, Ud-i). These are referred to as left order quasi-unimorphisms and right order quasi-unimorphisms respectively.

Definition 5.3 A Q-join valuation on a quasi-uniform join semilattice is a join valuation which is a right order quasi-unimorphism. A Q-join co-valuation on a quasiuniform join semilattice is a join co-valuation which is a left order quasi-unimorphism.

Remark: The fact that a co-join valuation is decreasing, while a Q-join co-valuation is increasing with respect to the associated orders is of course consistent, since the associated order of the left distance di is the opposite of the standard ordering on the reals.

Definition 5.4 A topological Riesz space is a Riesz space equipped with a topology. A topological Riesz space is determined by a quasi-uniformity iff the topology and order associated with the quasi-uniformity coincide respectively with the topology and order of the Riesz space.

The following theorem of [Sch02b] (Theorem 13) provides a solution to problem 7 of [Kun93], for the class of quasi-uniform join semilattices.

Theorem 5.5 If (X, U) is a quasi-uniform join semilattice for which U has a countable base, then

U is generated by a vjeightable U-invariant quasi-metric ^ there exists a Q-join co-valuation on (X, U).

We obtain the following corollary:

Corollary 5.6 Let (E, T) be a topological Riesz space determined by a quasi-uniformity U, where U has a countable base, and let C- be its negative cone.

If the quasi-uniform Riesz space (E, U) is quasi-normalizable via a quasi-norm ||-|| such that d||.|| is U-invariant then:

The quasi-uniform Riesz space (X, U) is norm-weightable iff there exists a Q-join co-valuation f on (C-,UlC-), where f satisfies (1) - (4).

Proof. Let (E, T) be a topological Riesz space determined by a quasi-uniformity U, where U has a countable base, and let C- be its negative cone.

We assume that the quasi-uniform Riesz space (E, U) is norm-weightable by a quasi-norm ||-||. In that case we know that the quasi-metric negative cone (C-, dyyc-) of this Riesz space is U-invariant and weightable by |H|C-, where dyyc- induces the trace quasi-uniformity UC-. Hence, by Theorem 5.5, we obtain that ||-||c- is a Q-join co-valuation on (C-,UC-) satisfying (1) — (4). Indeed, it follows from the proof of Theorem 5.5 (cf. proof of Theorem 13, [Sch02a]) that a Q-join valuation is obtained via any weighting function of the space (C- ,dyyc-).

To show the converse, we assume that there exists a Q-join co-valuation f on (C-,UC-) which satisfies (1) — (4). Again by Theorem 5.5, we obtain that there exists a weightable U-invariant quasi-metric d on C-, say with a fading weighting f, where d induces UC-. From the proof of Theorem 5.5 (cf. [Sch02a]) we know that this quasi-metric df is defined by: Vx,y G C-.df (x,y) = f (y) — f (x U y). Hence, from (1) — (4), we obtain that the fading weighting f is a quasi-norm on C-.

By Proposition 4.4, we can extend f to a quasi-norm f on E. Let dj be the quasi-metric generated by this quasi-norm.

It suffices to show that df and dy^y are equivalent, since df is U-invariant and order-convex.

By assumption, we know that the quasi-uniform Riesz space (E, U) is quasi-normalizable via a quasi-norm ||-|| such that (E,dyy) is U-invariant.

So we obtain that dy^y (x, y) = dy^y (x U y, y) = Hy — (x U y) || = || (y — (x U y))-1| = dH (0, (y — (x U y))-). _

Similarly we obtain that df (x,y) = df (xUy,y) = f (y—(xUy)) = f ((y—(xUy))-) = df (0, (y — (x U y))-). In each case, the one but last equality follows by Proposition 4.4.

Since df as well as dy^y induce the quasi-uniformity UC- on C- and hence are equivalent, the desired result follows.

Conclusion: We have continued the research on the mathematical analysis of the dual complexity space (cf. also [RS96] and [RS98]).

Connections between partial metrics and valuations have been indicated first in [O'N97]. According to [BSh97], "the existence of deep connections between partial metrics and valuations is well known in Domain Theory", a claim which is motivated in [BSh97] by examples and the generation of partial metrics from valuations.

Recently it was proved in [Sch02a] that partial metric spaces correspond dually, in the context of Domain Theory, to semivaluation spaces.

We have introduced the notion of a norm-weightable Riesz space and we have shown that the dual complexity space is the negative cone of a biBanach norm-weightable Riesz space. A characterization of norm-weightable Riesz spaces in terms of semivaluation spaces has been obtained. We have shown that the quasi-norm of an element of such a Riesz space is the quasi-norm of its projection on the negative cone. In particular, quasi-norms are determined by partial metrics, thereby justifying, in this context, an analogy formulated in [O'N97]. Finally quasi-uniform Riesz spaces have been defined and characterized in the setting of semivaluation spaces.

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Department of Computer Science National University of Ireland, Cork, University

College Cork Western Road Cork Ireland E-mail: okeeffem@student.cs.ucc.ie

Escuela de Caminos, Departamento de Matemática Aplicada, Universidad Politecnica

de Valencia, Apartado 22012, 46071 Valencia, Spain. E-mail: sromague mat.upv.es

Department of Computer Science National University of Ireland, Cork, University

College Cork Western Road Cork Ireland E-mail: m.schellekens@cs.ucc.ie