Once a monomial ordering is fixed, the "terms" of a polynomial (product of a monomial with its nonzero coefficient) are naturally ordered by decreasing monomials (for this order).
It can be instinctive to use cyclic orders for symmetric functions, for example as in: where writing the final monomial as would distract from the pattern.
It first compares the dot product of the exponent sequences of the monomials with this weight vector, and in case of a tie uses some other fixed monomial order.
A monomial of variables in this semiring is a linear map, represented in classical arithmetic as a linear function of the variables with integer coefficients.
