2. CHAINS AND TRACES 11
Definition 2.6 (Catenation). Let x, y, z α β. The catenation of two
(α, β)-traces Λ = (x, y, w) and Λ = (y, z, w ) is defined by
Λ#Λ := (x, z, w + w ).
Let u D(x, y) and u D(y, z) and suppose that u and u are constant near the
ends ±1 D. For 0 λ 1 sufficiently close to one the λ-catenation of u and
u is the map u#λu D(x, z) defined by
(u#λu )(ζ) :=



u
ζ+λ
1+λζ
, for Re ζ 0,
u
ζ−λ
1−λζ
, for Re ζ 0.
Lemma 2.7. If u D(x, y) and u D(y, z) are as in Definition 2.6 then
Λu#λu = Λu#Λu .
Thus the catenation of two (α, β)-traces is again an (α, β)-trace.
Proof. This follows directly from the definitions.
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