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isAlmostLexModule -- whether a monomial module over an exterior algebra is almost lex

Synopsis

Description

Let $\{g_1,g_2,\ldots,g_r\}$ be a graded basis of F. A monomial submodule $M=\oplus_{i=1} ^{r}{I_ig_i}$ of F is almost lex if $I_i$ is a lex ideal of E 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)

o3 = ideal (e e , e e )
             1 2   1 3

o3 : Ideal of E
 
i4 : I_2=ideal(e_1*e_2,e_1*e_3,e_2*e_3)

o4 = ideal (e e , e e , e e )
             1 2   1 3   2 3

o4 : Ideal of E
 
i5 : M=createModule({I_1,I_2},F)

o5 = image | e_1e_3 e_1e_2 0      0      0      |
           | 0      0      e_2e_3 e_1e_3 e_1e_2 |

                             2
o5 : E-module, submodule of E
 
i6 : isAlmostLexModule M

o6 = false
 
i7 : I_3=ideal(e_1*e_2,e_1*e_3, e_1*e_4)

o7 = ideal (e e , e e , e e )
             1 2   1 3   1 4

o7 : Ideal of E
 
i8 : isAlmostLexModule createModule({I_1,I_3},F)

o8 = true

See also

Ways to use isAlmostLexModule :

For the programmer

The object isAlmostLexModule is a method function.