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

- Usage:
    `isStronglyStableModule M`
- Inputs:
  - M, a monomial module over an exterior algebra
- Outputs:
  - a Boolean value, 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 variable(s)

    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

## See also

- isAlmostStronglyStableModule -- whether a monomial module over an exterior algebra is almost strongly stable
- stronglyStableModule -- compute the smallest strongly stable module containing a given monomial module

## Ways to use isStronglyStableModule:

- `isStronglyStableModule(Module)`

## For the programmer

The object isStronglyStableModule is a method function.

The source of this document is in /build/macaulay2-88fgJW/macaulay2-1.25.11+ds/M2/Macaulay2/packages/ExteriorModules.m2:822:0.