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