--- mring.spad.pamphlet.orig 2007-06-01 09:26:36.000000000 +0200 +++ mring.spad.pamphlet 2007-07-22 12:14:26.000000000 +0200 @@ -9,12 +9,12 @@ \eject \tableofcontents \eject -\section{domain MRING MonoidRing} -<>= -)abbrev domain MRING MonoidRing +\section{category MRCAT MonoidRingCategory} +<>= +)abbrev category MRCAT MonoidRingCategory ++ Authors: Stephan M. Watt; revised by Johannes Grabmeier ++ Date Created: January 1986 -++ Date Last Updated: 14 December 1995, Mike Dewar +++ Date Last Updated: 22 July 2007, Franz Lehner ++ Basic Operations: *, +, monomials, coefficients ++ Related Constructors: Polynomial ++ Also See: @@ -23,26 +23,13 @@ ++ indeterminates ++ References: ++ Description: -++ \spadtype{MonoidRing}(R,M), implements the algebra +++ \spadtype{MonoidRingCategory}(R,M) defines 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 {\em f(a)g(b)} -++ such that {\em 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: Ring, M: Monoid): MRcategory == MRdefinition where +MonoidRingCategory(R: Ring, M: Monoid):Category == MRCdefinition where Term ==> Record(coef: R, monom: M) - MRcategory ==> Join(Ring, RetractableTo M, RetractableTo R) with + MRCdefinition == 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 @@ -82,6 +69,38 @@ ++ among all those with non-zero coefficients. reductum : % -> % ++ reductum(f) is f minus its leading monomial. +@ +\section{domain MRING MonoidRing} +<>= +)abbrev domain MRING MonoidRing +++ Authors: Stephan M. Watt; revised by Johannes Grabmeier +++ Date Created: January 1986 +++ Date Last Updated: 14 December 1995, Mike Dewar +++ Basic Operations: *, +, monomials, coefficients +++ Related Constructors: Polynomial +++ Also See: +++ AMS Classifications: +++ Keywords: monoid ring, group ring, polynomials in non-commuting +++ indeterminates +++ References: +++ 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 {\em f(a)g(b)} +++ such that {\em 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: Ring, M: Monoid): MonoidRingCategory(R,M) == MRdefinition where + Term ==> Record(coef: R, monom: M) MRdefinition ==> add Ex ==> OutputForm @@ -395,6 +414,7 @@ <<*>>= <> +<> <> <> @