# Algebraic K-theory. Proc. Oberwolfach 1980 by R. Keith Dennis

By R. Keith Dennis

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**Example text**

Furthermore, assume that there exist some so E S and x,y E X* such that SO, x) = SO, y ) . T h e n for any s E S we have 6(s, x) = 6(s, y ) . e. S = G I and 6 = 6,. Then S(s0,x) = 6(s,,y) implies so@(x)= soQ(y), and h h CHAPTER 1. GROUP-MATRIX T Y P E AUTOMATA 44 thus we have Q(z) = Q ( y ) . Therefore, for any s E S (= have sQ(z) = sQ(y). = G) we The following is the main theorem in this section. 1 Let A = (Fn, X ,6 ,) be a strongly connected (n,G)automaton where n is a p r i m e number. T h e n A is not regular if and only if the following two conditions are satisfied: (1) For any Y E Q ( X * ) there exist some IJ f S ( n ) and G, i, j = 1 , 2 , .

In this case, for any Y E ' k ( X * )there exist a permutation 7 on {1,2, . . , n} and elements g i j E G, i, j = 1 , 2 , . . , n such that Y = (gPqePT(,))where e is the identity of G. Let m be a positive integer such that T~ = 1 where 1 is the identity permutation on {1,2, . . ,n}. Then there exist some elements hij E G, i , j = 1 , 2 , . . , n such that Y m = (hpqepq) E '@(X*). Notice that (epq)is the identity of ' k ( X * )and that ' k ( X * )is a finite monoid. Therefore, ' k ( X * )must be a group.

3 . j called the characteristic monoid of A . 1 Let A = ( S , X , 6 ) and B = ( T , X , y ) be two isomorphic automata. T h e n C ( A )= C ( B ) . Then for any x,y E X* and s E S , S ( s , x ) = S(s,y) if and only if y(p(s),x) = y ( p ( s ) , y). Since p is a bijective mapping, this means that, for any s E S , 6 ( s ,x) = 6(s, y) if and only if ~ ( sz) , = y(s, y). Hence C ( A )= C ( B ) . (G,X , 6,p) be a n (n,G)-automaton. T h e n the characteristic monoid of A is isomorphic to Q f ( X * ) . 2 Let A = Proof Obvious from the fact that, for any x,y E only if @(z)= @(y).