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By Samuel Eilenberg

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Show that A is infinite z f f n 5 I s I i 211 f o r some s E A . 2. Let JV' = ( Q , I , T ) be a E-automaton such that = ( E p ) * where p 2 0 is an integer. Show that card Q 2 p . 3. 1, 5 . 8. 6. Local Sets Let d = (0,I , T ) be a 2'-automaton and let Q set of edges in JV'. *B - Q"CQ" = { ( p , 0,q ) } be the 6. Local 27 Sets where the sets A c Q, B c Q, C c Qe are defined as We also consider the very fine morphism f: sz* + z*, (p, G, q ) f =G Let c E #-d. -d. 2) we deduce that c is a successful path in &' and that 1 c 1 # 1.

Obvious. (vi) (vii). Obvious. is complete (vii) 3 (i). Let ,d=(Q, i, 2') and s E P. T h u s su E A I a. 4 hold. If Ld-, is not complete, then s-*A = for some s E 2". T h e completion d,Lc of is then obtained by adjoining a sink state 0. l is that the condition s-'A # (3is no longer imposed. We shall call L ~ . , c the complete minimal automaton of A. I l l . 2 and Corollary 5 . 3 . 5. For any complete accessible Z-automaton ' A = (Q, i, T ) with behavior A , there exists a unique state-mapping p: Ld+- d .

Let L‘= {a,T } . Consider r a T h e behavior is the set A = (T u o+T’)* u T*O+ u T*LT+T and consists of all words s E {a, T}” which do not contain ment. 7. (TTU as a seg- Let S = {a, T ) . 8. u a%* = a*t*. Let S = {a, T } . T h e set A I >_ 0} = { ( T ” T ~ TZ is not recognizable. Indeed let A? be an automaton such that A = 1 &’ 1. For each n 2 0 we then have a successful path . tn dn en --* qn --* tn such that I c, I = a n , I d , I = T,. We claim that the states q,! must be distinct. Indeed if q,t = qm with m # 12, then is a successful path with I c,,d,,, 1 niteness of the set of states of LP’.

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