Groebner---Shirshov bases for replicated algebras

Abstract: We establish a universal approach to solution of the word problem in the varieties of di- and tri-algebras. This approach, for example, allows to apply Groebner---Shirshov bases method for Lie algebras to solve the ideal membership problem in free Leibniz algebras (Lie di-algebras). As another application, we prove an analogue of the Poincare---Birkhoff---Witt Theorem for universal enveloping associative tri-algebra of a Lie tri-algebra (CTD^!-algebra).