diff --git a/Algebraic_foundations/doc/Algebraic_foundations/CGAL/number_utils.h b/Algebraic_foundations/doc/Algebraic_foundations/CGAL/number_utils.h index 95da8f04182..ee13e024624 100644 --- a/Algebraic_foundations/doc/Algebraic_foundations/CGAL/number_utils.h +++ b/Algebraic_foundations/doc/Algebraic_foundations/CGAL/number_utils.h @@ -22,7 +22,7 @@ namespace CGAL { \ingroup PkgAlgebraicFoundations The template function `compare` compares the first argument with respect to -the second, i.e. it returns `CGAL::LARGER` if \f$ x\f$ is larger then \f$ y\f$. +the second, i.e.\ it returns `CGAL::LARGER` if \f$ x\f$ is larger then \f$ y\f$. In case the argument types `NT1` and `NT2` differ, `compare` is performed with the semantic of the type determined via @@ -138,9 +138,9 @@ namespace CGAL { /*! \ingroup PkgAlgebraicFoundations -The function `integral_division` (a.k.a. exact division or division without remainder) +The function `integral_division` (a.k.a.\ exact division or division without remainder) maps ring elements \f$ (x,y)\f$ to ring element \f$ z\f$ such that \f$ x = yz\f$ if such a \f$ z\f$ -exists (i.e. if \f$ x\f$ is divisible by \f$ y\f$). Otherwise the effect of invoking +exists (i.e.\ if \f$ x\f$ is divisible by \f$ y\f$). Otherwise the effect of invoking this operation is undefined. Since the ring represented is an integral domain, \f$ z\f$ is uniquely defined if it exists. diff --git a/Algebraic_foundations/doc/Algebraic_foundations/Concepts/AlgebraicStructureTraits--Gcd.h b/Algebraic_foundations/doc/Algebraic_foundations/Concepts/AlgebraicStructureTraits--Gcd.h index 403b80b16d9..acbc12ada90 100644 --- a/Algebraic_foundations/doc/Algebraic_foundations/Concepts/AlgebraicStructureTraits--Gcd.h +++ b/Algebraic_foundations/doc/Algebraic_foundations/Concepts/AlgebraicStructureTraits--Gcd.h @@ -9,7 +9,7 @@ The greatest common divisor (\f$ gcd\f$) of ring elements \f$ x\f$ and \f$ y\f$ ring element \f$ d\f$ (up to a unit) with the property that any common divisor of \f$ x\f$ and \f$ y\f$ also divides \f$ d\f$. (In other words: \f$ d\f$ is the greatest lower bound of \f$ x\f$ and \f$ y\f$ in the partial order of divisibility.) We demand the \f$ gcd\f$ to be -unit-normal (i.e. have unit part 1). +unit-normal (i.e.\ have unit part 1). \f$ gcd(0,0)\f$ is defined as \f$ 0\f$, since \f$ 0\f$ is the greatest element with respect to the partial order of divisibility. This is because an element \f$ a \in R\f$ is said to divide \f$ b \in R\f$, iff \f$ \exists r \in R\f$ such that \f$ a \cdot r = b\f$. diff --git a/Algebraic_foundations/doc/Algebraic_foundations/Concepts/AlgebraicStructureTraits--IntegralDivision.h b/Algebraic_foundations/doc/Algebraic_foundations/Concepts/AlgebraicStructureTraits--IntegralDivision.h index d3e153331b5..9fa38872fa7 100644 --- a/Algebraic_foundations/doc/Algebraic_foundations/Concepts/AlgebraicStructureTraits--IntegralDivision.h +++ b/Algebraic_foundations/doc/Algebraic_foundations/Concepts/AlgebraicStructureTraits--IntegralDivision.h @@ -7,7 +7,7 @@ Integral division (a.k.a. exact division or division without remainder) maps ring elements \f$ (x,y)\f$ to ring element \f$ z\f$ such that \f$ x = yz\f$ if such a \f$ z\f$ -exists (i.e. if \f$ x\f$ is divisible by \f$ y\f$). Otherwise the effect of invoking +exists (i.e.\ if \f$ x\f$ is divisible by \f$ y\f$). Otherwise the effect of invoking this operation is undefined. Since the ring represented is an integral domain, \f$ z\f$ is uniquely defined if it exists. diff --git a/Algebraic_foundations/doc/Algebraic_foundations/Concepts/AlgebraicStructureTraits--IsSquare.h b/Algebraic_foundations/doc/Algebraic_foundations/Concepts/AlgebraicStructureTraits--IsSquare.h index edc6438ace9..b4388a5208a 100644 --- a/Algebraic_foundations/doc/Algebraic_foundations/Concepts/AlgebraicStructureTraits--IsSquare.h +++ b/Algebraic_foundations/doc/Algebraic_foundations/Concepts/AlgebraicStructureTraits--IsSquare.h @@ -44,7 +44,7 @@ typedef Hidden_type second_argument; /// @{ /*! -returns true in case \f$ x\f$ is a square, i.e. \f$ x = y*y\f$. +returns true in case \f$ x\f$ is a square, i.e.\ \f$ x = y*y\f$. \post \f$ unit\_part(y) == 1\f$. */ diff --git a/Algebraic_foundations/doc/Algebraic_foundations/Concepts/AlgebraicStructureTraits--UnitPart.h b/Algebraic_foundations/doc/Algebraic_foundations/Concepts/AlgebraicStructureTraits--UnitPart.h index dae09367e66..cffc3ed7288 100644 --- a/Algebraic_foundations/doc/Algebraic_foundations/Concepts/AlgebraicStructureTraits--UnitPart.h +++ b/Algebraic_foundations/doc/Algebraic_foundations/Concepts/AlgebraicStructureTraits--UnitPart.h @@ -8,7 +8,7 @@ element. The mathematical definition of unit part is as follows: Two ring elements \f$ a\f$ and \f$ b\f$ are said to be associate if there exists an invertible ring element -(i.e. a unit) \f$ u\f$ such that \f$ a = ub\f$. This defines an equivalence relation. +(i.e.\ a unit) \f$ u\f$ such that \f$ a = ub\f$. This defines an equivalence relation. We can distinguish exactly one element of every equivalence class as being unit normal. Then each element of a ring possesses a factorization into a unit (called its unit part) and a unit-normal ring element diff --git a/Algebraic_foundations/doc/Algebraic_foundations/Concepts/IntegralDomainWithoutDivision.h b/Algebraic_foundations/doc/Algebraic_foundations/Concepts/IntegralDomainWithoutDivision.h index 3f13d2ca5df..ec744adc011 100644 --- a/Algebraic_foundations/doc/Algebraic_foundations/Concepts/IntegralDomainWithoutDivision.h +++ b/Algebraic_foundations/doc/Algebraic_foundations/Concepts/IntegralDomainWithoutDivision.h @@ -6,7 +6,7 @@ This is the most basic concept for algebraic structures considered within CGAL. A model `IntegralDomainWithoutDivision` represents an integral domain, -i.e. commutative ring with 0, 1, +, * and unity free of zero divisors. +i.e.\ commutative ring with 0, 1, +, * and unity free of zero divisors. Note: A model is not required to offer the always well defined integral division.