# zbMATH — the first resource for mathematics

##### Examples
 Geometry Search for the term Geometry in any field. Queries are case-independent. Funct* Wildcard queries are specified by * (e.g. functions, functorial, etc.). Otherwise the search is exact. "Topological group" Phrases (multi-words) should be set in "straight quotation marks". au: Bourbaki & ti: Algebra Search for author and title. The and-operator & is default and can be omitted. Chebyshev | Tschebyscheff The or-operator | allows to search for Chebyshev or Tschebyscheff. "Quasi* map*" py: 1989 The resulting documents have publication year 1989. so: Eur* J* Mat* Soc* cc: 14 Search for publications in a particular source with a Mathematics Subject Classification code (cc) in 14. "Partial diff* eq*" ! elliptic The not-operator ! eliminates all results containing the word elliptic. dt: b & au: Hilbert The document type is set to books; alternatively: j for journal articles, a for book articles. py: 2000-2015 cc: (94A | 11T) Number ranges are accepted. Terms can be grouped within (parentheses). la: chinese Find documents in a given language. ISO 639-1 language codes can also be used.

##### Operators
 a & b logic and a | b logic or !ab logic not abc* right wildcard "ab c" phrase (ab c) parentheses
##### Fields
 any anywhere an internal document identifier au author, editor ai internal author identifier ti title la language so source ab review, abstract py publication year rv reviewer cc MSC code ut uncontrolled term dt document type (j: journal article; b: book; a: book article)
Complete systems of $ℬ$-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 $\left(M\right)$, $\left(S\right)$ and $P\left(M\right)$ for each finite monoid $M$.

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

There special identities are: $\left(M\right)$ ${\left(ab\right)}^{*}=1+a{\left(ba\right)}^{*}b$; $\left(S\right)$ ${\left(a+b\right)}^{*}={\left({a}^{*}b\right)}^{*}{a}^{*}$; $P\left(M\right)$ ${A}_{M}^{*}={\sum }_{m\in M}{\varphi }_{M}^{-1}\left(m\right)$, where ${A}_{M}$ is an alphabet in bijection with $M$, ${\varphi }_{M}:{A}_{M}^{*}\to M$ the natural monoid homomorphism, and ${\varphi }_{M}^{-1}\left(m\right)$ 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 $\left(M\right)$ and $\left(S\right)$; 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 $\left(M\right)$, $\left(S\right)$ together with the symmetric group identities.

##### MSC:
 68Q45 Formal languages and automata