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Diffstat (limited to 'src/algebra/catdef.spad.pamphlet')
| -rw-r--r-- | src/algebra/catdef.spad.pamphlet | 98 |
1 files changed, 78 insertions, 20 deletions
diff --git a/src/algebra/catdef.spad.pamphlet b/src/algebra/catdef.spad.pamphlet index ae0d4f50..c7263464 100644 --- a/src/algebra/catdef.spad.pamphlet +++ b/src/algebra/catdef.spad.pamphlet @@ -1,14 +1,77 @@ \documentclass{article} \usepackage{axiom} -\begin{document} + \title{\$SPAD/src/algebra catdef.spad} -\author{James Davenport, Lalo Gonzalez-Vega} +\author{James Davenport, Lalo Gonzalez-Vega, Gabriel Dos~Reis} + +\begin{document} + \maketitle \begin{abstract} \end{abstract} -\eject \tableofcontents \eject + +\section{Linear sets} + +<<category LLINSET LeftLinearSet>>= +)abbrev category LLINSET LeftLinearSet +++ Author: Gabriel Dos Reis +++ Date Created: May 31, 2009 +++ Date Last Modified: May 31, 2009 +++ Description: +++ A set is an \spad{R}-left linear set if it is stable by left-dilation +++ by elements in the ring \spad{R}. This category differs from +++ \spad{LeftModule} in that no other assumption (such as addition) +++ is made about the underlying set. +++ See Also: RightLinearSet. +LeftLinearSet(R: Rng): Category == SetCategory with + 0: % + ++ \spad{0} represents the origin of the linear set + zero?: % -> Boolean + ++ \spad{zero? x} holds is \spad{x} is the origin. + *: (R,%) -> % + ++ \spad{r*x} is the left-dilation of \spad{x} by \spad{r}. + +@ + +<<category RLINSET RightLinearSet>>= +)abbrev category RLINSET RightLinearSet +++ Author: Gabriel Dos Reis +++ Date Created: May 31, 2009 +++ Date Last Modified: May 31, 2009 +++ Description: +++ A set is an \spad{R}-right linear set if it is stable by right-dilation +++ by elements in the ring \spad{R}. This category differs from +++ \spad{RightModule} in that no other assumption (such as addition) +++ is made about the underlying set. +++ See Also: LeftLinearSet. +RightLinearSet(R: Rng): Category == SetCategory with + 0: % + ++ \spad{0} represents the origin of the linear set + zero?: % -> Boolean + ++ \spad{zero? x} holds is \spad{x} is the origin. + *: (%,R) -> % + ++ \spad{r*x} is the left-dilation of \spad{x} by \spad{r}. + +@ + +<<category LINSET LinearSet>>= +)abbrev category LINSET LinearSet +++ Author: Gabriel Dos Reis +++ Date Created: May 31, 2009 +++ Date Last Modified: May 31, 2009 +++ Description: +++ A set is an \spad{R}-linear set if it is stable by dilation +++ by elements in the ring \spad{R}. This category differs from +++ \spad{Module} in that no other assumption (such as addition) +++ is made about the underlying set. +++ See Also: LeftLinearSet, RightLinearSet. +LinearSet(R: Rng): Category == Join(LeftLinearSet R, RightLinearSet R) + +@ + + \section{category ABELGRP AbelianGroup} <<category ABELGRP AbelianGroup>>= )abbrev category ABELGRP AbelianGroup @@ -29,13 +92,10 @@ ++ \spad{-(-x) = x} ++ \spad{x+(-x) = 0} -- following domain must be compiled with subsumption disabled -AbelianGroup(): Category == CancellationAbelianMonoid with - --operations - "-": % -> % ++ -x is the additive inverse of x. - "-": (%,%) -> % ++ x-y is the difference of x and y - ++ i.e. \spad{x + (-y)}. - -- subsumes the partial subtraction from previous - "*": (Integer,%) -> % ++ n*x is the product of x by the integer n. +AbelianGroup(): Category == Join(CancellationAbelianMonoid, LeftLinearSet Integer) with + -: % -> % ++ \spad{-x} is the additive inverse of \spad{x} + "-": (%,%) -> % ++ \spad{x-y} is the difference of \spad{x} + ++ and \spad{y} i.e. \spad{x + (-y)}. add (x:% - y:%):% == x+(-y) subtractIfCan(x:%, y:%):Union(%, "failed") == (x-y) :: Union(%,"failed") @@ -923,12 +983,10 @@ IntegralDomain(): Category == ++ \spad{ (a*b)*x = a*(b*x) } ++ \spad{ (a+b)*x = (a*x)+(b*x) } ++ \spad{ a*(x+y) = (a*x)+(a*y) } -LeftModule(R:Rng):Category == AbelianGroup with - --operations - "*": (R,%) -> % ++ r*x returns the left multiplication of the module element x - ++ by the ring element r. +LeftModule(R:Rng):Category == Join(AbelianGroup, LeftLinearSet R) @ + \section{category LINEXP LinearlyExplicitRingOver} <<category LINEXP LinearlyExplicitRingOver>>= )abbrev category LINEXP LinearlyExplicitRingOver @@ -972,7 +1030,7 @@ LinearlyExplicitRingOver(R:Ring): Category == Ring with ++ \spad{(a*b)*x = a*(b*x)} ++ \spad{(a+b)*x = (a*x)+(b*x)} ++ \spad{a*(x+y) = (a*x)+(a*y)} -Module(R:CommutativeRing): Category == BiModule(R,R) +Module(R:CommutativeRing): Category == Join(BiModule(R,R), LinearSet R) add if not(R is %) then x:%*r:R == r*x @@ -1502,12 +1560,10 @@ PrincipalIdealDomain(): Category == GcdDomain with ++ \spad{ x*(a*b) = (x*a)*b } ++ \spad{ x*(a+b) = (x*a)+(x*b) } ++ \spad{ (x+y)*x = (x*a)+(y*a) } -RightModule(R:Rng):Category == AbelianGroup with - --operations - "*": (%,R) -> % ++ x*r returns the right multiplication of the module element x - ++ by the ring element r. +RightModule(R:Rng):Category == Join(AbelianGroup, RightLinearSet R) @ + \section{category RING Ring} <<category RING Ring>>= )abbrev category RING Ring @@ -1524,7 +1580,6 @@ RightModule(R:Rng):Category == AbelianGroup with ++ The category of rings with unity, always associative, but ++ not necessarily commutative. ---Ring(): Category == Join(Rng,Monoid,LeftModule(%:Rng)) with Ring(): Category == Join(Rng,Monoid,LeftModule(%),CoercibleFrom Integer) with --operations characteristic: NonNegativeInteger @@ -1784,6 +1839,9 @@ VectorSpace(S:Field): Category == Module(S) with <<category CABMON CancellationAbelianMonoid>> <<category ABELGRP AbelianGroup>> <<category RNG Rng>> +<<category LLINSET LeftLinearSet>> +<<category RLINSET RightLinearSet>> +<<category LINSET LinearSet>> <<category LMODULE LeftModule>> <<category RMODULE RightModule>> <<category RING Ring>> |