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# initialModule -- compute the initial module of a given module

- Usage:
    `initialModule M`
- Inputs:
  - M, a module over an exterior algebra
- Outputs:
  - a module, the initial module of the module M with default monomial order

## 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 $\mathrm{deg}(e_{\sigma} g_i) = \mathrm{deg}(e_{\sigma}) + \mathrm{deg}(g_i).$ Any element $f$ of $F$ is a unique linear combination of monomials with coefficients in $K$. Let > be a monomial order on $E$. The largest monomial of $f$ is called the initial monomial of $f$ and it is denoted by $\mathrm{In}(f)$. If M is a graded submodule of $F$ then the submodule of initial terms of M, denoted by $\mathrm{In}(M)$, is the submodule of $F$ generated by the initial terms of elements of M.

Example:

    i1 : E=QQ[e_1..e_3,SkewCommutative=>true]

    o1 = E

    o1 : PolynomialRing, 3 skew commutative variable(s)

    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 : initialModule M

    o7 = image | 0         e_1e_2 e_1e_3 |
               | e_1e_2e_3 0      0      |

                                 2
    o7 : E-module, submodule of E

## See also

- isMonomialModule -- whether a module is monomial

## Ways to use initialModule:

- `initialModule(Module)`

## For the programmer

The object initialModule is a method function.

The source of this document is in /build/macaulay2-88fgJW/macaulay2-1.25.11+ds/M2/Macaulay2/packages/ExteriorModules.m2:932:0.