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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...
The notion of commutative integro-differential algebra was introduced for the algebraic study of boundary problems for linear ordinary differential equations. Its noncommutative analog achieves a similar purpose for linear systems of such equations. In both cases, free objects are crucial for analyzing the underlying algebraic structures, e.g. of the (matrix) functions.In this paper we apply the method...
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