Formations of finite monoids and formal languages: Eilenberg's variety theorem revisited

We present an extension of Eilenberg’s variety theorem, a wellknown result connecting algebra to formal languages. We prove that there is a bijective correspondence between formations of finite monoids and certain classes of languages, the formations of languages. Our result permits to treat classes of finite monoids which are not necessarily closed under taking submonoids, contrary to the original theory. We also prove a similar result for ordered monoids. This paper is the first step of a programme aiming at exploring the connections between the formations of finite groups and regular languages. The starting point is Eilenberg’s variety theorem [10], a celebrated result of the 1970’s which underscores the importance of varieties of finite monoids (also called pseudovarieties) in the study of formal languages. Since varieties of finite groups are special cases of varieties of finite monoids, varieties seems to be a natural structure to study languages recognized by finite groups. However, in finite group theory, varieties are challenged by another notion. Although varieties are incontestably a central notion, many results are better formulated in the setting of formations. This raised the question whether Eilenberg’s variety theorem could be extended to a “formation theorem”. The aim of this paper is to give a positive answer to this question. To our surprise, the resulting theorem holds not only for group formations but Departament d’Àlgebra, Universitat de València, C/ Dr. Moliner, 50 46100Burjassot(València), Spain LIAFA, Université Paris VII and CNRS, Case 7014, F-75205 Paris Cedex 13, France. Dpt. d’Estad́ıstica i Investigació Operativa, Universitat d’Alacant, Sant Vicent del Raspeig, Ap. Correus 99, E – 03080 Alacant. ∗The authors are supported by Proyecto MTM2010-19938-C03-01 from MICINN (Spain). The second author is supported by the project ANR 2010 BLAN 0202 02 FREC. The third author was supported by the Grant PAID-02-09 from Universitat Politècnica de València.