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diff --git a/boost/hana/fwd/concept/monoid.hpp b/boost/hana/fwd/concept/monoid.hpp new file mode 100644 index 0000000000..5486d3d1d8 --- /dev/null +++ b/boost/hana/fwd/concept/monoid.hpp @@ -0,0 +1,101 @@ +/*! +@file +Forward declares `boost::hana::Monoid`. + +@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_MONOID_HPP +#define BOOST_HANA_FWD_CONCEPT_MONOID_HPP + +#include <boost/hana/config.hpp> + + +BOOST_HANA_NAMESPACE_BEGIN + //! @ingroup group-concepts + //! @defgroup group-Monoid Monoid + //! The `Monoid` concept represents data types with an associative + //! binary operation that has an identity. + //! + //! Specifically, a [Monoid][1] is a basic algebraic structure typically + //! used in mathematics to construct more complex algebraic structures + //! like `Group`s, `Ring`s and so on. They are useful in several contexts, + //! notably to define the properties of numbers in a granular way. At its + //! core, a `Monoid` is a set `S` of objects along with a binary operation + //! (let's say `+`) that is associative and that has an identity in `S`. + //! There are many examples of `Monoid`s: + //! - strings with concatenation and the empty string as the identity + //! - integers with addition and `0` as the identity + //! - integers with multiplication and `1` as the identity + //! - many others... + //! + //! As you can see with the integers, there are some sets that can be + //! viewed as a monoid in more than one way, depending on the choice + //! of the binary operation and identity. The method names used here + //! refer to the monoid of integers under addition; `plus` is the binary + //! operation and `zero` is the identity element of that operation. + //! + //! + //! Minimal complete definition + //! --------------------------- + //! `plus` and `zero` satisfying the laws + //! + //! + //! Laws + //! ---- + //! For all objects `x`, `y` and `z` of a `Monoid` `M`, the following + //! laws must be satisfied: + //! @code + //! plus(zero<M>(), x) == x // left zero + //! plus(x, zero<M>()) == x // right zero + //! plus(x, plus(y, z)) == plus(plus(x, y), z) // associativity + //! @endcode + //! + //! + //! 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 `Monoid` + //! is automatically defined by setting + //! @code + //! plus(x, y) = (x + y) + //! zero<T>() = static_cast<T>(0) + //! @endcode + //! + //! > #### Rationale for not making `bool` a `Monoid` by default + //! > First, it makes no sense whatsoever to define an additive `Monoid` + //! > over the `bool` type. Also, it could make sense to define a `Monoid` + //! > with logical conjunction or disjunction. However, C++ allows `bool`s + //! > to be added, and the method names of this concept really suggest + //! > addition. In line with the principle of least surprise, no model + //! > is provided by default. + //! + //! + //! Structure-preserving functions + //! ------------------------------ + //! Let `A` and `B` be two `Monoid`s. A function `f : A -> B` is said + //! to be a [Monoid morphism][2] if it preserves the monoidal 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)) + //! f(zero<A>()) == zero<B>() + //! @endcode + //! Functions with these properties interact nicely with `Monoid`s, which + //! is why they are given such a special treatment. + //! + //! + //! [1]: http://en.wikipedia.org/wiki/Monoid + //! [2]: http://en.wikipedia.org/wiki/Monoid#Monoid_homomorphisms + template <typename M> + struct Monoid; +BOOST_HANA_NAMESPACE_END + +#endif // !BOOST_HANA_FWD_CONCEPT_MONOID_HPP |