By Samuel Eilenberg

ISBN-10: 0080873758

ISBN-13: 9780080873756

ISBN-10: 0122340027

ISBN-13: 9780122340024

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**Extra info for Automata, Languages and Machines. Volume B**

**Example text**

2 we then have elements u, ii E S such that b = au, a = bzi and uii is the identity on a ziu is the identity on For each s E S such that as =a b we then have b(iisu) = am = au =b Similarly, if bt = b, then a(utE) = a. It follows that u maps Ba bijectively onto Bb and yields the required isomorphism Ha w Hb I 7. The Holonomy Decomposition Theorem Let X = (Q, S ) be a ts and let h be a height function for X with Qh = n > 0. Given 0 < i 5 n let be a set of representatives of the equivalence classes of elements of A of height exactly i.

ULk Define T = L, L = L,, U ... ULk Then T is a subsemigroup of S and L u T = S . 5 yields X < Z o P with Z=(Q,L), If A, = Y=(Tul,,T) L , . Thus card L < card S and Qs, then s 217 If A, = Qt, then t Consequently $

O Y, is closed it follows that P < P, ... 0 F,, 0 II. Decomposition Theorems 42 Thus x < 8,0 . . 2 for X , in view of the fact that S' < s. Case 3. K > 1. Define Li = {S I s E S, QS c A i } for 1 5 i 4 k . Then Li is a left ideal in S and the maximality of the sets Ai implies s = L , u ... ULk Define T = L, L = L,, U ... ULk Then T is a subsemigroup of S and L u T = S . 5 yields X < Z o P with Z=(Q,L), If A, = Y=(Tul,,T) L , . Thus card L < card S and Qs, then s 217 If A, = Qt, then t Consequently $

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