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Complete systems of -rational identities. (English) Zbl 0737.68053

Two conjectures of Conway on rational expressions of languages are proved: the two following systems of identities are complete systems of identities (i.e. each rational identity is a consequence of the system):

1. The identities (M), (S) and P(M) for each finite monoid M.

2. The identities (M), (S) and P(G) for each finite group G.

There special identities are: (M) (ab) * =1+a(ba) * b; (S) (a+b) * =(a * b) * a * ; P(M) A M * = mM ϕ M -1 (m), where A M is an alphabet in bijection with M, ϕ M :A M * M the natural monoid homomorphism, and ϕ M -1 (m) represents a rational expression naturally associated to this language.

The considerable work done by the author in order to solve these conjectures has many byproducts: completeness of certain meta-rule systems; characterization fo aperiodic semigroups by deductibility of their rational expression from (M) and (S); formal proof of Schützenberger’s star-free theorem; deduction of the matrix semigroup identity from the semigroup identity; stability of identities under operations (subsemigroup, quotient, semidirect product), which allows to use the theorem of Krohn-Rhodes; completeness of (M), (S) together with the symmetric group identities.

68Q45Formal languages and automata