The operator / is left associative, which means that `w / f / g` is interpreted as `(w / f) / g`. The operator \ is right associative, so `g \ f \ w` is interpreted as `g \ (f \ w)`. Both operators have parsing precedence lower than that of @@, which means that the previous two expressions are equivalent to `w / g @@ f` and `g @@ f \ w`, respectively. See precedence of operators.

i1 : R = ZZ[a..d]; |

i2 : I = ideal"abc-d3,ab-d-1,a2+b2+c3-14d-3" 3 3 2 2 o2 = ideal (a*b*c - d , a*b - d - 1, c + a + b - 14d - 3) o2 : Ideal of R |

i3 : I/size o3 = {2, 3, 5} o3 : List |

i4 : (f->f+a*b-1)\I 3 3 2 2 o4 = {a*b*c - d + a*b - 1, 2a*b - d - 2, c + a + a*b + b - 14d - 4} o4 : List |

i5 : I/leadTerm/support/set//sum o5 = set {a, b, c} o5 : Set |