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ExteriorModules :: isMonomialModule

isMonomialModule -- whether a module is monomial

Synopsis

Description

Let $F$ a free module with homogeneous basis $\{g_1,g_2,\ldots,g_r\}.$ The elements $e_{\sigma}g_i$ with $e_{\sigma}$ a monomial of $E$ are called monomials of $F$ and $\mathrm{deg}(e_{\sigma} g_i) = \mathrm{deg}(e_{\sigma}) + \mathrm{deg}(g_i).$ A graded submodule M of $F$ is a monomial submodule if M is a submodule generated by monomials of $F$, i.e., $M=I_i g_i \oplus I_2 g_2 \oplus \cdots \oplus I_r g_r,$ where $I_i$ is a monomial ideal of $E$ for each $i.$

Example:

i1 : E=QQ[e_1..e_3,SkewCommutative=>true]

o1 = E

o1 : PolynomialRing, 3 skew commutative variables
 
i2 : F=E^{0,0}

      2
o2 = E

o2 : E-module, free
 
i3 : f_1=(e_1*e_2)*F_0

o3 = | e_1e_2 |
     |    0   |

      2
o3 : E
 
i4 : f_2=(e_1*e_3)*F_0+(e_2*e_3)*F_1

o4 = | e_1e_3 |
     | e_2e_3 |

      2
o4 : E
 
i5 : f_3=(e_1*e_2*e_3)*F_1

o5 = |     0     |
     | e_1e_2e_3 |

      2
o5 : E
 
i6 : M=image map(F,E^{-degree f_1,-degree f_2,-degree f_3},matrix {f_1,f_2,f_3})

o6 = image | e_1e_2 e_1e_3 0         |
           | 0      e_2e_3 e_1e_2e_3 |

                             2
o6 : E-module, submodule of E
 
i7 : isMonomialModule M

o7 = false

See also

Ways to use isMonomialModule :

For the programmer

The object isMonomialModule is a method function.