  Certain weight-based orders on the free associative algebra $R = k<x_1, ...,
x_t >$ can be specified by $t \times \infty$ arrays whose entries come from the
subring of nonnegative elements in a totally ordered field. Such an array $A$
satisfying certain additional conditions produces a partial order on $R$ which
is an admissible order on the quotient $R/I_A$, where $I_A$ is a homogeneous
binomial ideal called the {\em weight ideal} associated to the array and whose
structure is determined entirely by $A$. This article discusses the structure
of the weight ideals associated to two distinct sets of arrays whose elements
define admissible orders on the associated quotient algebra.
