Birkhoff's variety theorem in many sorts (Q1762479)

In this paper, Birkhoff's characterization of equational classes is proved to generalize the case of finitely many sorts: For every set \(S\) of sorts and every \(S\)-sorted signature \(\Sigma\), the equational classes of \(\Sigma\)-algebras are precisely the full subcategories of \(\Sigma\)-Alg closed under products, subalgebras, regular quotients,and directed unions. For infinitely sorted algebras, closure under directed unions needs to be added.