# isMonomialModule -- whether a module is monomial

## Synopsis

• Usage:
isMonomialModule M
• Inputs:
• M, a monomial submodule of the ambient module over an exterior algebra
• Outputs:
• , whether the module M is monomial

## 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