diff options
Diffstat (limited to 'src/algebra/catdef.spad.pamphlet')
| -rw-r--r-- | src/algebra/catdef.spad.pamphlet | 48 |
1 files changed, 24 insertions, 24 deletions
diff --git a/src/algebra/catdef.spad.pamphlet b/src/algebra/catdef.spad.pamphlet index c7263464..602e3eb6 100644 --- a/src/algebra/catdef.spad.pamphlet +++ b/src/algebra/catdef.spad.pamphlet @@ -138,7 +138,7 @@ AbelianMonoid(): Category == AbelianSemiGroup with ++ sample yields a value of type % zero?: % -> Boolean ++ zero?(x) tests if x is equal to 0. - "*": (NonNegativeInteger,%) -> % + *: (NonNegativeInteger,%) -> % ++ n * x is left-multiplication by a non negative integer add import RepeatedDoubling(%) @@ -176,8 +176,8 @@ import PositiveInteger ++ \spad{commutative("+":(%,%)->%)}\tab{30}\spad{ x+y = y+x } AbelianSemiGroup(): Category == SetCategory with --operations - "+": (%,%) -> % ++ x+y computes the sum of x and y. - "*": (PositiveInteger,%) -> % + +: (%,%) -> % ++ x+y computes the sum of x and y. + *: (PositiveInteger,%) -> % ++ n*x computes the left-multiplication of x by the positive integer n. ++ This is equivalent to adding x to itself n times. add @@ -234,10 +234,10 @@ import Boolean ++ \spadtype{BasicType} is the basic category for describing a collection ++ of elements with \spadop{=} (equality). BasicType(): Category == with - "=": (%,%) -> Boolean ++ x=y tests if x and y are equal. - "~=": (%,%) -> Boolean ++ x~=y tests if x and y are not equal. + =: (%,%) -> Boolean ++ x=y tests if x and y are equal. + ~=: (%,%) -> Boolean ++ x~=y tests if x and y are not equal. add - _~_=(x:%,y:%) : Boolean == not(x=y) + x:% ~= y:% == not(x=y) @ @@ -463,7 +463,7 @@ DifferentialExtension(R:Ring): Category == Ring with DivisionRing(): Category == Join(EntireRing, Algebra Fraction Integer) with - "**": (%,Integer) -> % + **: (%,Integer) -> % ++ x**n returns x raised to the integer power n. inv : % -> % ++ inv x returns the multiplicative inverse of x. @@ -548,10 +548,10 @@ EuclideanDomain(): Category == PrincipalIdealDomain with ++ \spad{quotient} and \spad{remainder}, ++ where the remainder is smaller (see \spadfunFrom{sizeLess?}{EuclideanDomain}) ++ than the divisor y. - "quo" : (%,%) -> % + quo : (%,%) -> % ++ x quo y is the same as \spad{divide(x,y).quotient}. ++ See \spadfunFrom{divide}{EuclideanDomain}. - "rem": (%,%) -> % + rem: (%,%) -> % ++ x rem y is the same as \spad{divide(x,y).remainder}. ++ See \spadfunFrom{divide}{EuclideanDomain}. extendedEuclidean: (%,%) -> Record(coef1:%,coef2:%,generator:%) @@ -692,7 +692,7 @@ EuclideanDomain(): Category == PrincipalIdealDomain with Field(): Category == Join(EuclideanDomain,UniqueFactorizationDomain, DivisionRing) with --operations - "/": (%,%) -> % + /: (%,%) -> % ++ x/y divides the element x by the element y. ++ Error: if y is 0. canonicalUnitNormal ++ either 0 or 1. @@ -875,9 +875,9 @@ GcdDomain(): Category == IntegralDomain with ++ \spad{rightInverse("*":(%,%)->%,inv)}\tab{30}\spad{ x*inv(x) = 1 } Group(): Category == Monoid with --operations - inv: % -> % ++ inv(x) returns the inverse of x. - "/": (%,%) -> % ++ x/y is the same as x times the inverse of y. - "**": (%,Integer) -> % ++ x**n returns x raised to the integer power n. + inv: % -> % ++ inv(x) returns the inverse of x. + /: (%,%) -> % ++ x/y is the same as x times the inverse of y. + **: (%,Integer) -> % ++ x**n returns x raised to the integer power n. unitsKnown ++ unitsKnown asserts that recip only returns ++ "failed" for non-units. conjugate: (%,%) -> % @@ -921,7 +921,7 @@ IntegralDomain(): Category == -- Join(CommutativeRing, Algebra(%:CommutativeRing), EntireRing) with Join(CommutativeRing, Algebra(%), EntireRing) with --operations - "exquo": (%,%) -> Union(%,"failed") + exquo: (%,%) -> Union(%,"failed") ++ exquo(a,b) either returns an element c such that ++ \spad{c*b=a} or "failed" if no such element can be found. unitNormal: % -> Record(unit:%,canonical:%,associate:%) @@ -946,7 +946,7 @@ IntegralDomain(): Category == if not (% has Field) then unitNormal(x) == [1$%,x,1$%]$UCA -- the non-canonical definition unitCanonical(x) == unitNormal(x).canonical -- always true - recip(x) == if zero? x then "failed" else _exquo(1$%,x) + recip(x) == if zero? x then "failed" else 1$% exquo x unit?(x) == (recip x case "failed" => false; true) if % has canonicalUnitNormal then associates?(x,y) == @@ -1062,7 +1062,7 @@ Monoid(): Category == SemiGroup with 1: constant -> % ++ 1 is the multiplicative identity. sample: constant -> % ++ sample yields a value of type % one?: % -> Boolean ++ one?(x) tests if x is equal to 1. - "**": (%,NonNegativeInteger) -> % ++ x**n returns the repeated product + **: (%,NonNegativeInteger) -> % ++ x**n returns the repeated product ++ of x n times, i.e. exponentiation. recip: % -> Union(%,"failed") ++ recip(x) tries to compute the multiplicative inverse for x @@ -1330,13 +1330,13 @@ import Boolean OrderedSet(): Category == SetCategory with --operations - "<": (%,%) -> Boolean + <: (%,%) -> Boolean ++ x < y is a strict total ordering on the elements of the set. - ">": (%, %) -> Boolean + >: (%, %) -> Boolean ++ x > y is a greater than test. - ">=": (%, %) -> Boolean + >=: (%, %) -> Boolean ++ x >= y is a greater than or equal test. - "<=": (%, %) -> Boolean + <=: (%, %) -> Boolean ++ x <= y is a less than or equal test. max: (%,%) -> % @@ -1651,9 +1651,9 @@ import PositiveInteger ++ \spad{commutative("*":(%,%)->%)}\tab{30}\spad{ x*y = y*x } SemiGroup(): Category == SetCategory with --operations - "*": (%,%) -> % ++ x*y returns the product of x and y. - "**": (%,PositiveInteger) -> % ++ x**n returns the repeated product - ++ of x n times, i.e. exponentiation. + *: (%,%) -> % ++ x*y returns the product of x and y. + **: (%,PositiveInteger) -> % ++ x**n returns the repeated product + ++ of x n times, i.e. exponentiation. add import RepeatedSquaring(%) x:% ** n:PositiveInteger == expt(x,n) @@ -1782,7 +1782,7 @@ UniqueFactorizationDomain(): Category == GcdDomain with ++ Vector Spaces (not necessarily finite dimensional) over a field. VectorSpace(S:Field): Category == Module(S) with - "/" : (%, S) -> % + / : (%, S) -> % ++ x/y divides the vector x by the scalar y. dimension: () -> CardinalNumber ++ dimension() returns the dimensionality of the vector space. |