Rota's Classification Problem, rewriting systems and Gröbner–Shirshov bases

In this paper we revisit Rota's Classification Problem on classifying algebraic identities for linear operators. We reformulate Rota's Classification Problem in the contexts of rewriting systems and Gröbner–Shirshov bases, through which Rota's Classification Problem amounts to the classification of operators, given by their defining operator identities, that give convergent rewriting systems or Gröbner–Shirshov bases. Relationship is established between the reformulations in terms of rewriting systems and that of Gröbner–Shirshov bases. We provide an effective condition that gives Gröbner–Shirshov operators and obtain a new class of Gröbner–Shirshov operators.