Description
A monomial module
M is lex if for all monomials $u,v$ of
F of the same degree with $v\in M$ and $u>v$ (> lex order) then $u\in M$. An equivalent definition of a lex submodule is the following one: a monomial submodule $M=\oplus_{i=1}^{r}{I_ig_i}$ of
F is lex if $I_i$ is a lex ideal of
E for each $i,$ and $(e_1,\dots, e_n)^{\rho_i + f_i - f_{i-1}} \subseteq I_{i-1}$ for $i = 2, \dots, r$ with $\rho_i = \mathrm{indeg}\ I_i.$
Example:
i1 : E = QQ[e_1..e_4, SkewCommutative => true]
o1 = E
o1 : PolynomialRing, 4 skew commutative variable(s)
|
i2 : F=E^{0,0}
2
o2 = E
o2 : E-module, free
|
i3 : I_1=ideal(e_1*e_2,e_1*e_3,e_2*e_3)
o3 = ideal (e e , e e , e e )
1 2 1 3 2 3
o3 : Ideal of E
|
i4 : I_2=ideal(e_1*e_2,e_1*e_3)
o4 = ideal (e e , e e )
1 2 1 3
o4 : Ideal of E
|
i5 : M=createModule({I_1,I_2},F)
o5 = image | e_2e_3 e_1e_3 e_1e_2 0 0 |
| 0 0 0 e_1e_3 e_1e_2 |
2
o5 : E-module, submodule of E
|
i6 : Malex=almostLexModule M
o6 = image | e_1e_4 e_1e_3 e_1e_2 e_2e_3e_4 0 0 |
| 0 0 0 0 e_1e_3 e_1e_2 |
2
o6 : E-module, submodule of E
|
i7 : isLexModule Malex
o7 = false
|
i8 : L=createModule({ideal(e_1*e_2,e_1*e_3*e_4),ideal(e_1*e_2*e_3*e_4)},F)
o8 = image | e_1e_2 e_1e_3e_4 0 |
| 0 0 e_1e_2e_3e_4 |
2
o8 : E-module, submodule of E
|
i9 : isLexModule L
o9 = true
|