/usr/share/axiom-20170501/src/algebra/XFALG.spad is in axiom-source 20170501-3.
This file is owned by root:root, with mode 0o644.
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1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 | )abbrev category XFALG XFreeAlgebra
++ Author: Michel Petitot petitot@lifl.fr
++ Date Created: 91
++ Date Last Updated: 7 Juillet 92
++ Fix History: compilation v 2.1 le 13 dec 98
++ Description:
++ This category specifies opeations for polynomials
++ and formal series with non-commutative variables.
XFreeAlgebra(vl,R) : Category == SIG where
vl : OrderedSet
R : Ring
WORD ==> OrderedFreeMonoid(vl) -- monoide libre
NNI ==> NonNegativeInteger
I ==> Integer
TERM ==> Record(k: WORD, c: R)
SIG ==> Join(Ring, XAlgebra(R), RetractableTo WORD) with
"*" : (vl,%) -> %
++ \spad{v * x} returns the product of a variable \spad{x}
++ by \spad{x}.
"*" : (%, R) -> %
++ \spad{x * r} returns the product of \spad{x} by \spad{r}.
++ Usefull if \spad{R} is a non-commutative Ring.
mindeg : % -> WORD
++ \spad{mindeg(x)} returns the little word which appears
++ in \spad{x}. Error if \spad{x=0}.
mindegTerm : % -> TERM
++ \spad{mindegTerm(x)} returns the term whose word is
++ \spad{mindeg(x)}.
coef : (%,WORD) -> R
++ \spad{coef(x,w)} returns the coefficient of the word \spad{w}
++ in \spad{x}.
coef : (%,%) -> R
++ \spad{coef(x,y)} returns scalar product of \spad{x} by \spad{y},
++ the set of words being regarded as an orthogonal basis.
lquo : (%,vl) -> %
++ \spad{lquo(x,v)} returns the left simplification of \spad{x}
++ by the variable \spad{v}.
lquo : (%,WORD) -> %
++ \spad{lquo(x,w)} returns the left simplification of \spad{x}
++ by the word \spad{w}.
lquo : (%,%) -> %
++ \spad{lquo(x,y)} returns the left simplification of \spad{x}
++ by \spad{y}.
rquo : (%,vl) -> %
++ \spad{rquo(x,v)} returns the right simplification of \spad{x}
++ by the variable \spad{v}.
rquo : (%,WORD) -> %
++ \spad{rquo(x,w)} returns the right simplification of \spad{x}
++ by \spad{w}.
rquo : (%,%) -> %
++ \spad{rquo(x,y)} returns the right simplification of \spad{x}
++ by \spad{y}.
monom : (WORD , R) -> %
++ \spad{monom(w,r)} returns the product of the word \spad{w}
++ by the coefficient \spad{r}.
monomial? : % -> Boolean
++ \spad{monomial?(x)} returns true if \spad{x} is a monomial
mirror : % -> %
++ \spad{mirror(x)} returns \spad{Sum(r_i mirror(w_i))} if
++ \spad{x} writes \spad{Sum(r_i w_i)}.
coerce : vl -> %
++ \spad{coerce(v)} returns \spad{v}.
constant? : % -> Boolean
++ \spad{constant?(x)} returns true if \spad{x} is constant.
constant : % -> R
++ \spad{constant(x)} returns the constant term of \spad{x}.
quasiRegular? : % -> Boolean
++ \spad{quasiRegular?(x)} return true if \spad{constant(x)} is zero
quasiRegular : % -> %
++ \spad{quasiRegular(x)} return \spad{x} minus its constant term.
if R has CommutativeRing then
sh : (%,%) -> %
++ \spad{sh(x,y)} returns the shuffle-product of \spad{x}
++ by \spad{y}.
++ This multiplication is associative and commutative.
sh : (%,NNI) -> %
++ \spad{sh(x,n)} returns the shuffle power of \spad{x} to
++ the \spad{n}.
map : (R -> R, %) -> %
++ \spad{map(fn,x)} returns \spad{Sum(fn(r_i) w_i)} if \spad{x}
++ writes \spad{Sum(r_i w_i)}.
varList : % -> List vl
++ \spad{varList(x)} returns the list of variables which
++ appear in \spad{x}.
if R has noZeroDivisors then noZeroDivisors
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