Varieties of formal series on trees and Eilenberg's theorem (Q1111587)

The authors prove a tree series version of Eilenberg's correspondence between varieties of languages and varieties of monoïds. More precisely, for each field K, they introduce the category \(K\)-\(\Sigma\)- Alg\({}_{fin}\) whose objects are (\(\Sigma\),\({\mathcal A})\), where \(\Sigma\) a ranked alphabet and \({\mathcal A}^ a \Sigma\)-algebra, which is also a finite dimensional K-vector space, satisfying an additional property. Then they define the notion of (E,M)-variety in \(K\)-\(\Sigma\)- Alg\({}_{fin}\), where E and M are appropriate classes of epimorphisms and monomorphisms in this category. And, finally, they prove that there is a bijection between varieties of tree series over K and (E,M)- varieties in \(K\)-\(\Sigma\)-Alg\({}_{fin}\).