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Diffstat (limited to 'src/algebra/free.spad.pamphlet')
| -rw-r--r-- | src/algebra/free.spad.pamphlet | 121 |
1 files changed, 72 insertions, 49 deletions
diff --git a/src/algebra/free.spad.pamphlet b/src/algebra/free.spad.pamphlet index 90588505..abfb0e65 100644 --- a/src/algebra/free.spad.pamphlet +++ b/src/algebra/free.spad.pamphlet @@ -164,6 +164,72 @@ ListMonoidOps(S, E, un): Exports == Implementation where g @ + +\section{A Category for Free Monoids} + +<<category FMONCAT FreeMonoidCategory>>= +)abbrev category FMONCAT FreeMonoidCategory +++ Free monoid on any set of generators +++ Author: Stephen M. Watt, Gabriel Dos Reis +++ Date Created: September 26, 2009 +++ Date Last Updated: September 26, 2009 +++ Description: +++ A free monoid on a set S is the monoid of finite products of +++ the form \spad{reduce(*,[si ** ni])} where the si's are in S, and the ni's +++ are nonnegative integers. The multiplication is not commutative. +FreeMonoidCategory(S: SetCategory): Category == Exports where + macro NNI == NonNegativeInteger + macro REC == Record(gen: S, exp: NonNegativeInteger) + macro Ex == OutputForm + + Exports == Join(Monoid, RetractableTo S) with + *: (S, %) -> % + ++ \spad{s * x} returns the product of \spad{x} by \spad{s} on the left. + *: (%, S) -> % + ++ \spad{x * s} returns the product of \spad{x} by \spad{s} on the right. + **: (S, NonNegativeInteger) -> % + ++ \spad{s ** n} returns the product of \spad{s} by itself \spad{n} times. + hclf: (%, %) -> % + ++ \spad{hclf(x, y)} returns the highest common left factor of + ++ \spad{x} and \spad{y}, + ++ i.e. the largest d such that \spad{x = d a} and \spad{y = d b}. + hcrf: (%, %) -> % + ++ hcrf(x, y) returns the highest common right factor of x and y, + ++ i.e. the largest d such that \spad{x = a d} and \spad{y = b d}. + lquo: (%, %) -> Union(%, "failed") + ++ lquo(x, y) returns the exact left quotient of x by y i.e. + ++ q such that \spad{x = y * q}, + ++ "failed" if x is not of the form \spad{y * q}. + rquo: (%, %) -> Union(%, "failed") + ++ rquo(x, y) returns the exact right quotient of x by y i.e. + ++ q such that \spad{x = q * y}, + ++ "failed" if x is not of the form \spad{q * y}. + divide: (%, %) -> Union(Record(lm: %, rm: %), "failed") + ++ divide(x, y) returns the left and right exact quotients of + ++ x by y, i.e. \spad{[l, r]} such that \spad{x = l * y * r}, + ++ "failed" if x is not of the form \spad{l * y * r}. + overlap: (%, %) -> Record(lm: %, mm: %, rm: %) + ++ overlap(x, y) returns \spad{[l, m, r]} such that + ++ \spad{x = l * m}, \spad{y = m * r} and l and r have no overlap, + ++ i.e. \spad{overlap(l, r) = [l, 1, r]}. + size : % -> NNI + ++ size(x) returns the number of monomials in x. + factors : % -> List Record(gen: S, exp: NonNegativeInteger) + ++ factors(a1\^e1,...,an\^en) returns \spad{[[a1, e1],...,[an, en]]}. + nthExpon : (%, Integer) -> NonNegativeInteger + ++ nthExpon(x, n) returns the exponent of the n^th monomial of x. + nthFactor : (%, Integer) -> S + ++ nthFactor(x, n) returns the factor of the n^th monomial of x. + mapExpon : (NNI -> NNI, %) -> % + ++ mapExpon(f, a1\^e1 ... an\^en) returns \spad{a1\^f(e1) ... an\^f(en)}. + mapGen : (S -> S, %) -> % + ++ mapGen(f, a1\^e1 ... an\^en) returns \spad{f(a1)\^e1 ... f(an)\^en}. + if S has OrderedSet then OrderedSet + +@ + + + \section{domain FMONOID FreeMonoid} <<domain FMONOID FreeMonoid>>= )abbrev domain FMONOID FreeMonoid @@ -175,55 +241,11 @@ ListMonoidOps(S, E, un): Exports == Implementation where ++ The free monoid on a set S is the monoid of finite products of ++ the form \spad{reduce(*,[si ** ni])} where the si's are in S, and the ni's ++ are nonnegative integers. The multiplication is not commutative. -FreeMonoid(S: SetCategory): FMcategory == FMdefinition where - NNI ==> NonNegativeInteger - REC ==> Record(gen: S, exp: NonNegativeInteger) - Ex ==> OutputForm - - FMcategory ==> Join(Monoid, RetractableTo S) with - *: (S, $) -> $ - ++ s * x returns the product of x by s on the left. - *: ($, S) -> $ - ++ x * s returns the product of x by s on the right. - **: (S, NonNegativeInteger) -> $ - ++ s ** n returns the product of s by itself n times. - hclf: ($, $) -> $ - ++ hclf(x, y) returns the highest common left factor of x and y, - ++ i.e. the largest d such that \spad{x = d a} and \spad{y = d b}. - hcrf: ($, $) -> $ - ++ hcrf(x, y) returns the highest common right factor of x and y, - ++ i.e. the largest d such that \spad{x = a d} and \spad{y = b d}. - lquo: ($, $) -> Union($, "failed") - ++ lquo(x, y) returns the exact left quotient of x by y i.e. - ++ q such that \spad{x = y * q}, - ++ "failed" if x is not of the form \spad{y * q}. - rquo: ($, $) -> Union($, "failed") - ++ rquo(x, y) returns the exact right quotient of x by y i.e. - ++ q such that \spad{x = q * y}, - ++ "failed" if x is not of the form \spad{q * y}. - divide: ($, $) -> Union(Record(lm: $, rm: $), "failed") - ++ divide(x, y) returns the left and right exact quotients of - ++ x by y, i.e. \spad{[l, r]} such that \spad{x = l * y * r}, - ++ "failed" if x is not of the form \spad{l * y * r}. - overlap: ($, $) -> Record(lm: $, mm: $, rm: $) - ++ overlap(x, y) returns \spad{[l, m, r]} such that - ++ \spad{x = l * m}, \spad{y = m * r} and l and r have no overlap, - ++ i.e. \spad{overlap(l, r) = [l, 1, r]}. - size : $ -> NNI - ++ size(x) returns the number of monomials in x. - factors : $ -> List Record(gen: S, exp: NonNegativeInteger) - ++ factors(a1\^e1,...,an\^en) returns \spad{[[a1, e1],...,[an, en]]}. - nthExpon : ($, Integer) -> NonNegativeInteger - ++ nthExpon(x, n) returns the exponent of the n^th monomial of x. - nthFactor : ($, Integer) -> S - ++ nthFactor(x, n) returns the factor of the n^th monomial of x. - mapExpon : (NNI -> NNI, $) -> $ - ++ mapExpon(f, a1\^e1 ... an\^en) returns \spad{a1\^f(e1) ... an\^f(en)}. - mapGen : (S -> S, $) -> $ - ++ mapGen(f, a1\^e1 ... an\^en) returns \spad{f(a1)\^e1 ... f(an)\^en}. - if S has OrderedSet then OrderedSet - - FMdefinition ==> ListMonoidOps(S, NonNegativeInteger, 1) add +FreeMonoid(S: SetCategory): FreeMonoidCategory(S) == FMdefinition where + macro NNI == NonNegativeInteger + macro REC == Record(gen: S, exp: NonNegativeInteger) + macro Ex == OutputForm + FMdefinition == ListMonoidOps(S, NonNegativeInteger, 1) add Rep := ListMonoidOps(S, NonNegativeInteger, 1) 1 == makeUnit() @@ -583,6 +605,7 @@ FreeAbelianGroup(S:SetCategory): Exports == Implementation where <<license>> <<domain LMOPS ListMonoidOps>> +<<category FMONCAT FreeMonoidCategory>> <<domain FMONOID FreeMonoid>> <<domain FGROUP FreeGroup>> <<category FAMONC FreeAbelianMonoidCategory>> |