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Posted On April 21, 2018 at 1:02 am by / Comments Off on Read e-book online Combinatorial Functors PDF

By J. N. Crossley, Anil Nerode (auth.)

ISBN-10: 364285933X

ISBN-13: 9783642859335

ISBN-10: 3642859356

ISBN-13: 9783642859359

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And converse/y. Proof The transformations F ~ Fand F ~ F~ both commute with composition so it suffices to show that if F, Gare strict finitary co mb inatorial functors then F = G if, and only if, F# = G#. But if F = G then for m: of finite dimension n, F ((m:) ) = G ( m:») so dirn F ( m:) ) = dimG(m:») so F#(nl=G#(n). 6, Fand Gare naturally equivalent on objects so F = G. D We note that the above corollary also holds when F#, G#, ... are strict combinatorial series as well as when they are strict combinatorial functions.

2 we produce a (Myhill) precombinatorial operator G: °

Then there is an elementary monomorphism pi Jrom 9J( into 9J( such that pi extends p and p(9J()nN=p. Proof Since 9)( is countable let mo , ml , ... be an enumeration of 9'R beginning with the elements of A. Let Ao=A and for i>O, let A i = cl {mo, ... , mi_I}' then A o c;: Al c;: ... and U Ai =9J(. Now set Po = P and then using the Duplication Lemma recursively define elementary isomorphisms Pi with domain Ai such that Pi extends Pj for j < i and p;(A;) n N =0. Then pi = UPi is the required isomorphism.

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Combinatorial Functors by J. N. Crossley, Anil Nerode (auth.)

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