Journal of Geometry. Vol. 5/1 1974. Birkh~user Verlag Basel

INDEPENDENT AXIOMS FOR CONVEXITY

Victor Bryant

Join-structures or Convexity Spaces generalise the geometry of Vector Spaces by means of axioms concerning line segments. Most other generalisations of this type are just particular examples of Convexity Bpaces. In the many papers on this subject the collection of axioms is too long: in this short note we exhibit an independent set of axioms for these structures.

Let X be a non-empty set, let a,b,.., be elements

of X and A,B,... subsets of X. We do not distinguish

between an element of X and the singleton subset which

it defines. Thus in X the notation ~ is redundant and

is replaced by ~ . Also we write A = B to mean A

meets B or A ~ B + ~. A join is a mapping

�9 : X x X + 2 X, i.e. it associates with each ordered

pair of elements of X a subset a.b (or simply ab) of X.

Given a join we can define a new operation

/ : X x X § 2 X by a/b = {x : a Cbx}. The operations

�9 , / easily extend to subsets, for example AB is

~J{ab : a cA,b CB}. In particular a(bc) and (ab)c are

subsets of X. Note that A/B = C if and only if A = BC.

The most common example of this situation is when

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X is a real vector space and

a.b = {%a + (I-%) b : O < X < I}, and in that case

a/b = {%a + (l-%)b : % > I}. The axioms which we con-

sider have a strong geometric motiviatlon based upon

this example.

DEFINITION. A pair (X,.) is a Convexity Space (or

Join-structure) if �9 is a join on X satisfying

(i) ab # r

(2) a/b ~ r

(3) aa = a = a/a

(4) ab = ba

(5) (ab)c = a(bc)

(6) a/b = c/d => ad = bc

for all a,b,c,dcX.

Some of the consequences of these axioms are

studied in E2-~ and E6~. Other similar axiomatic

approaches can be found, for example, in ~-3] - E53.

THEOREM. If �9 is a J@in on x satisfying

(I) a/b # r

(II) aa = a = a/a

(III) (ab)cCa(bc)

(IV) a/b = c/d => ad = bc

for all a,b,c,d ~X, then (X,-) is a Convexity Space.

Proof. Assume that (X,') satisfies I - IV. We show

that properties i, 4 and 5 hold and it will then follow

that (X,') is a Convexity Space.

(i) ah + r

For each a,bCX we have, by I, a/b ~ @. Thus

a/b = a/b and by IV ab = ba. Hence ab~ab ~ ba ~ r

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as required.

(4) ab = ba:

We show that for any a,b abcba and then 4 will

follow by symmetry. So let c Cab and by 1 above we

may choose d Ccb. Then b C d/c ~c/a and so by IV

da = cc. Thus by II and III

c = cc = daC(cb)a ~c(ba)

and so c cc(ba), c/c = ba and c cba as required.

(5) (ab)c = a(bc):

By III and 4 above we have

(ab)c ~a(bc) = (bc)acb(ca) = (ca)b~c(ab) = (ab)c

and the result follows. (Note that this is an example

of the construction in (iii) of ~I~.)

Hence (X,.) is a Convexity Space as claimed.

We conclude by giving examples to show that the

five properties I, IIi (aa = a), IIii (a/a = a), III

and IV are independent. The examples chosen are not

the simplest but they do illustrate why the particular

property fails.

EXAMPLE I. If X = {x ~ R : O $ x $ I} and �9

denotes the usual Vector Space join defined above,

then (X,') satisfies II, III and IV but 0/I = @.

EXAMPLE lli. If X = R and joins are defined by

a.b = {xC R : either x < min {a,b} or x -- b},

then clearly (X,-) satisfies I and IIii. Also

(ab)c = {x : x < min {y,c} some y Cab, or x -- c}

= {x : x < min {a,b,c}, x < min {b,c} or x = c}

= {x : x < min {b,c} or x = c}

= bcC a(bc)

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and so III holds. Furthermore ad = bc for all

a,b,c,dcX and so IV trivially holds. However

aa = {xCR : x ~ a} and so property lli fails.

EXAMPLE llii. If X = {O,I} and �9 is defined by

O.O = O, I.i = i, O.i = i.O = {O,I}, then (X,') satis-

fies I, lli, III and IV. However property llii fails

since I/i = {0,i}.

EXAMPLE III. Let X = R and let

a.b = f a if a = b

L R\{a,b} if a + b

Then in this case a/b = ab for each a,b and it is easy

to check that (X,') satisfies I, II and IV. For

example if ad + bc then exactly three of a,b,c and d

must be the same point a~d then a/b = ab # cd = c/d.

However property III fails for if a + b then

(ab)b = (R\{a,b})b = R~b

and a(bb) = ab = R\{a,b}

whence (ab)b~a(bb).

EXAMPLE IV. If X = R2\(0,0) and the join is

defined as the usual Vector Space join excluding (0,0),

then (X,') satisfies I, II and III. However

(2,2) C(O,I)/ (-2,O) rl (I,O)/ (0,-2)

and (O,i) . (O,-2)~(-2,O) . (I,O) = ~.

Thus property IV fails in this case.

We have thus shown that the properties I - IV

form independent axioms for a Convexity Space or Join-

structure. Sincere thanks are due to Roger Webster of

Sheffield for his original work concerning the axioms.

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REFERENCES

I. V. W. Bryant: Reducing classical axioms, Math. Gazette, 391 (1971) 38-40.

2. V. W. Bryant and R. J. Webster: Generalizations of the theorems of Radon, Helly and Caratheodory, Monatsh. Math., 73 (1969) 309-315.

3. J. W. Ellis: A general set-separation theorem, Duke Math. J., 19 (1952) 417-421.

4. A. Ghika: Separarea multimilor convexe in spatii lineate non- vectoriale, Acad. R.P. Romine Bul. Sti. Sect. Sti. Mat. Fiz., 7 (1955) 287-296.

5. V. Havel: Join systems and closure spaces, Comment Math. Univ. Corolinae, 7 (1966) 335-341.

6. W. Prenowitz: A contemporary approach to classical geometry, Amer. Math. Monthly, 68 (1961) Appendix 1-67.

Department of Pure Mathematics, The ~liversity of Sheffield, Sheffield $3 7RH, England.

(Eingegaugen am ~5. Februar 1974)

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