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\documentclass{article}
\usepackage{open-axiom}
\begin{document}
\title{\$SPAD/src/algebra attreg.spad}
\author{The Axiom Team}
\maketitle
\begin{abstract}
\end{abstract}
\eject
\tableofcontents
\eject
\section{category ATTREG AttributeRegistry}
<<category ATTREG AttributeRegistry>>=
)abbrev category ATTREG AttributeRegistry
++ This category exports the attributes in the AXIOM Library
AttributeRegistry(): Category == with
commutative("*")
++ \spad{commutative("*")} is true if it has an operation
++ \spad{"*": (D,D) -> D} which is commutative.
unitsKnown
++ \spad{unitsKnown} is true if a monoid (a multiplicative semigroup
++ with a 1) has \spad{unitsKnown} means that
++ the operation \spadfun{recip} can only return "failed"
++ if its argument is not a unit.
leftUnitary
++ \spad{leftUnitary} is true if \spad{1 * x = x} for all x.
rightUnitary
++ \spad{rightUnitary} is true if \spad{x * 1 = x} for all x.
noZeroDivisors
++ \spad{noZeroDivisors} is true if \spad{x * y \~~= 0} implies
++ both x and y are non-zero.
canonicalUnitNormal
++ \spad{canonicalUnitNormal} is true if we can choose a canonical
++ representative for each class of associate elements, that is
++ \spad{associates?(a,b)} returns true if and only if
++ \spad{unitCanonical(a) = unitCanonical(b)}.
canonicalsClosed
++ \spad{canonicalsClosed} is true if
++ \spad{unitCanonical(a)*unitCanonical(b) = unitCanonical(a*b)}.
arbitraryPrecision
++ \spad{arbitraryPrecision} means the user can set the
++ precision for subsequent calculations.
partiallyOrderedSet
++ \spad{partiallyOrderedSet} is true if
++ a set with \spadop{<} which is transitive,
++ but \spad{not(a < b or a = b)}
++ does not necessarily imply \spad{b<a}.
central
++ \spad{central} is true if, given an algebra over a ring R,
++ the image of R is the center
++ of the algebra, i.e. the set of members of the algebra which commute
++ with all others is precisely the image of R in the algebra.
noetherian
++ \spad{noetherian} is true if all of its ideals are finitely generated.
additiveValuation
++ \spad{additiveValuation} implies
++ \spad{euclideanSize(a*b)=euclideanSize(a)+euclideanSize(b)}.
multiplicativeValuation
++ \spad{multiplicativeValuation} implies
++ \spad{euclideanSize(a*b)=euclideanSize(a)*euclideanSize(b)}.
NullSquare
++ \axiom{NullSquare} means that \axiom{[x,x] = 0} holds.
++ See \axiomType{LieAlgebra}.
JacobiIdentity
++ \axiom{JacobiIdentity} means that
++ \axiom{[x,[y,z]]+[y,[z,x]]+[z,[x,y]] = 0} holds.
++ See \axiomType{LieAlgebra}.
canonical
++ \spad{canonical} is true if and only if distinct elements have
++ distinct data structures. For example, a domain of mathematical objects
++ which has the \spad{canonical} attribute means that two objects
++ are mathematically
++ equal if and only if their data structures are equal.
@
\section{License}
<<license>>=
--Copyright (c) 1991-2002, The Numerical ALgorithms Group Ltd.
--All rights reserved.
--
--Redistribution and use in source and binary forms, with or without
--modification, are permitted provided that the following conditions are
--met:
--
-- - Redistributions of source code must retain the above copyright
-- notice, this list of conditions and the following disclaimer.
--
-- - Redistributions in binary form must reproduce the above copyright
-- notice, this list of conditions and the following disclaimer in
-- the documentation and/or other materials provided with the
-- distribution.
--
-- - Neither the name of The Numerical ALgorithms Group Ltd. nor the
-- names of its contributors may be used to endorse or promote products
-- derived from this software without specific prior written permission.
--
--THIS SOFTWARE IS PROVIDED BY THE COPYRIGHT HOLDERS AND CONTRIBUTORS "AS
--IS" AND ANY EXPRESS OR IMPLIED WARRANTIES, INCLUDING, BUT NOT LIMITED
--TO, THE IMPLIED WARRANTIES OF MERCHANTABILITY AND FITNESS FOR A
--PARTICULAR PURPOSE ARE DISCLAIMED. IN NO EVENT SHALL THE COPYRIGHT OWNER
--OR CONTRIBUTORS BE LIABLE FOR ANY DIRECT, INDIRECT, INCIDENTAL, SPECIAL,
--EXEMPLARY, OR CONSEQUENTIAL DAMAGES (INCLUDING, BUT NOT LIMITED TO,
--PROCUREMENT OF SUBSTITUTE GOODS OR SERVICES; LOSS OF USE, DATA, OR
--PROFITS; OR BUSINESS INTERRUPTION) HOWEVER CAUSED AND ON ANY THEORY OF
--LIABILITY, WHETHER IN CONTRACT, STRICT LIABILITY, OR TORT (INCLUDING
--NEGLIGENCE OR OTHERWISE) ARISING IN ANY WAY OUT OF THE USE OF THIS
--SOFTWARE, EVEN IF ADVISED OF THE POSSIBILITY OF SUCH DAMAGE.
@
<<*>>=
<<license>>
<<category ATTREG AttributeRegistry>>
@
\eject
\begin{thebibliography}{99}
\bibitem{1} nothing
\end{thebibliography}
\end{document}
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