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diff --git a/boost/hana/fwd/concept/group.hpp b/boost/hana/fwd/concept/group.hpp new file mode 100644 index 0000000000..a7ec238d54 --- /dev/null +++ b/boost/hana/fwd/concept/group.hpp @@ -0,0 +1,111 @@ +/*! +@file +Forward declares `boost::hana::Group`. + +@copyright Louis Dionne 2013-2016 +Distributed under the Boost Software License, Version 1.0. +(See accompanying file LICENSE.md or copy at http://boost.org/LICENSE_1_0.txt) + */ + +#ifndef BOOST_HANA_FWD_CONCEPT_GROUP_HPP +#define BOOST_HANA_FWD_CONCEPT_GROUP_HPP + +#include <boost/hana/config.hpp> + + +BOOST_HANA_NAMESPACE_BEGIN + //! @ingroup group-concepts + //! @defgroup group-Group Group + //! The `Group` concept represents `Monoid`s where all objects have + //! an inverse w.r.t. the `Monoid`'s binary operation. + //! + //! A [Group][1] is an algebraic structure built on top of a `Monoid` + //! which adds the ability to invert the action of the `Monoid`'s binary + //! operation on any element of the set. Specifically, a `Group` is a + //! `Monoid` `(S, +)` such that every element `s` in `S` has an inverse + //! (say `s'`) which is such that + //! @code + //! s + s' == s' + s == identity of the Monoid + //! @endcode + //! + //! There are many examples of `Group`s, one of which would be the + //! additive `Monoid` on integers, where the inverse of any integer + //! `n` is the integer `-n`. The method names used here refer to + //! exactly this model. + //! + //! + //! Minimal complete definitions + //! ---------------------------- + //! 1. `minus`\n + //! When `minus` is specified, the `negate` method is defaulted by setting + //! @code + //! negate(x) = minus(zero<G>(), x) + //! @endcode + //! + //! 2. `negate`\n + //! When `negate` is specified, the `minus` method is defaulted by setting + //! @code + //! minus(x, y) = plus(x, negate(y)) + //! @endcode + //! + //! + //! Laws + //! ---- + //! For all objects `x` of a `Group` `G`, the following laws must be + //! satisfied: + //! @code + //! plus(x, negate(x)) == zero<G>() // right inverse + //! plus(negate(x), x) == zero<G>() // left inverse + //! @endcode + //! + //! + //! Refined concept + //! --------------- + //! `Monoid` + //! + //! + //! Concrete models + //! --------------- + //! `hana::integral_constant` + //! + //! + //! Free model for non-boolean arithmetic data types + //! ------------------------------------------------ + //! A data type `T` is arithmetic if `std::is_arithmetic<T>::%value` is + //! true. For a non-boolean arithmetic data type `T`, a model of `Group` + //! is automatically defined by setting + //! @code + //! minus(x, y) = (x - y) + //! negate(x) = -x + //! @endcode + //! + //! @note + //! The rationale for not providing a Group model for `bool` is the same + //! as for not providing a `Monoid` model. + //! + //! + //! Structure-preserving functions + //! ------------------------------ + //! Let `A` and `B` be two `Group`s. A function `f : A -> B` is said to + //! be a [Group morphism][2] if it preserves the group structure between + //! `A` and `B`. Rigorously, for all objects `x, y` of data type `A`, + //! @code + //! f(plus(x, y)) == plus(f(x), f(y)) + //! @endcode + //! Because of the `Group` structure, it is easy to prove that the + //! following will then also be satisfied: + //! @code + //! f(negate(x)) == negate(f(x)) + //! f(zero<A>()) == zero<B>() + //! @endcode + //! Functions with these properties interact nicely with `Group`s, which + //! is why they are given such a special treatment. + //! + //! + //! [1]: http://en.wikipedia.org/wiki/Group_(mathematics) + //! [2]: http://en.wikipedia.org/wiki/Group_homomorphism + template <typename G> + struct Group; +BOOST_HANA_NAMESPACE_END + +#endif // !BOOST_HANA_FWD_CONCEPT_GROUP_HPP |