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 variable(s)
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i2 : F=E^{0,0}
2
o2 = E
o2 : E-module, free
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i3 : f_1=(e_1*e_2)*F_0
o3 = | e_1e_2 |
| 0 |
2
o3 : E
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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
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i5 : f_3=(e_1*e_2*e_3)*F_1
o5 = | 0 |
| e_1e_2e_3 |
2
o5 : E
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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
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i7 : isMonomialModule M
o7 = false
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