Description
Let $\{g_1,g_2,\ldots,g_r\}$ be a graded basis of
F and let $M=\oplus_{i=1}^{r}{I_ig_i}$ be a monomial submodule of
F. The almost lex module associated to
M is the monomial module $M^{alex}=\oplus_{i=1}^{r}{J_ig_i}$ with $J_i=\mathrm{lexIdeal}\ I_i$ for each $i$, i.e., the lex ideal associated to $I_i$ for each $i.$
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 : 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
|