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 /*!
 @file
 Forward declares `boost::hana::Monoid`.
 
 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_MONOID_HPP
 #define BOOST_HANA_FWD_CONCEPT_MONOID_HPP
 
 #include <boost/hana/config.hpp>
 
 
 namespace boost { namespace hana {
     //! @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;
 }} // end namespace boost::hana
 
 #endif // !BOOST_HANA_FWD_CONCEPT_MONOID_HPP

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