By Carl Faith
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Additional resources for Algebra II. Ring Theory: Ring Theory
5. Each automaton image of the Moore is a homomorphic (in states) automaton. Proof. Let us c o n s t r u c t a new automaton 9=(AxB,X,B) by the automaton 9=(A,X,B) s e t t i n g : (a, b) <>x=(a°x, a»x); (a,b)*x= a*x. Define t h e mapping 0: AxB — » B as (a,b)^=b. Then ( ( a , b) ' x l ^ t a o x , a*x)^=a*x=(a, b)»x. Hence, 9 i s a Moore automaton. 5 means t h a t each automaton i s e q u i v a l e n t i n s t a t e s to a Moore automaton and t h e r e f o r e , any automaton can be modeled by a Moore automaton.
E automata having the g i v e n u n i v e r s a l p r o p e r t i e s . For example, i f the automaton B=(A,r,C) i s such t h a t f o r any automaton 9=(A,r,B) w i t h the same as i n B o p e r a t i o n ° there e x i s t s the 3 unique homomorphism from B i n t o 9 , then B i s isomorphic t o Atm (A,D. 2. Exactness of the universal automata; left and right reducibility It 1 2 i s c l e a r t h a t Atm (A, B) and Atm (r,B) are exact and l e f t and 3 r i g h t reduced, and Atm (A,D resentation, i s r i g h t reduced.
Factor automaton 9/p=(A/A , T,B/B ) correso o o o ponds t o the congruence p . Take i n T a set J of a l l cr, such t h a t aoo-6A Q , a*