# Representation and Monomial Orders

Let $A$ be the free algebra $K\langle x_1, \ldots, x_n\rangle$ of rank $n$ over a field $K$. A word in the underlying monoid of $A$ is simply an associative product of the letters (or variables) of $A$. For consistency with the commutative case, we will call these monoid words *monomials*. Elements of $A$, called noncommutative polynomials, are finite sums of terms, where a term is the product of a coefficient from $K$ and a monomial. The terms are sorted with respect to an admissible order $<$, which satisfies, for monomials $p,q,r$, the following conditions:

**(a)**
If $p<q$, then $pr < qr$ and $sp < sq$.

**(b)**
If $p=qr$ then $p > q$ and $p > r$. Currently Magma only supports the noncommutative graded-lexicographical order (*glex*), which first compares degrees and then uses a left-lexicographical comparison for degree-ties. There is no admissible lexicographic order in the noncommutative case.