A GALOIS THEORY FOR MONOIDS Dedicated to Manuela Sobral on the occasion of her seventieth birthday

We show that the adjunction between monoids and groups obtained via the Grothendieck group construction is admissible, relatively to surjective homomorphisms, in the sense of categorical Galois theory. The central extensions with respect to this Galois structure turn out to be the so-called special homogeneous surjections. Introduction An action of a monoid B on a monoid X can be de ned as a monoid homomorphism B Ñ EndpXq, where EndpXq is the monoid of endomorphisms of X. These actions were studied in [15], where it is shown that they are equivalent to a certain class of split epimorphisms, called Schreier split epimorphisms in the recent paper [13]. Some properties of Schreier split epimorphisms, as well as the closely related notions of special Schreier surjection and Schreier re exive relation, were then studied in [2] and [3], where the foundations for a cohomology theory of monoids are laid. Many typical properties of the category of groups, such as the Split Short Five Lemma or the fact that any internal re exive relation is transitive, remain valid in the category of monoids when, in the spirit of relative homological algebra, those properties are restricted to Schreier split epimorphisms and Schreier re exive relations. When an action B Ñ EndpXq factors through the group AutpXq of automorphisms of X, the corresponding split epimorphism is called homogeneous [2]. Some properties of homogeneous split epimorphisms and of the related notions of special homogeneous surjection and homogeneous re exive relation were also studied in [2] and [3]. The rst and the second author were supported by the Centro de Matemática da Universidade de Coimbra (CMUC), funded by the European Regional Development Fund through the program COMPETE and by the Portuguese Government through the FCT Fundação para a Ciência e a Tecnologia under the project PEst-C/MAT/UI0324/2013 and grants number PTDC/MAT/120222/2010 and SFRH/BPD/69661/2010. The third author is a Research Associate of the Fonds de la Recherche Scienti que FNRS. He would like to thank CMUC for its kind hospitality during his stays in Coimbra. Received by the editors 2014-02-05 and, in revised form, 2014-04-25. Transmitted by Walter Tholen. Published on 2014-05-01. 2010 Mathematics Subject Classi cation: 20M32, 20M50, 11R32, 19C09, 18F30.