K field, P = $K[x_i]$ i < n
$T^n = {x_i^{\alpha_i} | \alpha_i \in \mathbb{N}}$
monomial ideal $I =
each non zero polynomial has terms, which we want to order to make it easier to work with them.
let $\sigma$ be a complete ordering relation on $T^n$ i.e.
(1) $for t_1, t_2 \in T^n$ we have $t_1 <=_{\sigma} t_1 then\ \ t_1 == t_2$
(2) $for t_1, t_2 for all t_i in T^n$
(3)if $t_1 <= t_2 <= t_3$ then $t_1 <= t_3$
then the ordering $\sigma$ is called a term ordering. if
(a) "multiplicative" property: $t_1 <= t_2$ then $t_1 * t_3 <= t_2 * t_3$ for all $t_3 \in T^n$
(b) "well-ordering" property: $1 <= t_i$ for all $t_i \in T^n$
(a) $\sigma = lex$ - lexicographic term ordering and satisfies $x_1^{\alpha_1} * x_2^{\alpha_2} * ... * x_n^{\alpha_n} <=_{\sigma} x_1^{\beta_1} * x_2^{\beta_2} * ... * x_n^{\beta_n}$ <=> $\alpha_1 = \beta_1$
(b) degree-lexicographic term ordering^ $t_1 <= _{deglex} t_2$ <=> $deg(t_1) < deg(t_2)$ or $deg(t_1) = deg(t_2)$ and $t_1 <_{lex} t_2$
(c) deglexref - degree-reverse lexicographic term ordering $x_1^{\alpha_1} * x_2^{\alpha_2} * ... * x_n^{\alpha_n} <=_{drl} x_1^{\beta_1} * x_2^{\beta_2} * ... * x_n^{\beta_n}$ <=> smthing, check board pic
let V be a matrix with $V = (v_1, v_2, ... v_n)$ and det(V) != 0.
then let $x_1^{\alpha_1} * x_2^{\alpha_2} * ... * x_n^{\alpha_n}$ <= _{ord(V)} $x_1^{\beta_1} * x_2^{\beta_2} * ... * x_n^{\beta_n}$ <=> the first non-zero component of the vector $V * $log(t_2) - log(t_1)$ is positive
<=> $^{v_1} * (\alpha_1...\alpha_n) - v_2 * v_1$ - scalar product
check blackboard pic for more details
let V $ \in Mat_n(Z)l det(v) != 0$ \sigma = Ord(V). Then the ordering $\sigma$ is a term ordering if $1 <_{\sigma}$ for all t in T^n <=> $1 <=_{\sigma} x_i$ for all i = 1, n
when we study P-submodules, we need terms of P^r we need "terms" of P^r of $P^r: T^n