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Zbl 0737.68053
Krob, Daniel
Complete systems of $\cal B$-rational identities.
(English)
[J] Theor. Comput. Sci. 89, No.2, 207-343 (1991). ISSN 0304-3975

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):\par 1. The identities $(M)$, $(S)$ and $P(M)$ for each finite monoid $M$.\par 2. The identities $(M)$, $(S)$ and $P(G)$ for each finite group $G$.\par There special identities are: $(M)$ $(ab)\sp*=1+a(ba)\sp*b$; $(S)$ $(a+b)\sp*=(a\sp*b)\sp*a\sp*$; $P(M)$ $A\sp*\sb M=\sum\sb{m\in M}\varphi\sp{-1}\sb M(m)$, where $A\sb M$ is an alphabet in bijection with $M$, $\varphi\sb M: A\sp*\sb M\to M$ the natural monoid homomorphism, and $\varphi\sp{-1}\sb M(m)$ represents a rational expression naturally associated to this language.\par 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.
[C.Reutenauer (Montreal)]
MSC 2000:
*68Q45 Formal languages

Keywords: $\cal B$-rational expressions; matrix identities; + operation; maximal ideals of a semigroup

Cited in: Zbl 0856.08009 Zbl 0766.68081

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