Categorical Topology Proceedings of the International Conference, Be
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Vol. 521: G. Cherlin, Model Theoretic Algebra - Selected Topics. IV, 234 pages. 1976.
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f(x)+g(x)
The relation
h = f+g can then be c h a r a c t e r i z e d Recall
that,
in EBAN,
w i t h the usual whereas
A ~ C differs
f,g,h:
of all pairs
~ Nxll for
to EBAN and
A ÷ B given by
in EBAN by the following
operations
this
IEl.
(x e A) again belongs maps
of the
step towards in
such that ilf(x)+g(x)il
between
A x C consists
componentwise
The m a i n
of addition
If f and g are any maps A ÷ B in EBAN
will be called
of p r o p e r t i e s
considerations:
[x~y), x e A and y £ C,
and the norm II (x,Y)ll = max{ilxIi,llYli~,q
from A x C only in its norm which
is II (x,y)ll =
llxli+nlyil; since max{llxll,llyil} ~ nlxli+llyll, the m a p j: A@C ÷ ArC, w i t h identity effect,
belongs
to EBAN.
Formally,
j is defined
by the specifications
p j u = IA, qju = OA, pjv = OC, qjv = 1 C where u: A ÷ A~C and v: C ÷ A@C are the coproduct and q: A r C ÷ C the product
projections. P
A
A A is the diagonal
embedding
m a y be v i e w e d as the subspace (x,x) =
(y,z),
it consists
also be r e p r e s e n t e d
by A (2) w h i c h
the p u l l b a c k
A
(A@A) of all
(x,(x,x)), is o b t a i n e d
(x,(y,z))
W i t h these maps,
pullback
we o b v i o u s l y
diagram. have
Since P
such that
w i t h norm 211xll. Hence, from A by doubling
We let iA: A (2) ÷ A and a (2) A : A (2) ÷ A @ A be the vertical map in the c o r r e s p o n d i n g
diagram
A@A
and JA the m a p just discussed.
of A x
of all
consider
and p: ArC ÷ A
.~ AXA A
where
Now, >
embe
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