Automata, Languages and Machines. Volume B by Samuel Eilenberg

Machine Theory

By Samuel Eilenberg

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4. Let X Xv Y o Z where X is a cts while Y and Z are ts’s. Show that there is a complete ts Z’ c Z such that X Y o Z‘ where y’ is the restriction of y . 5. t: Q X xQ y -+ Q y be the projection. Prove that XOY-qY n-’ State necessary and suficient conditions for Y

Since card b = card a, the inclusion b c as implies b = as. Condition b a implies that a c bt for some t E: S u 1,. By the argument above, we have a = bt and thus a = ast. Since st defines a permutation of a, some power (st)”, n > 1, is the identity on a. Define S = t(st)”-l. Then qsS = q for all q E a. This implies qsSs = qs and therefore q‘is = q’ for all q’ E as = b. 2. 6) all the singletons of Q. In this set A we have the preorder and the equivalence as defined above (for all subsets of 9).

Thus t y17 @ T and card T < card S.

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