-- -*- M2-comint -*- {* hash: 1961767751 *}

i1 : M = monoid [a,b,c,Degrees=>{2,3,4}]

o1 = M

o1 : GeneralOrderedMonoid

i2 : degrees M

o2 = {{2}, {3}, {4}}

o2 : List

i3 : M_0 * M_1^6

        6
o3 = a*b

o3 : M

i4 : a

o4 = a

o4 : M

i5 : use M

o5 = M

o5 : GeneralOrderedMonoid

i6 : a * b^6

        6
o6 = a*b

o6 : M

i7 : options M

o7 = OptionTable{DegreeLift => null                     }
                 DegreeMap => null
                 DegreeRank => 1
                 Degrees => {{2}, {3}, {4}}
                 Global => true
                 Heft => {1}
                 Inverses => false
                 Join => null
                 Local => false
                 MonomialOrder => {MonomialSize => 32  }
                                  {GRevLex => {2, 3, 4}}
                                  {Position => Up      }
                 SkewCommutative => {}
                 Variables => {a, b, c}
                 WeylAlgebra => {}

o7 : OptionTable

i8 : R = ZZ[x,y, Degrees => {-1,-2}, Heft => {-1}]

o8 = R

o8 : PolynomialRing

i9 : degree \ gens R

o9 = {{-1}, {-2}}

o9 : List

i10 : transpose vars R

o10 = {1} | x |
      {2} | y |

              2       1
o10 : Matrix R  <--- R

i11 : R = ZZ/101[x,dx,y,dy,WeylAlgebra => {x=>dx, y=>dy}]

o11 = R

o11 : PolynomialRing

i12 : dx*x

o12 = x*dx + 1

o12 : R

i13 : dx*x^10

       10        9
o13 = x  dx + 10x

o13 : R

i14 : dx*y^10

          10
o14 = dx*y

o14 : R

i15 : R = ZZ[x,y,z,SkewCommutative=>{x,y}]

o15 = R

o15 : PolynomialRing

i16 : x*y

o16 = x*y

o16 : R

i17 : y*x

o17 = -x*y

o17 : R

i18 : x*z-z*x

o18 = 0

o18 : R

i19 : QQ[x][y]

o19 = QQ[x][y]

o19 : PolynomialRing

i20 : oo.FlatMonoid

o20 = [y, x, Degrees => {{1}, {0}}, Heft => {2:1}, MonomialOrder => {MonomialSize => 32}, DegreeRank => 2]
                         {0}  {1}                                   {GRevLex => {1}    }
                                                                    {Position => Up    }
                                                                    {GRevLex => {1}    }

o20 : GeneralOrderedMonoid

i21 : QQ[x][y][z]

o21 = QQ[x][y][z]

o21 : PolynomialRing

i22 : oo.FlatMonoid

o22 = [z, y, x, Degrees => {{1}, {0}, {0}}, Heft => {3:1}, MonomialOrder => {MonomialSize => 32}, DegreeRank => 3]
                            {0}  {1}  {0}                                   {GRevLex => {1}    }
                            {0}  {0}  {1}                                   {Position => Up    }
                                                                            {2:(GRevLex => {1})}

o22 : GeneralOrderedMonoid

i23 : QQ[x][y,Join => false]

o23 = QQ[x][y]

o23 : PolynomialRing

i24 : oo.FlatMonoid

o24 = [y, x, Degrees => {2:1}, Heft => {1}, MonomialOrder => {MonomialSize => 32}, DegreeRank => 1]
                                                             {GRevLex => {1}    }
                                                             {Position => Up    }
                                                             {GRevLex => {1}    }

o24 : GeneralOrderedMonoid

i25 : A = QQ[x];

i26 : B = A[y,Join => false,DegreeMap => x -> 7*x]

o26 = B

o26 : PolynomialRing

i27 : B.FlatMonoid

o27 = [y, x, Degrees => {1, 7}, Heft => {1}, MonomialOrder => {MonomialSize => 32}, DegreeRank => 1]
                                                              {GRevLex => {1}    }
                                                              {Position => Up    }
                                                              {GRevLex => {1}    }

o27 : GeneralOrderedMonoid

i28 : degrees A^{-1,-2}

o28 = {{1}, {2}}

o28 : List

i29 : degrees (B**A^{-1,-2})

o29 = {{7}, {14}}

o29 : List

i30 : B = A[y,Join => false,DegreeMap => x -> 7*x,
           DegreeLift => x -> apply(x, d -> lift(d/7,ZZ))]

o30 = B

o30 : PolynomialRing

i31 : matrix {{x_B}}

o31 = | x |

              1       1
o31 : Matrix B  <--- B

i32 : degrees oo

o32 = {{{0}}, {{7}}}

o32 : List

i33 : lift(matrix {{x_B}},A)

o33 = | x |

              1       1
o33 : Matrix A  <--- A

i34 : degrees oo

o34 = {{{0}}, {{1}}}

o34 : List

i35 : R = QQ[a..d, Weights=>{1,2,3,4}]

o35 = R

o35 : PolynomialRing

i36 : monoid R

o36 = [a..d, Degrees => {4:1}, Heft => {1}, MonomialOrder => {Weights => {1..4} }, DegreeRank => 1]
                                                             {MonomialSize => 32}
                                                             {GRevLex => {4:1}  }
                                                             {Position => Up    }

o36 : GeneralOrderedMonoid

i37 :