## Complete systems of $$\mathcal B$$-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: $$(M)$$ $$(ab)^*=1+a(ba)^*b$$; $$(S)$$ $$(a+b)^*=(a^*b)^*a^*$$; $$P(M)$$ $$A^*_ M=\sum_{m\in M}\varphi^{-1}_ M(m)$$, where $$A_ M$$ is an alphabet in bijection with $$M$$, $$\varphi_ M: A^*_ M\to M$$ the natural monoid homomorphism, and $$\varphi^{-1}_ M(m)$$ 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.

### MSC:

 68Q45 Formal languages and automata
Full Text:

### References:

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