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1 | /*! |
2 | @file | |
3 | Forward declares `boost::hana::Monoid`. | |
4 | ||
5 | @copyright Louis Dionne 2013-2016 | |
6 | Distributed under the Boost Software License, Version 1.0. | |
7 | (See accompanying file LICENSE.md or copy at http://boost.org/LICENSE_1_0.txt) | |
8 | */ | |
9 | ||
10 | #ifndef BOOST_HANA_FWD_CONCEPT_MONOID_HPP | |
11 | #define BOOST_HANA_FWD_CONCEPT_MONOID_HPP | |
12 | ||
13 | #include <boost/hana/config.hpp> | |
14 | ||
15 | ||
16 | BOOST_HANA_NAMESPACE_BEGIN | |
17 | //! @ingroup group-concepts | |
18 | //! @defgroup group-Monoid Monoid | |
19 | //! The `Monoid` concept represents data types with an associative | |
20 | //! binary operation that has an identity. | |
21 | //! | |
22 | //! Specifically, a [Monoid][1] is a basic algebraic structure typically | |
23 | //! used in mathematics to construct more complex algebraic structures | |
24 | //! like `Group`s, `Ring`s and so on. They are useful in several contexts, | |
25 | //! notably to define the properties of numbers in a granular way. At its | |
26 | //! core, a `Monoid` is a set `S` of objects along with a binary operation | |
27 | //! (let's say `+`) that is associative and that has an identity in `S`. | |
28 | //! There are many examples of `Monoid`s: | |
29 | //! - strings with concatenation and the empty string as the identity | |
30 | //! - integers with addition and `0` as the identity | |
31 | //! - integers with multiplication and `1` as the identity | |
32 | //! - many others... | |
33 | //! | |
34 | //! As you can see with the integers, there are some sets that can be | |
35 | //! viewed as a monoid in more than one way, depending on the choice | |
36 | //! of the binary operation and identity. The method names used here | |
37 | //! refer to the monoid of integers under addition; `plus` is the binary | |
38 | //! operation and `zero` is the identity element of that operation. | |
39 | //! | |
40 | //! | |
41 | //! Minimal complete definition | |
42 | //! --------------------------- | |
43 | //! `plus` and `zero` satisfying the laws | |
44 | //! | |
45 | //! | |
46 | //! Laws | |
47 | //! ---- | |
48 | //! For all objects `x`, `y` and `z` of a `Monoid` `M`, the following | |
49 | //! laws must be satisfied: | |
50 | //! @code | |
51 | //! plus(zero<M>(), x) == x // left zero | |
52 | //! plus(x, zero<M>()) == x // right zero | |
53 | //! plus(x, plus(y, z)) == plus(plus(x, y), z) // associativity | |
54 | //! @endcode | |
55 | //! | |
56 | //! | |
57 | //! Concrete models | |
58 | //! --------------- | |
59 | //! `hana::integral_constant` | |
60 | //! | |
61 | //! | |
62 | //! Free model for non-boolean arithmetic data types | |
63 | //! ------------------------------------------------ | |
64 | //! A data type `T` is arithmetic if `std::is_arithmetic<T>::%value` is | |
65 | //! true. For a non-boolean arithmetic data type `T`, a model of `Monoid` | |
66 | //! is automatically defined by setting | |
67 | //! @code | |
68 | //! plus(x, y) = (x + y) | |
69 | //! zero<T>() = static_cast<T>(0) | |
70 | //! @endcode | |
71 | //! | |
72 | //! > #### Rationale for not making `bool` a `Monoid` by default | |
73 | //! > First, it makes no sense whatsoever to define an additive `Monoid` | |
74 | //! > over the `bool` type. Also, it could make sense to define a `Monoid` | |
75 | //! > with logical conjunction or disjunction. However, C++ allows `bool`s | |
76 | //! > to be added, and the method names of this concept really suggest | |
77 | //! > addition. In line with the principle of least surprise, no model | |
78 | //! > is provided by default. | |
79 | //! | |
80 | //! | |
81 | //! Structure-preserving functions | |
82 | //! ------------------------------ | |
83 | //! Let `A` and `B` be two `Monoid`s. A function `f : A -> B` is said | |
84 | //! to be a [Monoid morphism][2] if it preserves the monoidal structure | |
85 | //! between `A` and `B`. Rigorously, for all objects `x, y` of data | |
86 | //! type `A`, | |
87 | //! @code | |
88 | //! f(plus(x, y)) == plus(f(x), f(y)) | |
89 | //! f(zero<A>()) == zero<B>() | |
90 | //! @endcode | |
91 | //! Functions with these properties interact nicely with `Monoid`s, which | |
92 | //! is why they are given such a special treatment. | |
93 | //! | |
94 | //! | |
95 | //! [1]: http://en.wikipedia.org/wiki/Monoid | |
96 | //! [2]: http://en.wikipedia.org/wiki/Monoid#Monoid_homomorphisms | |
97 | template <typename M> | |
98 | struct Monoid; | |
99 | BOOST_HANA_NAMESPACE_END | |
100 | ||
101 | #endif // !BOOST_HANA_FWD_CONCEPT_MONOID_HPP |