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# stronglyStableModule -- compute the smallest strongly stable module containing a given monomial module

- Usage:
    `stronglyStableModule M`
- Inputs:
  - M, a monomial module over an exterior algebra
- Outputs:
  - a module, the smallest strongly stable submodule of the ambient module containing M

## Description

Let $F$ be a free module with homogeneous basis $\{g_1,g_2,\ldots,g_r\}$ and let $M$ be a monomial submodule of F. This method allows the construction of the smallest strongly stable submodule of F containing M. It is useful, although it does not preserve invariants. In fact, the computation by hand of a strongly stable submodule implies some tedious calculations overall in the case when the elements of the homogeneous basis of F have different degrees. Furthermore, it is worth pointing out that such methods are analogous to the Macaulay2 function borel that computes the smallest borel ideal containing a given ideal.

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 : isStronglyStableModule M

    o6 = false

    i7 : Mss=stronglyStableModule M

    o7 = image | e_1e_2 e_1e_3e_4 0         0         0         |
               | 0      0         e_1e_3e_4 e_1e_2e_4 e_1e_2e_3 |

                                 2
    o7 : E-module, submodule of E

    i8 : isStronglyStableModule Mss

    o8 = true

## See also

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

## Ways to use stronglyStableModule:

- `stronglyStableModule(Module)`

## For the programmer

The object stronglyStableModule is a method function.

The source of this document is in /build/macaulay2-88fgJW/macaulay2-1.25.11+ds/M2/Macaulay2/packages/ExteriorModules.m2:844:0.