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authordos-reis <gdr@axiomatics.org>2011-03-12 01:01:24 +0000
committerdos-reis <gdr@axiomatics.org>2011-03-12 01:01:24 +0000
commit4a92a0cc2bca6807ea4362797cfe6939bad4889b (patch)
tree5394491ae9aa1433eb2d3d8971205875ec557750 /src/algebra/catdef.spad.pamphlet
parent6672aec6078bc810e4fb018168a8ea1d9959a186 (diff)
downloadopen-axiom-4a92a0cc2bca6807ea4362797cfe6939bad4889b.tar.gz
* algebra/catdef.spad.pamphlet (OrderedAbelianGroup): Export
negative?, abs, sign, with default implementations. (OrderedAbelianMonoid): Export positive?, with default. (OrderedRing): Don't export them.
Diffstat (limited to 'src/algebra/catdef.spad.pamphlet')
-rw-r--r--src/algebra/catdef.spad.pamphlet55
1 files changed, 28 insertions, 27 deletions
diff --git a/src/algebra/catdef.spad.pamphlet b/src/algebra/catdef.spad.pamphlet
index 7dfc10fc..fdb915fa 100644
--- a/src/algebra/catdef.spad.pamphlet
+++ b/src/algebra/catdef.spad.pamphlet
@@ -1271,7 +1271,7 @@ Monoid(): Category == SemiGroup with
)abbrev category OAGROUP OrderedAbelianGroup
++ Author:
++ Date Created:
-++ Date Last Updated:
+++ Date Last Updated: March 10, 2011
++ Basic Functions:
++ Related Constructors:
++ Also See:
@@ -1283,9 +1283,26 @@ Monoid(): Category == SemiGroup with
++ the ordering.
OrderedAbelianGroup(): Category ==
- Join(OrderedCancellationAbelianMonoid, AbelianGroup)
-
+ Join(OrderedCancellationAbelianMonoid, AbelianGroup) with
+ negative?: % -> Boolean
+ ++ \spad{negative?(x)} holds when \spad{x} is less than \spad{0}.
+ sign: % -> Integer
+ ++ \spad{sign(x)} is \spad{1} if \spad{x} is positive,
+ ++ \spad{-1} if \spad{x} is negative, and \spad{0} otherwise.
+ abs: % -> %
+ ++ \spad{abs(x)} returns the absolute value of \spad{x}.
+ add
+ negative? x == x < 0
+ sign x ==
+ positive? x => 1
+ negative? x => -1
+ 0
+ abs x ==
+ positive? x => x
+ negative? x => -x
+ 0
@
+
\section{category OAMON OrderedAbelianMonoid}
<<category OAMON OrderedAbelianMonoid>>=
)abbrev category OAMON OrderedAbelianMonoid
@@ -1303,7 +1320,12 @@ OrderedAbelianGroup(): Category ==
++ preserves the ordering.
OrderedAbelianMonoid(): Category ==
- Join(OrderedAbelianSemiGroup, AbelianMonoid)
+ Join(OrderedAbelianSemiGroup, AbelianMonoid) with
+ positive?: % -> Boolean
+ ++ \spad{positive?(x)} holds when \spad{x} is greater
+ ++ than \spad{0}.
+ add
+ positive? x == 0 < x
@
\section{category OAMONS OrderedAbelianMonoidSup}
@@ -1456,7 +1478,7 @@ OrderedMonoid(): Category == Join(OrderedSemiGroup, Monoid)
)abbrev category ORDRING OrderedRing
++ Author:
++ Date Created:
-++ Date Last Updated:
+++ Date Last Updated: March 10, 2011
++ Basic Functions:
++ Related Constructors:
++ Also See:
@@ -1470,28 +1492,7 @@ OrderedMonoid(): Category == Join(OrderedSemiGroup, Monoid)
++ Axiom:
++ \spad{0<a and b<c => ab< ac}
-OrderedRing(): Category == Join(OrderedAbelianGroup,Ring,Monoid) with
- positive?: % -> Boolean
- ++ positive?(x) tests whether x is strictly greater than 0.
- negative?: % -> Boolean
- ++ negative?(x) tests whether x is strictly less than 0.
- sign : % -> Integer
- ++ sign(x) is 1 if x is positive, -1 if x is negative, 0 if x equals 0.
- abs : % -> %
- ++ abs(x) returns the absolute value of x.
- add
- positive? x == x>0
- negative? x == x<0
- sign x ==
- positive? x => 1
- negative? x => -1
- zero? x => 0
- error "x satisfies neither positive?, negative? or zero?"
- abs x ==
- positive? x => x
- negative? x => -x
- zero? x => 0
- error "x satisfies neither positive?, negative? or zero?"
+OrderedRing(): Category == Join(OrderedAbelianGroup,Ring,Monoid)
@