The monomials are kept ordered since, in this case, the decomposition of a polynomial into the leading monomial and its remaining polynomial, which is frequently needed during the Gröbner bases algorithm, can be implemented more efficiently.
We also implemented the quotient of two polynomials, which is to be understood as ``exact division without remainder''. This is of course only possible if the divisor is invertible. In the case of polynomials over a field, which is the case we concentrated on, the invertible elements are the elements of the ground field (i.e. the constant polynomials). When attempting to divide by a non-constant polynomial, GRÖBNER will detect this and handle the error properly. In the case of polynomials over a domain that is no field, an error might be detected when trying to invert the divisor in the case where it is a constant that is not invertible. Implementing exact division in this fashion considerably simplifies the parser for input of polynomials.