By Masami Ito

ISBN-10: 9810247273

ISBN-13: 9789810247270

Even if there are a few books facing algebraic thought of automata, their contents consist more often than not of Krohn–Rhodes concept and similar subject matters. the themes within the current ebook are fairly various. for instance, automorphism teams of automata and the partly ordered units of automata are systematically mentioned. in addition, a few operations on languages and exact periods of normal languages linked to deterministic and nondeterministic directable automata are handled. The e-book is self-contained and accordingly doesn't require any wisdom of automata and formal languages.

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**Example text**

Here, h is an arbitrary element in H . 6 Direct product of automata In this section, we will deal with direct products of group-matrix type automata and their automorphism groups. 1 Let A = (S,X , 6 ) and B = (T,X , y) be two automata. The direct product, A x B,is the automaton A x B = ( S x T , X , 6 x y) where 6 x y ( ( s , t ) , a ) = ( S ( s , a ) , y ( t , a ) )for any ( s , t ) E S x T and a E X . Before introducing the notion of the direct product of groupmatrix type automata, we define the product of two group-matrices.

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).

Then fij = x g i k h k j fzT'. n # 0 and k=l h r j # 0 , we have Hi C_ girHrg,' and Hr & hrjHjhG1. e. Hi C_ f i j H j f i l . This completes the proof of (gpqHq)(hpqHq) E G(H(,)). such that fij = C g i k h f i j = g i r h r j . 4 Let H ( n ) = { H I ,Hz, H3,. . , H,} be a family of subgroups of a given finite group G satisfying the SP-condition. Then a vector ( f p H p )is called a generalized group-vector of order n over ( H ( n ) , G )and denoted ( f p H p )E 6 ( H ( n ) )if (fp) E G^, holds.

### Algebraic Theory of Automata and Languag by Masami Ito

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