# isStronglyStableModule -- whether a monomial module over an exterior algebra is strongly stable

## Synopsis

• Usage:
isStronglyStableModule M
• Inputs:
• M, a monomial module over an exterior algebra
• Outputs:
• , whether the module M is strongly stable

## Description

Let $\{g_1,g_2,\ldots,g_r\}$ be a graded basis of F with $deg(g_i)=f_i,\ i=1,\ldots,r.$ A monomial submodule $M=\oplus_{i=1}^{r}{I_ig_i}$ of F is strongly stable if it is almost strongly stable and $(x_1,\ldots,x_n)^{(f_{i+1}-f_i)} I_{i+1}$ belongs to $I_i$ for $i=1,\ldots,r-1.$ A monomial ideal $I$ of $E$ is called strongly stable if for each monomial $e_{\sigma} \in I$ and each $j \in \sigma$ one has $e_ie_{\sigma \setminus \{j\}} \in I$ for all $i<j.$

Example:

 i1 : E = QQ[e_1..e_4, SkewCommutative => true] o1 = E o1 : PolynomialRing, 4 skew commutative variables i2 : F=E^{0,0} 2 o2 = E o2 : E-module, free i3 : I_1=ideal(e_1*e_2) o3 = ideal(e e ) 1 2 o3 : Ideal of E i4 : I_2=ideal(e_1*e_2*e_3,e_1*e_2*e_4,e_1*e_3*e_4) o4 = ideal (e e e , e e e , e e e ) 1 2 3 1 2 4 1 3 4 o4 : Ideal of E i5 : M=createModule({I_1,I_2},F) o5 = image | e_1e_2 0 0 0 | | 0 e_1e_3e_4 e_1e_2e_4 e_1e_2e_3 | 2 o5 : E-module, submodule of E i6 : isAlmostStronglyStableModule M o6 = true i7 : isStronglyStableModule M o7 = false