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/*! @file Forward declares `boost::hana::Group`. Copyright Louis Dionne 2013-2022 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 namespace boost { namespace hana { //! @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(), 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() // right inverse //! plus(negate(x), x) == zero() // 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::%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()) == zero() //! @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 struct Group; }} // end namespace boost::hana #endif // !BOOST_HANA_FWD_CONCEPT_GROUP_HPP