Equational descriptions of languages

This paper is a survey on the equational descriptions of languages. The first part is devoted to Birkhoff’s and Reiterman’s theorems on equational descriptions of varieties. Eilenberg’s variety theorem and its successive generalizations form the second part. The more recent results on equational descriptions of lattices of languages are presented in the third part of the paper. Equations have been used for a long time in mathematics to provide a concise description of various mathematical objects. This article roughly follows a historical approach to present such equational descriptions for formal languages, ranging over a period of 45 years: from Schützenberger’s characterization of star-free languages [36] to the following recent result of [18]: Every lattice of languages admits an equational description. This evolution was made possible by a gradual abstraction of the notion of equation. The story really starts in 1935 with Birkhoff’s theorem on equational classes [6]. It holds for any kind of universal algebra, but I will present it only for monoids. 1 Varieties and identities A Birkhoff variety of monoids is a class of (possibly infinite) monoids which is closed under taking submonoids, homomorphic images (also called quotients) and arbitrary direct products. Birkhoff’s theorem states that a class of monoids is a Birkhoff variety if and only if it can be defined by a set of identities. Formally, an identity is an equality between two words of the free monoid Σ on a countable alphabet Σ. A monoid M satisfies the identity u = v if and only if φ(u) = φ(v) for every morphism φ : Σ → M . For instance, the identity xy = yx defines the variety of commutative monoids and x = x defines the variety of idempotent monoids. It is easy to see that if E is a set of identities, the class of monoids satisfying all the identities of E is a variety, denoted JEK, and called the Birkhoff variety defined by E. The difficult part of Birkhoff’s theorem is to prove that the converse also holds: for each Birkhoff variety V, there exists a set of identities E such that V = JEK. Note that the set E might be infinite. For instance, if LIAFA, CNRS and Univ. Paris-Diderot, Case 7014, 75205 Paris Cedex 13, France. The author acknowledges support from the project ANR 2010 BLAN 0202 02 FREC.