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++ Authors: Stephan M. Watt; revised by Johannes Grabmeier
++ Date Created: January 1986
++ Date Last Updated: 14 December 1995, Mike Dewar
++ Description:
++ \spadtype{MonoidRing}(R,M), implements the algebra
++ of all maps from the monoid M to the commutative ring R with
++ finite support.
++ Multiplication of two maps f and g is defined
++ to map an element c of M to the (convolution) sum over f(a)g(b)
++ such that ab = c. Thus M can be identified with a canonical
++ basis and the maps can also be considered as formal linear combinations
++ of the elements in M. Scalar multiples of a basis element are called
++ monomials. A prominent example is the class of polynomials
++ where the monoid is a direct product of the natural numbers
++ with pointwise addition. When M is
++ \spadtype{FreeMonoid Symbol}, one gets polynomials
++ in infinitely many non-commuting variables. Another application
++ area is representation theory of finite groups G, where modules
++ over \spadtype{MonoidRing}(R,G) are studied.
MonoidRing(R,M) : SIG == CODE where
R : Ring
M : Monoid
Term ==> Record(coef: R, monom: M)
SIG ==> Join(Ring, RetractableTo M, RetractableTo R) with
monomial : (R, M) -> %
++ monomial(r,m) creates a scalar multiple of the basis element m.
coefficient : (%, M) -> R
++ coefficient(f,m) extracts the coefficient of m in f with respect
++ to the canonical basis M.
coerce: List Term -> %
++ coerce(lt) converts a list of terms and
++ coefficients to a member of the domain.
terms : % -> List Term
++ terms(f) gives the list of non-zero coefficients combined
++ with their corresponding basis element as records.
++ This is the internal representation.
map : (R -> R, %) -> %
++ map(fn,u) maps function fn onto the coefficients
++ of the non-zero monomials of u.
monomial? : % -> Boolean
++ monomial?(f) tests if f is a single monomial.
coefficients: % -> List R
++ coefficients(f) lists all non-zero coefficients.
monomials: % -> List %
++ monomials(f) gives the list of all monomials whose
++ sum is f.
numberOfMonomials: % -> NonNegativeInteger
++ numberOfMonomials(f) is the number of non-zero coefficients
++ with respect to the canonical basis.
if R has CharacteristicZero then CharacteristicZero
if R has CharacteristicNonZero then CharacteristicNonZero
if R has CommutativeRing then Algebra(R)
if (R has Finite and M has Finite) then Finite
if M has OrderedSet then
leadingMonomial : % -> M
++ leadingMonomial(f) gives the monomial of f whose
++ corresponding monoid element is the greatest
++ among all those with non-zero coefficients.
leadingCoefficient : % -> R
++ leadingCoefficient(f) gives the coefficient of f, whose
++ corresponding monoid element is the greatest
++ among all those with non-zero coefficients.
reductum : % -> %
++ reductum(f) is f minus its leading monomial.
CODE ==> add
Ex ==> OutputForm
Cf ==> coef
Mn ==> monom
Rep := List Term
coerce(x: List Term): % == x :: %
monomial(r:R, m:M) ==
r = 0 => empty()
[[r, m]]
if (R has Finite and M has Finite) then
size() == size()$R ** size()$M
index k ==
-- use p-adic decomposition of k
-- coefficient of p**j determines coefficient of index(i+p)$M
i:Integer := k rem size()
p:Integer := size()$R
n:Integer := size()$M
ans:% := 0
for j in 0.. while i > 0 repeat
h := i rem p
-- we use index(p) = 0$R
if h ^= 0 then
c : R := index(h :: PositiveInteger)$R
m : M := index((j+n) :: PositiveInteger)$M
--ans := ans + c *$% m
ans := ans + monomial(c, m)$%
i := i quo p
ans
lookup(z : %) : PositiveInteger ==
-- could be improved, if M has OrderedSet
-- z = index lookup z, n = lookup index n
-- use p-adic decomposition of k
-- coefficient of p**j determines coefficient of index(i+p)$M
zero?(z) => size()$% pretend PositiveInteger
liTe : List Term := terms z -- all non-zero coefficients
p : Integer := size()$R
n : Integer := size()$M
res : Integer := 0
for te in liTe repeat
-- assume that lookup(p)$R = 0
l:NonNegativeInteger:=lookup(te.Mn)$M
ex : NonNegativeInteger := (n=l => 0;l)
co : Integer := lookup(te.Cf)$R
res := res + co * p ** ex
res pretend PositiveInteger
random() == index( (1+(random()$Integer rem size()$%) )_
pretend PositiveInteger)$%
0 == empty()
1 == [[1, 1]]
terms a == (copy a) pretend List(Term)
monomials a == [[t] for t in a]
coefficients a == [t.Cf for t in a]
coerce(m:M):% == [[1, m]]
coerce(r:R): % ==
-- coerce of ring
r = 0 => 0
[[r, 1]]
coerce(n:Integer): % ==
-- coerce of integers
n = 0 => 0
[[n::R, 1]]
- a == [[ -t.Cf, t.Mn] for t in a]
if R has noZeroDivisors
then
(r:R) * (a:%) ==
r = 0 => 0
[[r*t.Cf, t.Mn] for t in a]
else
(r:R) * (a:%) ==
r = 0 => 0
[[rt, t.Mn] for t in a | (rt:=r*t.Cf) ^= 0]
if R has noZeroDivisors
then
(n:Integer) * (a:%) ==
n = 0 => 0
[[n*t.Cf, t.Mn] for t in a]
else
(n:Integer) * (a:%) ==
n = 0 => 0
[[nt, t.Mn] for t in a | (nt:=n*t.Cf) ^= 0]
map(f, a) == [[ft, t.Mn] for t in a | (ft:=f(t.Cf)) ^= 0]
numberOfMonomials a == #a
retractIfCan(a:%):Union(M, "failed") ==
((#a) = 1) and ((a.first.Cf) = 1) => a.first.Mn
"failed"
retractIfCan(a:%):Union(R, "failed") ==
((#a) = 1) and ((a.first.Mn) = 1) => a.first.Cf
"failed"
if R has noZeroDivisors then
if M has Group then
recip a ==
lt := terms a
#lt ^= 1 => "failed"
(u := recip lt.first.Cf) case "failed" => "failed"
--(u::R) * inv lt.first.Mn
monomial((u::R), inv lt.first.Mn)$%
else
recip a ==
#a ^= 1 or a.first.Mn ^= 1 => "failed"
(u := recip a.first.Cf) case "failed" => "failed"
u::R::%
mkTerm(r:R, m:M):Ex ==
r=1 => m::Ex
r=0 or m=1 => r::Ex
r::Ex * m::Ex
coerce(a:%):Ex ==
empty? a => (0$Integer)::Ex
empty? rest a => mkTerm(a.first.Cf, a.first.Mn)
reduce(_+, [mkTerm(t.Cf, t.Mn) for t in a])$List(Ex)
if M has OrderedSet then -- we mean totally ordered
-- Terms are stored in decending order.
leadingCoefficient a == (empty? a => 0; a.first.Cf)
leadingMonomial a == (empty? a => 1; a.first.Mn)
reductum a == (empty? a => a; rest a)
a = b ==
#a ^= #b => false
for ta in a for tb in b repeat
ta.Cf ^= tb.Cf or ta.Mn ^= tb.Mn => return false
true
a + b ==
c:% := empty()
while not empty? a and not empty? b repeat
ta := first a; tb := first b
ra := rest a; rb := rest b
c :=
ta.Mn > tb.Mn => (a := ra; concat_!(c, ta))
ta.Mn < tb.Mn => (b := rb; concat_!(c, tb))
a := ra; b := rb
not zero?(r := ta.Cf+tb.Cf) =>
concat_!(c, [r, ta.Mn])
c
concat_!(c, concat(a, b))
coefficient(a, m) ==
for t in a repeat
if t.Mn = m then return t.Cf
if t.Mn < m then return 0
0
if M has OrderedMonoid then
-- we use that multiplying an ordered list of monoid elements
-- by a single element respects the ordering
if R has noZeroDivisors then
a:% * b:% ==
+/[[[ta.Cf*tb.Cf, ta.Mn*tb.Mn]$Term
for tb in b ] for ta in reverse a]
else
a:% * b:% ==
+/[[[r, ta.Mn*tb.Mn]$Term
for tb in b | not zero?(r := ta.Cf*tb.Cf)]
for ta in reverse a]
else -- M hasn't OrderedMonoid
-- we cannot assume that mutiplying an ordered list of
-- monoid elements by a single element respects the ordering:
-- we have to order and to collect equal terms
ge : (Term,Term) -> Boolean
ge(s,t) == t.Mn <= s.Mn
sortAndAdd : List Term -> List Term
sortAndAdd(liTe) == -- assume liTe not empty
liTe := sort(ge,liTe)
m : M := (first liTe).Mn
cf : R := (first liTe).Cf
res : List Term := []
for te in rest liTe repeat
if m = te.Mn then
cf := cf + te.Cf
else
if not zero? cf then res := cons([cf,m]$Term, res)
m := te.Mn
cf := te.Cf
if not zero? cf then res := cons([cf,m]$Term, res)
reverse res
if R has noZeroDivisors then
a:% * b:% ==
zero? a => a
zero? b => b -- avoid calling sortAndAdd with []
+/[sortAndAdd [[ta.Cf*tb.Cf, ta.Mn*tb.Mn]$Term
for tb in b ] for ta in reverse a]
else
a:% * b:% ==
zero? a => a
zero? b => b -- avoid calling sortAndAdd with []
+/[sortAndAdd [[r, ta.Mn*tb.Mn]$Term
for tb in b | not zero?(r := ta.Cf*tb.Cf)]
for ta in reverse a]
else -- M hasn't OrderedSet
-- Terms are stored in random order.
a = b ==
#a ^= #b => false
brace(a pretend List(Term)) =$Set(Term) brace(b pretend List(Term))
coefficient(a, m) ==
for t in a repeat
t.Mn = m => return t.Cf
0
addterm(Tabl: AssociationList(M,R), r:R, m:M):R ==
(u := search(m, Tabl)) case "failed" => Tabl.m := r
zero?(r := r + u::R) => (remove_!(m, Tabl); 0)
Tabl.m := r
a + b ==
Tabl := table()$AssociationList(M,R)
for t in a repeat
Tabl t.Mn := t.Cf
for t in b repeat
addterm(Tabl, t.Cf, t.Mn)
[[Tabl m, m]$Term for m in keys Tabl]
a:% * b:% ==
Tabl := table()$AssociationList(M,R)
for ta in a repeat
for tb in (b pretend List(Term)) repeat
addterm(Tabl, ta.Cf*tb.Cf, ta.Mn*tb.Mn)
[[Tabl.m, m]$Term for m in keys Tabl]
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