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Boolean Metric Spaces and Boolean AlgebraicVarietiesAntonio Avilés a ba Departamento de Matemáticas , Universidad de Murcia , Murcia, Spainb Departamento de Matemáticas , Universidad de Murcia, Facultad de Matemáticas,Campus de Espinardo , Murcia, 30100, SpainPublished online: 21 Oct 2011.

To cite this article: Antonio Avilés (2004) Boolean Metric Spaces and Boolean Algebraic Varieties, Communications inAlgebra, 32:5, 1805-1822, DOI: 10.1081/AGB-120029903

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Boolean Metric Spaces and Boolean Algebraic Varieties#

Antonio Aviles*

Departamento de Matematicas, Universidad de Murcia,Murcia, Spain

ABSTRACT

The concepts of Boolean metric space and convex combination are used tocharacterize polynomial maps An �!Am in a class of commutative Von

Neumann regular rings including p-rings, that we have called CFG-rings. In thoserings, the study of the category of algebraic varieties (i.e., sets of solutions to afinite number of polynomial equations with polynomial maps as morphisms) isequivalent to the study of a class of Boolean metric spaces, that we call here

CFG-spaces.

Key Words: Boolean metric; Boolean distance; Algebraic variety; Boolean

domain.

Mathematics Subject Classification: Primary 13AXX; Secondary 06E30; 16E50;

51F99.

#Communicated by R. Wisbauer.*Correspondence: Antonio Aviles, Departamento de Matematicas, Universidad de Murcia,

Facultad de Matematicas, Campus de Espinardo, Murcia 30100, Spain; Fax: +34-968-364182; E-mail: avileslo@um.es.

COMMUNICATIONS IN ALGEBRA�

Vol. 32, No. 5, pp. 1805–1822, 2004

1805

DOI: 10.1081/AGB-120029903 0092-7872 (Print); 1532-4125 (Online)

Copyright # 2004 by Marcel Dekker, Inc. www.dekker.com

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NOTATIONS AND CONVENTIONS

Throughout this work, ðB;þ; �Þ will be a Boolean ring where the operationa _ b ¼ aþ bþ ab is the analogue for set union, the order a � b , ab ¼ a is theanalogue for set inclusion and for each a 2 B, �aa ¼ aþ 1 is the analogue for theset complement of a.

All rings will be commutative with identity. Regular ring will mean herecommutative Von Neumann regular ring, i.e., a (commutative) ring for which anyprincipal ideal is generated by an idempotent, also known as absolutely flat rings,see Goodearl (1979) and Popescu and Vraciu (1976). Unless otherwise stated, A willbe a regular ring, BðAÞ will denote the set of the idempotent elements of A ande : A�!BðAÞ will be the map that sends each a 2 A to the only idempotenteðaÞ 2 BðAÞ such that aA ¼ eðaÞA. The set BðAÞ has a structure of Boolean ring withproduct inherited from A and with the sum a ~þþ b ¼ ða� bÞ2. For a1; . . . ; an 2 BðAÞwith aiaj ¼ 0 for i 6¼ j, it holds a ¼ a1 þ � � � þ an ¼ a1 ~þþ � � � ~þþ an ¼ a1 _ � � � _ an.In this case we will denote a by a1 � � � � � an.

Given a prime p 2 Z, a p-ring is a ring A for which px ¼ 0 and xp ¼ x for allx 2 A. In particular, a Boolean ring is a 2-ring. Any p-ring is a regular ring witheðxÞ ¼ xp�1.

An algebraic variety over a ring A is a set U � An which is the set of solutionsto a finite number of polynomial equations. If U � An and V � Am are algebraicvarieties, a map f : U �!V is called a polynomial map if there are polynomialsf1; . . . ; fm 2 A½X1; . . . ;Xn� such that fðxÞ ¼ ðf1ðxÞ; . . . ; fmðxÞÞ. When A ¼ B is aBoolean ring the usual terms are Boolean domain and Boolean transformation,see Rudeanu (1974, 2001).

INTRODUCTION

Boolean metric spaces (Definition 1.1) appeared in several works in the 1950sand 1960s Blumenthal (1952, 1961), Ellis (1951, 1956), Hararay et al. (1982) andMelter (1964a), where some authors investigated the analogue for some topics inGeometry such as betweeness, motions or topology in some of those spaces, asBoolean algebras and some rings where a suitable Boolean metric could be defined.In some papers, Batbedat (1971), Melter (1964b), Melter and Rudeanu (1974) andZemmer (1956), a special attention was paid to p-rings, that admit a metric spacestructure over its ring of idempotents. In fact, if A is a regular ring, then An is aBoolean metric space over BðAÞ with the distance

dððx1; . . . ; xnÞ; ðy1; . . . ; ynÞÞ ¼ eðx1 � y1Þ _ � � � _ eðxn � ynÞ:

We will show the close relation that exists between the theory of Boolean metricspaces and the Algebraic Geometry over CFG-rings. We define a regular ring A to bean CFG-ring if there are x1; . . . ; xn in A such that any element in A is of the formP

aixi where a1; . . . ; an 2 BðAÞ and a1 � � � � � an ¼ 1.

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In Secs. 1 and 2 we develop some tools concerning the structure of Booleanmetric spaces, while in Secs. 3 and 4 the main results are exposed. Namely, inSec. 3, we prove that if A is a CFG-ring and U is a subset of An, then U is an alge-braic variety if and only if there are x1; . . . ; xn in U such that any element of x 2 U isof the form x ¼ Pn

1 aixi where a1; . . . ; an 2 BðAÞ and a1 � � � � � an ¼ 1, if and only ifthere is distance-preserving bijection from U onto an algebraic variety V � Am. Also,if U � An and V � Am are algebraic varieties over A and f : U �!V is a map, thefollowing are equivalent:

(1) f is a polynomial map.(2) dðfðxÞ; fðyÞÞ � dðx; yÞ for all x; y in U .(3) fðPn

1 aixiÞ ¼Pn

1 aifðxiÞ for all x1; . . . ; xn in U and for all a1; . . . ; an inBðAÞ with a1 � � � � � an ¼ 1.

Thus, the category of algebraic varieties over an CFG-ring is equivalent to thecategory of those Boolean metric spaces over BðAÞ that are isometric to somealgebraic variety, that we have called CFG-spaces. Some special cases of theseimplications were known for A ¼ B a Boolean ring: that 1 is equivalent to 3 whenU ¼ Bn, V ¼ B is in Theorem 4.2 in Rudeanu (1974), and that 1 is equivalent to 2when U ¼ V ¼ B was observed in Melter (1982).

In Sec. 4, we present a classification of the Boolean metric spaces over a Booleanring B, which is a classification of the algebraic varieties over a CFG-ring. We associ-ate to each of those spaces a finite decreasing sequence of nonzero elements of B suchthat two spaces are isometric if and only if they have the same associated sequence.

1. BOOLEAN METRIC SPACES

1.1. Basic Definitions and Examples

Definition 1.1. Let X be a set. A map d : X � X�!B is said to be a Boolean metricif the following axioms hold, for all x; y; z 2 X :

(1) dðx; yÞ ¼ 0 if and only if x ¼ y.(2) dðx; yÞ ¼ dðy; xÞ.(3) dðx; zÞ � dðx; yÞ _ dðy; zÞ.

In that case, we will say that ðX;dÞ is a metric space over B.

In the above definition, axiom 3 can be substituted by any of the following:

ð30Þ dðx; zÞdðy; zÞ � dðx; yÞ.ð300Þ ðx; zÞ þ dðz; yÞ � dðx; yÞ.Some suitable subsets of modules possess structure of Boolean metric space. We

have called these subsets metrizable. Recall that the annihilator of an element x of amodule over the ring A is the ideal AnnðxÞ ¼ fa 2 A : ax ¼ 0g.

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Definition 1.2. Let A be a regular ring, M a module over A and X a subset of A.The set X will be said to be metrizable if for each x; y 2 X the ideal Annðx� yÞ isa principal ideal of A.

If X is a metrizable subset of M, for each x; y 2 M the ideal Annðx� yÞ has aunique idempotent generator, say axy 2 BðAÞ. Then, the map dðx; yÞ ¼ axy is aBoolean metric on X, called the modular metric on X. Triangle inequality followsfrom

Annðx� yÞ \ Annðy� zÞ � Annðx� zÞ:

For every a 2 A we have AnnðaÞ ¼ eðaÞA, so A is a metrizable subset of itselfand its modular metric is given by dðx; yÞ ¼ eðx� yÞ. This is the same metric on A

as defined in Melter (1964a).Furthermore, for every n 2 N, An is also a metrizable subset of itself and its

modular metric is given by

dððx1; . . . ; xnÞ; ðy1; . . . ; ynÞÞ ¼ eðx1 � y1Þ _ � � � _ eðxn � ynÞ:

This is a particular case of the following general construction:

Definition 1.3. Let ðX1;d1Þ; . . . ; ðXn;dnÞ be metric spaces over B. ThenðX1 � � � � � Xn;dÞ is also a metric space over B with

dððx1; . . . ; xnÞ; ðy1; . . . ; ynÞÞ :¼ d1ðx1; y1Þ _ � � � _ dnðxn; ynÞ:

This space will be called the product space of the spaces ðXi;diÞ and d will be calledthe product metric of the metrics di.

The formation of products is compatible with modular metrics:

Proposition 1.4. Let Si be a metrizable subset of the A-module Mi, for i ¼ 1; . . . ; n.Then S ¼ S1 � � � � � Sn is a metrizable subset of M1 � � � � �Mn and the modularmetric in S equals the product metric of the modular metrics in the Si’s.

Proof. Call di the modular metric in Si. For each x ¼ ðx1; . . . ; xnÞ andy ¼ ðy1; . . . ; ynÞ in S

Annðx� yÞ ¼\ni¼1

Annðxi � yiÞ ¼\ni¼1

diðx; yÞA

¼�Yn

i¼1

diðx; yÞ�A ¼

�_ni¼1

diðx; yÞ�A: &

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Definition 1.5. Let X and Y be Boolean metric spaces over B. A map f : X�! Y issaid to be

(1) Contractive if dðfðxÞ; fðyÞÞ � dðx; yÞ for all x; y 2 X.(2) An immersion if dðfðxÞ; fðyÞÞ ¼ dðx; yÞ for all x; y 2 X.(3) An isometry if it is a bijective immersion.

Contractive maps play the role of morphisms in the category of Boolean metricspaces over B, while isometries are the isomorphisms. Observe that immersions arealways into.

Theorem 1.6. Every metric space X over B is isometric to a metrizable subset of aB-module. Furthermore, if we fix x0 2 X there is a metrizable subset S of a B-moduleM such that 0 2 S and an isometry g : X�! S such that gðx0Þ ¼ 0.

Proof. We define f : X�!BX by fðxÞ ¼ ðdðx; zÞÞz2X. To prove that fðXÞ ismetrizable and that f : X�! fðXÞ is an isometry, it is enough to see thatAnnðfðxÞ � fðyÞÞ ¼ dðx; yÞB for all x; y 2 X. If a 2 AnnðfðxÞ þ fðyÞÞ then,

aðdðx; zÞ þ dðy; zÞÞz2X ¼ 0;

so for z ¼ x, we have adðy; xÞ ¼ 0 and therefore a � dðx; yÞ. Conversely, supposea 2 dðx; yÞB, then

aðdðx; zÞ þ dðz; yÞÞ � adðx; yÞ ¼ 0;

for all z 2 X, so a 2 AnnðfðxÞ þ fðyÞÞ. For the last assertion, take h : fðXÞ �!fðXÞ þ fðx0Þ given by hðxÞ ¼ xþ fðx0Þ. Then, h is an isometry between fðXÞ andthe metrizable set S ¼ fðXÞ þ fðx0Þ because AnnðhðxÞ � hðyÞÞ ¼ Annðx� yÞ for allx; y. Hence, the map g ¼ h f is an isometry between X and S that verifiesgðx0Þ ¼ 0. &

1.2. Convex Combinations and Convex Closures

Unless otherwise stated, X will be a metric space over B.

Definition 1.7. Let x1; . . . ; xn 2 X and let a1; . . . ; an 2 B such that a1 � � � � � an ¼ 1.We will say that x 2 X is a convex combination of x1; . . . ; xn with coefficientsa1; . . . ; an if aidðx; xiÞ ¼ 0 for i ¼ 1; . . . ; n.

Proposition 1.8. If x 2 X is a convex combination of x1; . . . ; xn with coefficientsa1; . . . ; an, then for all y 2 X

dðx; yÞ ¼Mni¼1

aidðxi; yÞ

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Proof. For all i ¼ 1; . . . ; n, since aidðx; xiÞ ¼ 0, we have aidðxi; yÞ ¼aiðdðx; xiÞ þ dðxi; yÞÞ � aidðx; yÞ � aiðdðx; xiÞ _ dðxi; yÞÞ ¼ aidðxi; yÞ, so aidðx; yÞ ¼aidðxi; yÞ and hence, we have dðx; yÞ ¼ ðPi aiÞdðx; yÞ ¼

Pi aidðxi; yÞ. &

Proposition 1.9. If x and y are convex combinations of x1; . . . ; xn with coefficientsa1; . . . ; an, then x ¼ y.

Proof. By Proposition 1.8

dðx; yÞ ¼Xni¼1

aidðx; xiÞ ¼Xni¼1

ai

Xnj¼1

ajdðxjxiÞ ¼Xni¼1

Xnj¼1

aiajdðxj; xiÞ

Note that if i 6¼ j then aiaj ¼ 0 and if i ¼ j then dðxj; xiÞ ¼ 0, so all the terms in theabove sum are 0, and therefore dðx; yÞ ¼ 0. &

Lemma 1.10. Let S be a metrizable subset of an A-module M. Then,

convðSÞ ¼na1x1 þ � � � þ anxn 2 M : xi 2 S ai 2 BðAÞ

Mi

ai ¼ 1o

is also a metrizable subset of M.

Proof. Take x; y 2 convðSÞ, x ¼ Pn

1 aixi and y ¼ Pm

1 bjyj. Call cij ¼ aibj.Then,

Li;j cij ¼ 1 and x ¼ P

i;j cijxi and y ¼ Pi;j cijyj. Hence, Annðx� yÞ ¼

AnnðPi;j cijðxi � yjÞÞ ¼P

i;j cijAnnðxi � yjÞ which is a principal ideal becauseevery Annðxi � yjÞ is principal (recall that, for regular rings, any finitely generatedideal is principal). &

The following proposition will show that, when X is a metrizable subset of amodule, convex combinations in ðX;dÞ are exactly the corresponding linearcombinations in the module.

Proposition 1.11. Let S be a metrizable subset of an A-module, x; x1; . . . ; xn 2 S

and a1; . . . ;an 2 B such thatLn

i¼1 ai ¼ 1. Then, x is a convex combination ofx1; . . . ; xn with coefficients a1; . . . ; an if and only if x ¼ a1x1 þ � � � þ anxn.

Proof. Suppose x ¼ a1x1 þ � � � þ anxn. We must check that, for each i 2 f1; . . . ; ng,aidðx; xiÞ ¼ 0. It is clear that ai 2 Annðx� xiÞ ¼ dðx; xiÞA, so ai � dðx; xiÞ andhence, aidðx; xiÞ ¼ 0.

Conversely, suppose x 2 S is a convex combination of x1; . . . ; xn with coefficientsa1; . . . ; an . Let y ¼ Pn

1 aixi 2 convðSÞ, which is metrizable, by Lemma 1.10. Theimplication that we have already proved, tells us that y is a convex combinationof x1; . . . ; xn with coefficients a1; . . . ; an in convðSÞ. The same holds for x, so byProposition 1.9, x ¼ y. &

In general, in any metric space X, we will denote byPn

i¼1 aixi or bya1x1 þ � � � þ anxn the convex combination of x1; . . . ; xn with coefficients a1; . . . ; an,if it exists.

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Recall that Theorem 1.6 allows us to identify any metric space X over B witha metrizable subset of a B-module, and then, by Proposition 1.11, convex combi-nations are just the corresponding linear combinations in the module and the metricis the modular metric.

Contractive maps can be characterized as those that preserve convexcombinations.

Theorem 1.12. For a map f : X�! Y between two metric spaces ðX;dÞ and ðY ;d0Þthe following are equivalent:

(1) f is contractive.(2) For all x; x1; . . . ; xn 2 X and a1; . . . ; an 2 B with

Ln1 ai ¼ 1, if x ¼ P

aixi,then fðxÞ ¼ P

aifðxiÞ.

Proof. (1 ) 2) Let x ¼ Pi aixi. Then, for every i, we have 0 ¼ aidðx; xiÞ

aidðfðxÞ; fðxiÞÞ, so fðxÞ ¼ PaifðxiÞ.

ð2 ) 1Þ Given x; y 2 X, x ¼ dðx; yÞxþ dðx; yÞy. Hence, by our assumptionfðxÞ ¼ dðx; yÞfðxÞ þ dðx; yÞfðyÞ and making use of Proposition 1.8 we have finally

dðfðxÞ; fðyÞÞ ¼ dðx; yÞdðfðxÞ; fðyÞÞ þ dðx; yÞdðfðyÞ; fðyÞÞ¼ dðx; yÞdðfðxÞ; fðyÞÞ

and therefore dðfðxÞ; fðyÞÞ � dðx; yÞ. &

Given x1; . . . ; xn 2 X and a1; . . . ; an 2 B withL

ai ¼ 1, there may exist noconvex combination of the xi’s with coefficients ai’s. So we have the next definition:

Definition 1.13. A metric space ðX;dÞ over B is said to be convex if given anyx1; . . . ; xn 2 X and any a1; . . . ; an 2 B with

Lai ¼ 1, there exists in X the convex

combination of the xi’s with coefficients the ai’s.

This notion of convexity is different from the defined in Blumenthal and Penning(1961).

Definition 1.14. A convex closure of a metric space X is a convex metric spaceY � X such that any element in Y is a convex combination of elements of X.

Observe that every metric space X over B has a convex closure, because X isisometric to a metrizable subset S of B-module and in this case, the set convðSÞ ofLemma 1.10 is a convex closure of S.

Theorem 1.15. Let X � X and Y � Y be convex closures. Each contractive mapf : X �! Y extends to a unique contractive map �ff : X �! Y . Furthermore,

(1) �ff is immersion if and only if f is, and if f is isometry, so is �ff .(2) For two contractive maps f : X�! Y and g : Y �!Z we have gf ¼ �gg�ff .

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Proof. For each element x 2 X, choose an expression of x as a convex combinationof elements of X, like x ¼ P

i aixi. If we want �ff to be contractive it must be definedlike �ffðxÞ ¼ P

i aifðxiÞ 2 Y . This proves uniqueness. For existence we must checkthat, so defined, �ff is contractive. We take x; y 2 X, and their correspondingexpressions x ¼ P

aixi and y ¼ Pbjyj with xi; yj 2 X:

dð�ffðxÞ; �ffðyÞÞ ¼M

aibjdðfðxiÞ; fðyjÞÞ �M

aibjdðxi; yjÞ ¼ dðx; yÞ

If f is immersion then the inequality turns into an equality, and we deduce that �ff isan immersion. Property (2) is trivial and from this, using f�1, we deduce that if f isisometry so is �ff . &

As a corollary, we get that the convex closure of a metric space is unique, up toisometry, since if X � X1;X2 are two convex closures of X, then 1X extend to anisometry f : X1 �!X2.

In the sequel convðXÞ will denote a convex closure of X. We finish by statingsome elementary properties of convex spaces and convex closures.

Let X and Y be convex metric spaces over B and U � X. Then, the followinghold:

(1) The set of all convex combinations of elements of U in X is a convex closureof U (in this situation, the notation convðUÞ will refer to this set).

(2) If f : X�! Y is contractive, then fðconvðUÞÞ ¼ convðfðUÞÞ.(3) If X1; . . . ;Xn are metric spaces over B, then convðX1Þ � � � � � convðXnÞ is a

convex closure of X1 � � � � � Xn.

2. CFG-SPACES

Definition 2.1. A metric space X over B will be said to be a CFG-space (convexfinitely generated space) if it is the convex closure of a finite subspace.

Observe that

(1) If X is a CFG-space and f : X�! Y is contractive, then fðXÞ is a CFG-space.

(2) The product of a finite number of CFG-spaces is a CFG-space.

For technical reasons, it is convenient to work with pointed metric spaces. ðX; 0Þis said to be a (pointed) metric space if X is a metric space over B and 0 2 X. Amap f : ðX; 0Þ�!ðY ; 00Þ will mean a map f : X�! Y such that fð0Þ ¼ 00, andexpressions like x 2 ðX; 0Þ will mean simply x 2 X.

Throughout this section, we fix a convex metric space ðX; 0Þ. By Theorem 1.6,it is not restrictive to suppose that X is a metrizable convex subset of aB-module M and that 0 is the zero element of M. In ðX; 0Þ we will use the followingnotations:

� For x 2 X, jxj :¼ dð0; xÞ.� If x1; . . . ; xn 2 X and a1; . . . ; an 2 B are such that aiaj ¼ 0 whenever i 6¼ j,

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then we have an element of X:

a1x1 þ � � � þ anxn ¼ a00þ a1x1 þ � � � þ anxn

where a0 ¼ 1þ a1 þ � � � an (note that the right expression represents anelement of X since a0 � � � � � an ¼ 1 and X is a convex space). Such acombination will be called an orthogonal combination. As a particular case,ax ¼ axþ �aa0 for x 2 X, a 2 B.

� For x 2 X, Bx :¼ fax : a 2 Bg ¼ convð0; xÞ.� For x; y 2 X, xZ y :¼ dðx; yÞx.

Note that any contractive map f : ðX; 0Þ�!ðY ; 00Þ preserves orthogonalcombinations. In the following lemma, we state some elementary properties:

Lemma 2.2. Let x; y 2 X and a; b 2 B. Then:

(1) The maps jj : ðX; 0Þ�!ðB; 0Þ and xZ : ðX; 0Þ�!ðX; 0Þ are contractive,so both preserve orthogonal combinations.

(2) ax ¼ bx if and only if aþ b 2 jxjB (if and only if aþ jxjB ¼ bþ jxjB).(3) ax ¼ 0 if and only if a � jxj, and ax ¼ x if and only if a jxj.(4) The operation ðZÞ is commutative.

Proof. For Property 1, the function xZ can be expressed as a composition ofy 7!dðx; yÞ, b 7! �bb and b 7! bx and all of them are contractive.

Property 2. Suppose X is a metrizable subset of a B-module. Then, ax ¼ bx ifand only if aþ b 2 Annðx� 0Þ ¼ dðx; 0ÞB.

Property 3 follows from 2.

For Property 4, xZ y ¼ yZ x if and only if dðx; yÞxþ dðx; yÞ0 ¼ dðx; yÞyþdðx; yÞ0. This equality is easily checked verifying that the distance between thetwo terms is 0 using Proposition 1.8. &

Lemma 2.3. Bx \ By ¼ BðxZ yÞ for all x; y 2 X.

Proof. Just by the definition of Z we have BðxZ yÞ � Bx, and symmetrically, sinceZ is commutative, BðxZ yÞ � By, so one inclusion is proved. Now supposeu 2 Bx \ By. Then u ¼ ax ¼ by, and if we call c ¼ ab then cx ¼ bax ¼ bu ¼ bby ¼u ¼ aax ¼ au ¼ aby ¼ cy. Thus, cx ¼ u ¼ cy, and that implies c 2 Annðx� yÞ ¼dðx; yÞB (suppose X is a subset of a module) and u ¼ cx ¼ cdðx; yÞx ¼ cðxZ yÞ.

&

Proposition 2.4. For two elements x; y 2 X the following are equivalent:

(1) xZ y ¼ 0.(2) Bx \ By ¼ f0g.(3) dðx; yÞ ¼ jxj _ jyj.

In this case, x and y will be said to be orthogonal and we will write x ? y.

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Proof. (1,2) is a direct consequence of Lemma 2.3. For (1,3), we have (1) ifand only if 0 ¼ jxZ yj ¼ dðx; yÞjxj and, by symmetry, if and only if dðx; yÞjxj ¼ 0 ¼dðx; yÞjyj, which is equivalent to jxj; jyj � dðx; yÞ, and jxj _ jyj � dðx; yÞ. Theconverse of the latter inequality is always true by Axiom 3 of Boolean metricspaces. &

For x; y 2 ðB; 0Þ, we have xZ y ¼ dðx; yÞx ¼ ðxþ yþ 1Þx ¼ xy, so this conceptof orthogonality corresponds to disjointness in B.

Definition 2.5. A finite subset R � X will be said to be orthogonal if every twodifferent elements in R are orthogonal, and 0 62 R. If, moreover, X¼ convðR[f0gÞ,R will be said to be a reference system or a referential of ðX; 0Þ.

Proposition 2.6. Let R ¼ fx1; . . . ; xng be a reference system of ðX; 0Þ and x 2 X.There is a unique tuple ða1; . . . ; anÞ 2 Bn satisfying the three following properties:

(1) aiaj ¼ 0 whenever i 6¼ j.(2)

Pn1 aixi ¼ x.

(3) ai � jxij for i ¼ 1; . . . ; n.

Such a tuple will be called the tuple of coordinates of x with respect to R.

Proof. Uniqueness: IfPn

1 aixi ¼Pn

1 bixi in those conditions, multiplying byaibj, i 6¼ j, we obtain aibjxi ¼ aibjxj 2 Bxi \ Bxj ¼ f0g so for each i, aixi ¼aið

Pj bjÞxi ¼ aibixi, and symmetrically bixi ¼ aibixi ¼ aixi, so by Lemma 2.2

ai þ bi 2 jxijB, and also ai þ bi 2 jxijB since the ai and bi’s are assumed to verifyProperty 3. So ai þ bi ¼ 0 for all i.

Existence: Since X ¼ convf0; x1; . . . ; xng, we can find b1; . . . ; bn 2 B verifying1 and 2. Now set ai ¼ jxijbi. The ai’s satisfy trivially 1 and 3. Using Lemma 2.2, wededuce from ai þ bi ¼ jxijbi 2 jxijB that aixi ¼ bixi for all i. So

Pn1 aixi ¼

Pn1 bixi ¼ x.

&

Proposition 2.7. Let R ¼ fx1; . . . ; xng be a referential of ðX; 0Þ and ðY ; 00Þ aconvex metric space. Then, f :R �! Y is extensible to a (unique) contractive mapff :ðX; 0Þ �! ðY ; 00Þ if and only if jfðxiÞj � jxij for i ¼ 1; . . . ; n.

Proof. Define f on R [ f0g by fð0Þ ¼ 00. By Theorem 1.15, f admits such anextension if and only if it is contractive. If f is contractive, it is clear thatjfðxiÞj � jxij for i ¼ 1; . . . ; n, so one way is proved. Conversely, supposejfðxiÞj � jxij for every i. Then, for all i 6¼ j, since xi and xj are orthogonal, we havelast equality in dðfðxiÞ; fðxjÞÞ � jfðxiÞj _ jfðxjÞj � jxij _ jxjj ¼ dðxi; xjÞ. &

We check now that any CFG-space possesses a reference system.

Theorem 2.8. Suppose X¼ convf0;x1; . . . ;xng and that the set fx1; . . . ;xsg isorthogonal. Then, there exist asþ1; . . . ; an 2B such that fx1; . . . ; xs; asþ1xsþ1; . . . ; anxngnf0g is a referential of ðX; 0Þ.

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Proof. Let r ¼ cardfði; jÞ : xi Z xj 6¼ 0g. We make induction on r. We suppose thatthe theorem holds for any value lower than r > 0. We take xi; xj with xi Z xj 6¼ 0and suppose, without loss of generality that i; s < j. Let a :¼ dðxi; xjÞ. Sinceaxj Z xi ¼ adðxi; xjÞxi ¼ 0, we have axj ? xi. Also, xj ¼ aðaxjÞ þ �aaxi, and from thiswe deduce that convf0; xi; xjg ¼ convf0; xi; axjg and therefore:

X ¼ convf0; x1; . . . ; xj�1; axj; xjþ1; . . . ; xng

Making use of the induction hypothesis, the proof is complete (in this system ofgenerators there is at least one orthogonal pair more, since xi ? axj). &

Corollary 2.9. Let fx1; . . . ; xsg be an orthogonal subset of the CFG-space ðX; 0Þ.Then, there exist xsþ1; . . . ; xn 2 X such that fx1; . . . ; xng is a referential of ðX; 0Þ.

In particular, any CFG-space ðX; 0Þ has a reference system.

Definition 2.10. For U � X, U? ¼ fx 2 X : x ? y 8y 2 Ug.

Proposition 2.11. For two CFG-spaces ðU ; 0Þ � ðX; 0Þ, the space U? is aCFG-space and convðU [ U?Þ ¼ X.

Proof. Let fx1; . . . ; xmg be a reference system of ðU ; 0Þ that we can extend to areference system fx1; . . . ; xng of ðX; 0Þ. We prove that U? ¼ convf0; xmþ1; . . . ; xng.Take x 2 U?, x ¼ Pn

1 aixi. Then, for j ¼ 1; . . . ;m we have 0 ¼ xj Z x ¼Pn1 aiðxi Z xjÞ ¼ ajxj. Hence, x ¼ Pm

1 aixi. &

Proposition 2.12. Let ðU ; 0Þ � ðX; 0Þ be two CFG-spaces, ðY ; 00Þ a convex metricspace and f : ðU ; 0Þ�!ðY ; 00Þ and g : ðU?; 0Þ�!ðY ; 00Þ contractive maps. Then,there is a unique contractive map f ? g : ðX; 0Þ�!ðY ; 00Þ that extends f and g.

Proof. We take a referential of ðU ; 0Þ and another one of ðU?; 0Þ. The union is areferential of ðX; 0Þ. Applying Proposition 2.7 the proposition is proved. &

3. ALGEBRAIC GEOMETRY OVER CFG-RINGS

Definition 3.1. A regular ring A is said to be a CFG-ring if, equipped with itsmodular metric, it is a CFG-space over BðAÞ.

In this case, An (which is a metrizable A-module for which the product metricand the modular metric coincide) is a CFG-space over BðAÞ too. If p is a primenumber, any p-ring is a CFG-ring because if A is a p-ring, then A ¼ conv

f0; 1; . . . ;p� 1g. A proof of this fact can be found in Zemmer (1956, Corollary 1).There are CFG-rings that are not p-rings. For instance, take K a finite field and Oa set. Then, KO is regular and it is easy to see that the set of constant tuples constitutea finite system of generators of KO, so KO is a CFG-ring. The aim of this section is toprove Theorem 3.8.

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Lemma 3.2. Let R be a ring and f : Rn �!R a polynomial function. For everyx1; . . . ; xm 2 Rn and every e1; . . . ; en 2 BðRÞ such that e1 � � � � � � � � en ¼ 1, we havefðPi eixiÞ ¼

Pi eifðxiÞ.

Proof. Let S be the set of all maps g : Rn �!R verifying the conclusion of thelemma. It is straightforward to check that the projections pi : Rn �!R are in S (thatproves the lemma for the polynomials X1; . . . ;Xn), that constant maps are in S, andthat the sums and products of maps in S lie in S. Any polynomial map is a sum ofproducts of constants and the variables Xi’s. &

Lemma 3.3. Let A be a CFG-ring. A map f : An �!Am is contractive if and onlyif it is a polynomial map.

Proof. f is contractive if and only if all its components are, and the same holdsabout f being a polynomial map, so we can assume m ¼ 1. The ‘‘if’’ part is a con-sequence of Lemma 3.2, Theorem 1.12, and Proposition 1.11. For the ‘‘only if’’ part,we assume that fð0Þ ¼ 0 (it is sufficient to prove this case, just considering for anarbitrary f , the composition h f where h : A�!A is hðxÞ ¼ xþ fð0Þ). Takefx1; . . . ; xrg a referential of ðAn; 0Þ. We prove first the case n ¼ 1:

Case n¼ 1. Consider the polynomial giðxÞ ¼ xQ

j 6¼iðx� xjÞ for i ¼ 1; . . . ; r. Then,

eðgiðxiÞÞ ¼ eðxiÞYj 6¼i

eðxi � xjÞ ¼ eðxiÞYj 6¼i

dðxi; xjÞ

¼ eðxiÞYj 6¼i

jxij _ jxjj� � ¼ jxij ¼ eðxiÞ

Therefore, for each i ¼ 1; . . . ; n there is a unit ai of A such that giðxiÞ ¼ aieðxiÞ.Consider the polynomial map g : A�!A given by

gðxÞ ¼Xr

i¼1

a�1i fðxiÞgiðxÞ

Since A ¼ convf0; x1; . . . ; xrg and f and g are contractive, we prove that gðxÞ andfðxÞ coincide for x ¼ 0; x1; . . . ; xr . It is clear that gð0Þ ¼ 0 ¼ fð0Þ becausegið0Þ ¼ 0. For x ¼ xj,

gðxjÞ ¼Xr

i¼1

a�1i fðxiÞgiðxjÞ

and since giðxjÞ ¼ 0 whenever i 6¼ j,

giðxjÞ ¼ a�1j fðxjÞgjðxjÞ ¼ fðxjÞeðxjÞ

and this equals fðxjÞ because jfðxjÞj � jxjj ¼ eðxjÞ.

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General Case. As a consequence of the case n ¼ 1, we find that e : A�!A is a poly-nomial map, and therefore, for v 2 An, the map dð ; vÞ : An �!A is polynomial too,since if v ¼ ða1; . . . ; anÞ then dðx; vÞ ¼ eðx1 � a1Þ _ � � � _ eðxn � anÞ (recall thatx _ y ¼ xþ y� xy for x; y 2 BðAÞ). Hence, we can construct polynomial maps fori ¼ 1; . . . ; r given by GiðxÞ ¼ jxjQj 6¼i dðx; xjÞ. We define GðxÞ ¼ Pr

i¼1 fðxiÞ GiðxÞand we are going to see that G and f coincide on f0; x1; . . . ; xrg, so that, since bothare contractive and this set generates An, that will prove that f ¼ G, so f is a poly-nomial map. It is clear that Gð0Þ ¼ 0 ¼ fð0Þ because Gið0Þ ¼ 0 for all i. SinceGiðxjÞ ¼ 0 if i 6¼ j,

GðxiÞ ¼ fðxiÞGiðxiÞ ¼ fðxiÞjxijYi6¼j

dðxi; xjÞ

¼ fðxiÞjxijYj 6¼i

jxij _ jxjj ¼ fðxiÞjxij ¼ fðxiÞ

where the last equality follows from the fact that jfðxiÞj � jxij. &

Lemma 3.4. Let ðX; 0Þ ¼ convðHÞ be a convex metric space with 0 2 H, and letf : ðX; 0Þ�!ðY ; 00Þ be contractive. Then

f�1ð00Þ ¼ conv�0; jfðxÞjx : x 2 H

�

Proof. One inclusion is clear because all the elements that appear in the right termare in the convex set f�1ð00Þ. For the converse, if x 2 f�1ð00Þ, in particular it is in X,so it can be expressed like x ¼ Pn

i¼1 aixi with xi 2 H, and aiaj ¼ 0 whenever i 6¼ j.Then,

0 ¼ jfðxÞj ¼ a1jfðx1Þj � � � � � anjfðxnÞj;

and therefore aijfðxiÞj ¼ 0, so ai ¼ aijfðxiÞj for all i ¼ 1; . . . ; n. Finally,

x ¼Xni¼1

aixi ¼Xni¼1

aijfðxiÞjxi 2 conv�0; jfðx1Þjx1; . . . ; jfðxnÞjxn

�: &

Note that a convex subset of a CFG-space need not be a CFG-space. Forinstance, B ¼ convf0; 1g is a CFG-space, and those ideals of B that are not finitelygenerated are convex subsets that are not CFG-spaces.

Lemma 3.5. Let X be a CFG-space. Then Y � X is a CFG-space if and only ifthere exists a contractive map f : X�!B such that Y ¼ f�1ð0Þ.

Proof. One way is a direct consequence of Lemma 3.4. For the converse, supposeY is a CFG-space. If Y ¼ ; it is trivial and if not, take 00 2 Y and fu1; . . . ; ukg areferential in ðY ; 00Þ that we extend to a referential of ðX; 00Þ, fu1; . . . ; ung. ByProposition 2.7 we can define a contractive map f : ðX; 00Þ �!ðB; 0Þ such that

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fðuiÞ ¼ 0 if i � k and fðuiÞ ¼ juij if i > k. It is clear that Y � f�1ð0Þ and for the otherinclusion suppose x 2 f�1ð0Þ has coordinates ða1; . . . ; anÞ. Then

0 ¼ fðxÞ ¼ f

�Xni¼1

aiui

�¼

Mni¼1

aifðuiÞ ¼Mn

i¼kþ1

aijuij ¼Mni¼kþ1

ai

so ai ¼ 0 for i > k, and x ¼ Pk

i¼1 aiui 2 convf00; u1; . . . ; ukg ¼ Y . &

Corollary 3.6. If Y ;Z are CFG-spaces contained in the space X, then Y \ Z is aCFG-space.

Proof. If Y ¼ f�1ð0Þ and Z ¼ g�1ð0Þ with f ; g : convðY [ ZÞ�!B contractivemaps, then Y \ Z ¼ ðf _ gÞ�1ð0Þ. &

Corollary 3.7. Let f : X�! Y be a contractive map between CFG-spaces. If Z � Y

is a CFG-space, then f�1ðZÞ is a CFG-space too.

Proof. Let g : Y �!B be such that K ¼ g�1ð0Þ. Then f�1ðKÞ ¼ ðg fÞ�1ð0Þ, so it isa CFG-space. &

Theorem 3.8. Let A be a CFG-ring.

(1) A subset U � An is an algebraic variety if and only if U is a CFG-metricsubspace of An.

(2) A map f : U �!V between two algebraic varieties is a polynomial map ifand only if it is contractive.

Proof. If U is an algebraic variety then, U ¼ Tk1 f

�1k ð0Þ where fi : A

n �!A arepolynomial maps, and therefore, by Lemma 3.3, contractive maps. Using Lemma3.5 and Corollary 3.6 we deduce that U is a CFG-space. Conversely, if U is aCFG-space, by Lemma 3.5, there is a contractive map f : An �!BðAÞ withU ¼ f�1ð0Þ. Then U ¼ g�1ð0Þ where g is the composition An �!BðAÞ ,!A, that iscontractive and therefore polynomial, again by Lemma 3.3.

Suppose f : U �!V is contractive, choose some u 2 U and considerf : ðU ; uÞ�!ðV ; fðuÞÞ and k : ðU?; uÞ�!ðV ; fðuÞÞ the constant map. Since U andAn are CFG-spaces and V is convex, we can consider, by Proposition 2.12,f ? k : An �!Am that is contractive, and therefore, a polynomial map, that extendsf . The converse is a direct consequence of Lemma 3.3. &

4. STRUCTURE THEOREM FOR CFG-SPACES

We shall classify now CFG-spaces up to isomorphism. Reference systems do notgive good isomorphism invariants, since they are not unique up to isometry (forinstance, f1g and fa; �aag are non-isometric referentials of ðB; 0Þ). The right concept

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for this purpose is the following:

Definition 4.1. A referential fx1; . . . ; xng of ðX; 0Þ is said to be a base of ðX; 0Þ ifjx1j jx2j � � � jxnj.

We will prove that there exists a base for any pointed CFG-space ðX; 0Þ, and thatthey are unique in the sense of Theorem 4.6 below. We prove uniqueness first, andexistence afterwards.

Definition 4.2. Let k > 0 be an integer and X a metric space over B. The k-ideal of X(denoted by IkðXÞ) is the ideal of B generated byn Y

0�i<j�k

dðui; ujÞ : u0; . . . ; uk 2 X

o:

If IkðXÞ is principal, we will denote by akðXÞ its generator.

Lemma 4.3. If X ¼ convðHÞ, then IkðHÞ ¼ IkðXÞ for all k 2 N.

Proof. If U is any Boolean metric space, the map fU : Ukþ1 �!B givenby fUðu0; . . . ; ukÞ ¼

Q0�i<j�k dðui; ujÞ is contractive because it is a composition of

distance functions and a polynomial function (the product). With this notation,IkðUÞ is the ideal generated by the image of fU , and

ImðfXÞ ¼ fXðXkþ1Þ ¼ fXðconvðHkþ1ÞÞ ¼ convðfXðHkþ1ÞÞ¼ convðImðfHÞÞ

so both images generate the same ideal. &

Lemma 4.4. Let X be a CFG-space. Then IkðXÞ is principal for all k 2 N and thereexists n 2 N such that IkðXÞ ¼ 0 for all k n. Hence, akðXÞ exists for all k 2 N andakðXÞ ¼ 0 for k n.

Proof. Suppose X ¼ convðHÞ with H finite. Then, IkðXÞ ¼ IkðHÞ is always a finitelygenerated ideal of B, so it is principal, and if we take n ¼ cardðHÞ, 0 ¼ IkðHÞ ¼ IkðXÞif k n. &

Lemma 4.5. Let fx1; . . . ; xng be a base of ðX; 0Þ. Then, akðXÞ ¼ jxkj for k � n andakðXÞ ¼ 0 if k > n.

Proof. By Lemma 4.3, akðXÞ ¼ akðHÞ where H ¼ f0; x1; . . . ; xng. If k > n, it istrivial that akðHÞ ¼ 0. If k � n, call yi’s to the reordering of the xi’s such that0 ¼ jy0j �jy1j � � � � � jynj (yr ¼ xn�rþ1 if r > 0). For i < j we have dðyi; yjÞ ¼jyij _ jyjj ¼ jyjj, by orthogonality. We wonder whether IkðHÞ ¼ jxkjBð¼ jyn�kþ1jBÞ.One inclusion is because

jyn�kþ1j ¼Y

ni>n�k

jyij ¼Y

ni>jn�k

dðyi; yjÞ

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is one of the generators of IkðHÞ. For the other inclusion, we shall check thatall the generators of IkðHÞ are in the ideal jyn�kþ1jB. Take U ¼ fu0; . . . ; ukg � H.By a cardinality argument, there must exist indices r< s � n� kþ 1 such thatyr ; ys 2 U , so

Y0�i<j�k

dðui; ujÞ � dðyr ; ysÞ ¼ jysj � jyn�kþ1j: &

Theorem 4.6. If fx1; . . . ; xng is a base of ðX; 0Þ and fy1; . . . ; ymg is a base of ðX; 00Þ,then n ¼ m and jxij ¼ jyij for i ¼ 1; . . . ; n. Moreover, there exists an isometryf : ðX; 0Þ�!ðX; 00Þ such that fðxiÞ ¼ yi for i ¼ 1; . . . ; n.

Proof. By Lemma 4.5, we know that n ¼ maxfk : akðXÞ 6¼ 0g ¼ m andjxij ¼ aiðXÞ ¼ jyij. About the last assertion, there exists a contractive mapf : ðX; 0Þ �! ðX; 00Þ such that fðxiÞ ¼ yi for i ¼ 1; . . . ; n, by virtue of Proposition2.7. It is an isometry because we can find its inverse in an analogue wayg : ðX; 00Þ �!ðX; 0Þ with gðyiÞ ¼ xi. &

Lemma 4.7. If ðV ; 0Þ � ðX; 0Þ are CFG-spaces and V? ¼ f0g, then V ¼ X.

Proof. A reference system of ðV ; 0Þ, fx1; . . . ; xmg can be extended to a referential ofðX; 0Þ, fx1; . . . ; xng. Then, xmþ1; . . . ; xn 2 V? ¼ f0g, so V ¼ convf0; x1; . . . ; xmg ¼convf0; x1; . . . ; xng ¼ X. &

Lemma 4.8. Let X be a CFG-space and f : X�!B contractive. Then, there existsu 2 X such that fðuÞ ¼ maxffðxÞ : x 2 Xg.

Proof. Suppose X ¼ convfx0; . . . ; xng. The set fðXÞ � B is closed under theoperation ð_Þ, because it is convex and a _ b ¼ aaþ �aab. Hence, there exists u 2 X

such that fðuÞ ¼ fðx0Þ _ � � � _ fðxnÞ. If x 2 X, we express it as a convex combinationx ¼ P

i aixi and fðxÞ ¼ Li aifðxiÞ ¼

Wi aifðxiÞ �

Wi fðxiÞ ¼ fðuÞ. &

Theorem 4.9. Any CFG-space ðX; 0Þ possesses a base.

Proof. We define by recursion a sequence ðxnÞ1n¼1 in X and a sequence ðUnÞ1n¼1 ofCFG-spaces contained in X:

� x1 is such that jx1j ¼ maxfjxj : x 2 Xg; U1 ¼ convf0; x1g.� Given xi and Ui for i < n, we take xn such that jxnj ¼ maxfjxj : x 2 U?

n�1gand Un :¼ convf0; x1; . . . ; xng.

Note that those maximums exist by virtue of Lemma 4.8, since U?n�1 is a CFG-space

by Proposition 2.11. The xi’s form an orthogonal set and verify jxij jxjj wheneveri < j. Therefore, fx1; . . . ; xngnf0g is a base of ðUn; 0Þ. Since X is a CFG-space, byLemma 4.4, there must exist some k > 0 with 0 ¼ akðXÞ akðUkÞ ¼ jxkj. So, takingr the largest integer such that jxrj 6¼ 0, we have, just by the definition of xrþ1 ¼ 0,

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that U?r ¼ f0g. Therefore, by Lemma 4.7, Ur ¼ X and we have already shown that

fx1; . . . ; xrgnf0g is a base of ðUr ; 0Þ. &

Theorem 4.10. Two CFG-spaces X and Y are isometric if and only ifakðXÞ ¼ akðY Þ for all k 2 N.

Proof. Suppose akðXÞ ¼ akðY Þ for all k. Choose 0 2 X, 00 2 Y and basesfx1; . . . ; xng and fy1; . . . ; ymg of ðX; 0Þ and ðY ; 00Þ, respectively. Then,n ¼ maxfk : akðXÞ ¼ akðY Þ 6¼ 0g ¼ m and we can construct an isometry like in theproof of Theorem 4.6. &

ACKNOWLEDGMENTS

The author wishes to thank Professors Juan Martınez and Manuel Saorın, of theUniversity of Murcia, and Sergiu Rudeanu, of the University of Bucharest, for theirsupport and stimulus, and for their help in the redaction of this article. He wassupported by FPU grant of SEEU-MECD, Spain.

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Batbedat, A. (1971). Distance Booleenne sur un 3-anneau. Enseignement Math.17:165–185.

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