1. Math
  2. Advanced Math
  3. this is 1 question please answer all parts will give...

Question: this is 1 question please answer all parts will give...

Question details

I. Automata, Semigroups, and Groups By a semigroup we mean a set A with an associative operation. (There does not need to be an identity ele- ment, nor do elements necessarily have inverses.) Every group is a semigroup, though the converse is clearly false. With every semigroup A we associate an automaton M- M (A) called the automaton of the semigroup A. The alphabet of M is A, the set of states also is A, and the next-state function is α(s, a)-sa [ora(s, a)-s + if the operation of the semigroup is denoted additively] 1 Describe M(24). That is, give the table of its next- state function, as well as its state diagram 2 Describe M(S3) If Mis a machine and S is the set of states of M, the state transition functions of M (defined in Exercise H6 of this chapter) are functions from S to S. In the next exercise you will be asked to show that「 7,- Txv; that is, the composite of two transition functions is a transition function. Since the composition of func- tions is associative [fo g h)-(f that the set of all transition functions, with the opera- tion o, is a semigroup. It is denoted by (M) and called the semigroup of the machine M. o g) ° h], it followsrove that Ty Ty T 4 Let Mi be the machine of the example in Exercise H above. Give the table of the semigroup »(Mı. Does » (Mi) have an identity element? Is »(Mı) a group? 5 Let M2 be the machine of Exercise H3. How many distinct functions are there in (M3)? Give the table of ,(Ma). 1s ,(M3) a group? (Why?) 6 Find the table of (M) if M is the machine whose state diagram is Co 0

This is 1 question, please answer all parts. Will give thumbs up for good answers

Solution by an expert tutor
Blurred Solution
This question has been solved
Subscribe to see this solution