By Masami Ito

ISBN-10: 9810247273

ISBN-13: 9789810247270

Even though there are a few books facing algebraic concept of automata, their contents consist usually of Krohnâ€“Rhodes idea and similar themes. the themes within the current publication are really various. for instance, automorphism teams of automata and the partly ordered units of automata are systematically mentioned. furthermore, a few operations on languages and detailed periods of normal languages linked to deterministic and nondeterministic directable automata are handled. The ebook is self-contained and for that reason doesn't require any wisdom of automata and formal languages.

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**Example text**

Therefore, A x A' holds and also A' is strongly connected. 6, G(A) M G(A') holds. This means that A' is regular. 1 Let A = ( S , X , S ) be a perfect automaton and let G be a group such that G M G(A). Then A is isomorphic to some (1,G)-automaton. 3. 5) Let A = ( S ,X, 6 ) be a strongly connected automaton. T h e n if IS1 = IG(A)I, A i s a permutation automaton. Proof By JSJ= JG(A)J,A is isomorphic to some (1,G)-automaton. 2, A is a permutation automaton. 1 Let A = X , 6,) be a regular ( n ,G)-automaton.

O)@(Z)= ( 0 , . . , O , O t ( g ' ) , 0 , . . , 0) and O ( ( g g ' , 0 , . . , O ) ) = O ( ( e ,0 , . . , O)YZ)= O ( ( e ,0 , . . , O ) ) @ ( Y Z=) (0,. . ,0, ht,O,. . ,O)@(YZ)= (0,. . ,O,Ot(gg'), 0 , . . , O ) . Hence the ( t ,t)entries of @ ( Y )@(Z) , and @(YZ) are h F I O t ( g ) ,h r l O t ( g ' ) and hFIOt (gg'), respectively. Notice that @ ( Y Z )= @(Y)@(Z). Consequently, hF1@t(gg') = h ; ' O t ( g ) h F 1 O t ( g ' ) and cp(gg') = cp(g)'p(g'). h) (-+) Assume that there exist cp, ki, i = 1,2,, .

N. By the fact that E is a regular system in G,, we can see that for any g E G there exist some group-matrices (ypq), ($,) E E" such that yT(+(i) = g and yb(i)T(j) = e. Thus we have @i(g)h,lhj = @ j ( g ) for any g E G and i , j = 1 , 2 , . . ,n. , 0)) = (0, . . , o ) , t = 1 , 2, . . ,n. P u t cp(g) = h ; l @ d g ) , g E G and ki = hF1hi E G , i = 1 , 2 , . . , n. Then we have h ; ' O j ( g ) = h;l&(g)h;'hj = k i l @ ( g ) k j for any g E Go a n d i , j = 1 , 2 , . . ,n. First we prove that cp is an automorphism of G.

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