By Masami Ito

Even supposing there are a few books facing algebraic concept of automata, their contents consist more often than not of Krohn–Rhodes concept and similar themes. the subjects within the current booklet are fairly diverse. for instance, automorphism teams of automata and the in part ordered units of automata are systematically mentioned. additionally, a few operations on languages and precise periods of normal languages linked to deterministic and nondeterministic directable automata are handled. The ebook is self-contained and as a result doesn't require any wisdom of automata and formal languages.

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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. = n ( G Jand G(A) M G hold. Then Consequently, we have the following result: T h e determination of any distinct strongly connected automata whose automorphism groups are isomorphic to a given finite group G is equivalent t o that of any distinct regular group-matrix type automata of each positive integer's order o n G . Notice that we do not consider two isomorphic automata as distinct ones.

1 , j = 1 , 2 , .. , r , 2,r 2 1 holds. Then we have gQ(u1)Q(u2) . . * ( a [ ) = gQ(bl)Q(b2) . . Q(b,) for any g E G,. l))Q’(E(a2>) . . ) . for any 6 E G^,. x(<(bd G,Q’(t(a1))Q’(E(a2)). . Q ’ ( t ( a l ) >= Q’(t@l))Q’(t@2)). . Q ’ ( t ( b ) ) must hold. Thus, we obtain the first part. As for the second part, we can prove it in a similar way. Consequently, Q, is well defined. Next, we can prove easily that Q, is a surjective mapping of Q(X)*(= Q(X*)) onto Q’(X)*(= Q’(X*)). Moreover, it is easily seen that @ is a homomorphism.

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. 2, C ( A )forms a group. (+) For any Y E '@(X*), Y is of the form gpqepT(q) where g i j E G, i, j = 1 , 2 , . . , n and 7 is a permutation on {1,2,. . , n}. Because, if' it is not the case, Y m# (epq)for any positive integer m. This contradicts the assumption that C ( A )is a group.