12.3Linearity

We have already remarked that the operation t 7!t u is strict, i.e. preserves ?. Better than this it is linear. Let us look now at what that can mean. Let E be the function from A ! B to B de ned by

E(f) = f(a) where a is a given object of A.

Let us work out Tr(E): we have to nd all the 2 E(f) with f minimal. Now 2 E(f) = f(a) i there exists some a a such that (a ; ) 2 f. So the minimal f are the singletons f(a ; )g with a a, a nite, and the objects of Tr(E) are of the form

(f(a ; )g; ) with 2 jBj, a a, a nite.

A stable function F from A to B is linear precisely when Tr(F ) consists of pairs (f g; ) with 2 jAj and 2 jBj.

12.3.1Characterisation in terms of preservation

Let us look at some of the properties of linear functions.

ii) if a = a0 [ a00, then F (a) = F (a0) [ F (a00), so 2 F (a0) or 2 F (a00); so, if a is not a singleton, we can nd a decomposition a = a0 [ a00 which contradicts the minimality of a.

Properties (i) and (ii) combine with preservation of ltered unions (Lin):

if A A, and for all a1; a2 2 A, a1 [ a2 2 A,

SS

then F ( A) = fF (a) : a 2 Ag

Observe that this condition is in the spirit of coherence spaces, which must be closed under pairwise-bounded unions. So we can de ne linear stable functions from A to B by (Lin) and (St):

if a1 [ a2 2 A then F (a1 \ a2) = F (a1) \ F (a2)

the monotonicity of F being a consequence of (Lin).

12.3.2Linear implication

We strayed from the trace to give a characterisation in terms of preservation. Returning to it, if we know that F is linear, we can discard the singleton symbols in Tr(F ):

Trlin(F ) = f( ; ) : 2 F ( )g

The set of all the Trlin(F ) as F varies over stable linear functions from A to B forms a coherence space A ( B (linear implication), with jA ( Bj = jAj jBj and ( ; ) _^ ( 0; 0) (mod A ( B) if

i)_^ 0 (mod A) ) _^ 0 (mod B)

ii)^_ 0 (mod B) ) ^_ 0 (mod A)

in which we introduce the abbreviation:

^_ 0 (mod A) for :( _^ 0) or = 0

for incoherence.