On syntactic semirings of regular languages

We recall first some classical results concerning syntactic monoids of regular languages: Schutzenberger theorem characterizing the so-called star-free languages, Eilenberg correspondence relating the so-called varieties of languages and pseudovarieties of finite monoids. We introduce the notion of the syntactic semiring of a given regular language and we will clarify its relationship to the syntactic monoid. We present examples showing the equational independence of those two structures. Further we show the possibility of translations of certain language equations into calculations in (finite) syntactic semirings. For instance, we can compute roots of a given language. The syntactic semiring of a given language is also a structure in which the syntactic monoid and the so-called universal automaton of this language are hidden. We mention algorithms for calculations of those structures. Finally, we present Eilenberg-type theorem for the so-called pseudovarieties of semiring homomorphisms and we formulate Reitermann theorem for this setting. We add also several illustrative examples. All the mentioned results are from a series of author's papers; they are available at http://www.math.muni.cz/~polak.