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<!DOCTYPE html><html lang="en"><head><meta charset="utf-8"><meta name="viewport" content="width=device-width, initial-scale=1.0"><meta name="generator" content="rustdoc"><meta name="description" content="API documentation for the Rust `general` mod in crate `alga`."><meta name="keywords" content="rust, rustlang, rust-lang, general"><title>alga::general - Rust</title><link rel="stylesheet" type="text/css" href="../../normalize.css"><link rel="stylesheet" type="text/css" href="../../rustdoc.css" id="mainThemeStyle"><link rel="stylesheet" type="text/css" href="../../dark.css"><link rel="stylesheet" type="text/css" href="../../light.css" id="themeStyle"><script src="../../storage.js"></script><noscript><link rel="stylesheet" href="../../noscript.css"></noscript><link rel="shortcut icon" href="../../favicon.ico"><style type="text/css">#crate-search{background-image:url("../../down-arrow.svg");}</style></head><body class="rustdoc mod"><!--[if lte IE 8]><div class="warning">This old browser is unsupported and will most likely display funky things.</div><![endif]--><nav class="sidebar"><div class="sidebar-menu">&#9776;</div><a href='../../alga/index.html'><div class='logo-container'><img src='../../rust-logo.png' alt='logo'></div></a><p class='location'>Module general</p><div class="sidebar-elems"><div class="block items"><ul><li><a href="#structs">Structs</a></li><li><a href="#traits">Traits</a></li></ul></div><p class='location'><a href='../index.html'>alga</a></p><script>window.sidebarCurrent = {name: 'general', ty: 'mod', relpath: '../'};</script><script defer src="../sidebar-items.js"></script></div></nav><div class="theme-picker"><button id="theme-picker" aria-label="Pick another theme!"><img src="../../brush.svg" width="18" alt="Pick another theme!"></button><div id="theme-choices"></div></div><script src="../../theme.js"></script><nav class="sub"><form class="search-form js-only"><div class="search-container"><div><select id="crate-search"><option value="All crates">All crates</option></select><input class="search-input" name="search" autocomplete="off" spellcheck="false" placeholder="Click or press ‘S’ to search, ‘?’ for more options…" type="search"></div><a id="settings-menu" href="../../settings.html"><img src="../../wheel.svg" width="18" alt="Change settings"></a></div></form></nav><section id="main" class="content"><h1 class='fqn'><span class='out-of-band'><span id='render-detail'><a id="toggle-all-docs" href="javascript:void(0)" title="collapse all docs">[<span class='inner'>&#x2212;</span>]</a></span><a class='srclink' href='../../src/alga/general/mod.rs.html#15-200' title='goto source code'>[src]</a></span><span class='in-band'>Module <a href='../index.html'>alga</a>::<wbr><a class="mod" href=''>general</a></span></h1><div class='docblock'><p>Fundamental algebraic structures.</p>
<p>For most applications requiring an abstraction over the reals, <code>RealField</code>
should be sufficient.</p>
<h2 id="algebraic-properties" class="section-header"><a href="#algebraic-properties">Algebraic properties</a></h2>
<p>The goal of algebraic structures is to allow elements of sets to be combined together using one
or several operators. The number and properties of those operators characterize the algebraic
structure. Abstract operators are usually noted <code>∘</code>, <code>+</code>, or <code>×</code>. The last two are preferred
when their behavior conform with the usual meaning of addition and multiplication of reals.
Let <code>Self</code> be a set. Here is a list of the most common properties those operator may fulfill:</p>
<pre><code class="language-notrust">(Closure) a, b ∈ Self ⇒ a ∘ b ∈ Self,
(Divisibility) ∀ a, b ∈ Self, ∃! r, l ∈ Self such that l ∘ a = b and a ∘ r = b
(Invertibility) ∃ e ∈ Self, ∀ a ∈ Self, ∃ r, l ∈ Self such that l ∘ a = a ∘ r = e
If the right and left inverse are equal they are usually noted r = l = a⁻¹.
(Associativity) ∀ a, b, c ∈ Self, (a ∘ b) ∘ c = a ∘ (b ∘ c)
(Neutral Elt.) ∃ e ∈ Self, ∀ a ∈ Self, e ∘ a = a ∘ e = a
(Commutativity) ∀ a, b ∈ Self, a ∘ b = b ∘ a
</code></pre>
<h2 id="identity-elements" class="section-header"><a href="#identity-elements">Identity elements</a></h2>
<p>Two traits are provided that allow the definition of the additive and
multiplicative identity elements:</p>
<ul>
<li><code>IdentityAdditive</code></li>
<li><code>IdentityMultiplicative</code></li>
</ul>
<h2 id="abstractgroup-like-structures" class="section-header"><a href="#abstractgroup-like-structures">AbstractGroup-like structures</a></h2>
<p>These structures are provided for both the addition and multiplication.</p>
<p>These can be derived automatically by <code>alga_traits</code> attribute from <code>alga_derive</code> crate.</p>
<pre><code class="language-notrust"> AbstractMagma
|
_______/ \______
/ \
divisibility associativity
| |
V V
AbstractQuasigroup AbstractSemigroup
| |
identity identity
| |
V V
AbstractLoop AbstractMonoid
| |
associativity invertibility
\______ _______/
\ /
|
V
AbstractGroup
|
commutativity
|
V
AbstractGroupAbelian
</code></pre>
<p>The following traits are provided:</p>
<ul>
<li>(<code>Abstract</code>|<code>Additive</code>|<code>Multiplicative</code>)<code>Magma</code></li>
<li>(<code>Abstract</code>|<code>Additive</code>|<code>Multiplicative</code>)<code>Quasigroup</code></li>
<li>(<code>Abstract</code>|<code>Additive</code>|<code>Multiplicative</code>)<code>Loop</code></li>
<li>(<code>Abstract</code>|<code>Additive</code>|<code>Multiplicative</code>)<code>Semigroup</code></li>
<li>(<code>Abstract</code>|<code>Additive</code>|<code>Multiplicative</code>)<code>Monoid</code></li>
<li>(<code>Abstract</code>|<code>Additive</code>|<code>Multiplicative</code>)<code>Group</code></li>
<li>(<code>Abstract</code>|<code>Additive</code>|<code>Multiplicative</code>)<code>GroupAbelian</code></li>
</ul>
<h2 id="ring-like-structures" class="section-header"><a href="#ring-like-structures">Ring-like structures</a></h2>
<p>These can be derived automatically by <code>alga_traits</code> attribute from <code>alga_derive</code> crate.</p>
<pre><code class="language-notrust"> GroupAbelian Monoid
\________ ________/
\ /
|
V
Ring
|
commutativity_of_mul
|
V
RingCommutative GroupAbelian
\_______ ___________/
\ /
|
V
Field
</code></pre>
<p>The following traits are provided:</p>
<ul>
<li><code>Ring</code></li>
<li><code>RingCommutative</code></li>
<li><code>Field</code></li>
</ul>
<h2 id="module-like-structures" class="section-header"><a href="#module-like-structures">Module-like structures</a></h2>
<pre><code class="language-notrust"> GroupAbelian RingCommutative
\______ _____/
\ /
| |
V V
Module&lt;Scalar&gt; Field
\______ _____/
\ /
| |
V V
VectorSpace&lt;Scalar&gt;
</code></pre>
<p>The following traits are provided:</p>
<ul>
<li><code>Module</code></li>
<li><code>VectorSpace</code></li>
</ul>
<h1 id="quickcheck-properties" class="section-header"><a href="#quickcheck-properties">Quickcheck properties</a></h1>
<p>Functions are provided to test that algebraic properties like
associativity and commutativity hold for a given set of arguments.</p>
<p>These tests can be automatically derived by <code>alga_quickcheck</code> attribute from <code>alga_derive</code> crate.</p>
<p>For example:</p>
<div class='information'><div class='tooltip ignore'>ⓘ<span class='tooltiptext'>This example is not tested</span></div></div><div class="example-wrap"><pre class="rust rust-example-rendered ignore">
<span class="kw">use</span> <span class="ident">algebra</span>::<span class="ident">general</span>::<span class="ident">SemigroupMultiplicative</span>;
<span class="macro">quickcheck</span><span class="macro">!</span> {
<span class="kw">fn</span> <span class="ident">prop_mul_is_associative</span>(<span class="ident">args</span>: (<span class="ident">i32</span>, <span class="ident">i32</span>, <span class="ident">i32</span>)) <span class="op">-</span><span class="op">&gt;</span> <span class="ident">bool</span> {
<span class="ident">SemigroupMultiplicative</span>::<span class="ident">prop_mul_is_associative</span>(<span class="ident">args</span>)
}
}</pre></div>
</div><h2 id='structs' class='section-header'><a href="#structs">Structs</a></h2>
<table><tr class='module-item'><td><a class="struct" href="struct.Additive.html" title='alga::general::Additive struct'>Additive</a></td><td class='docblock-short'><p>The addition operator, commonly symbolized by <code>+</code>.</p>
</td></tr><tr class='module-item'><td><a class="struct" href="struct.Id.html" title='alga::general::Id struct'>Id</a></td><td class='docblock-short'><p>The universal identity element wrt. a given operator, usually noted <code>Id</code> with a
context-dependent subscript.</p>
</td></tr><tr class='module-item'><td><a class="struct" href="struct.Multiplicative.html" title='alga::general::Multiplicative struct'>Multiplicative</a></td><td class='docblock-short'><p>The multiplication operator, commonly symbolized by <code>×</code>.</p>
</td></tr></table><h2 id='traits' class='section-header'><a href="#traits">Traits</a></h2>
<table><tr class='module-item'><td><a class="trait" href="trait.AbstractField.html" title='alga::general::AbstractField trait'>AbstractField</a></td><td class='docblock-short'><p>A field is a commutative ring, and an Abelian group under both operators.</p>
</td></tr><tr class='module-item'><td><a class="trait" href="trait.AbstractGroup.html" title='alga::general::AbstractGroup trait'>AbstractGroup</a></td><td class='docblock-short'><p>A group is a loop and a monoid at the same time.</p>
</td></tr><tr class='module-item'><td><a class="trait" href="trait.AbstractGroupAbelian.html" title='alga::general::AbstractGroupAbelian trait'>AbstractGroupAbelian</a></td><td class='docblock-short'><p>An Abelian group is a <strong>commutative</strong> group.</p>
</td></tr><tr class='module-item'><td><a class="trait" href="trait.AbstractLoop.html" title='alga::general::AbstractLoop trait'>AbstractLoop</a></td><td class='docblock-short'><p>A loop is a quasigroup with an unique <strong>identity element</strong>, e.</p>
</td></tr><tr class='module-item'><td><a class="trait" href="trait.AbstractMagma.html" title='alga::general::AbstractMagma trait'>AbstractMagma</a></td><td class='docblock-short'><p>A magma is an algebraic structure which consists of a set equipped with a binary operation, ∘,
which must be closed.</p>
</td></tr><tr class='module-item'><td><a class="trait" href="trait.AbstractModule.html" title='alga::general::AbstractModule trait'>AbstractModule</a></td><td class='docblock-short'><p>A module combines two sets: one with an Abelian group structure and another with a
commutative ring structure.</p>
</td></tr><tr class='module-item'><td><a class="trait" href="trait.AbstractMonoid.html" title='alga::general::AbstractMonoid trait'>AbstractMonoid</a></td><td class='docblock-short'><p>A monoid is a semigroup equipped with an identity element, e.</p>
</td></tr><tr class='module-item'><td><a class="trait" href="trait.AbstractQuasigroup.html" title='alga::general::AbstractQuasigroup trait'>AbstractQuasigroup</a></td><td class='docblock-short'><p>A quasigroup is a magma which that has the <strong>divisibility property</strong> (or Latin square property).
<em>A set with a closed binary operation with the divisibility property.</em></p>
</td></tr><tr class='module-item'><td><a class="trait" href="trait.AbstractRing.html" title='alga::general::AbstractRing trait'>AbstractRing</a></td><td class='docblock-short'><p>A <strong>ring</strong> is the combination of an Abelian group and a multiplicative monoid structure.</p>
</td></tr><tr class='module-item'><td><a class="trait" href="trait.AbstractRingCommutative.html" title='alga::general::AbstractRingCommutative trait'>AbstractRingCommutative</a></td><td class='docblock-short'><p>A ring with a commutative multiplication.</p>
</td></tr><tr class='module-item'><td><a class="trait" href="trait.AbstractSemigroup.html" title='alga::general::AbstractSemigroup trait'>AbstractSemigroup</a></td><td class='docblock-short'><p>A semigroup is a quasigroup that is <strong>associative</strong>.</p>
</td></tr><tr class='module-item'><td><a class="trait" href="trait.AdditiveGroup.html" title='alga::general::AdditiveGroup trait'>AdditiveGroup</a></td><td class='docblock-short'><p>[Alias] Algebraic structure specialized for one kind of operation.</p>
</td></tr><tr class='module-item'><td><a class="trait" href="trait.AdditiveGroupAbelian.html" title='alga::general::AdditiveGroupAbelian trait'>AdditiveGroupAbelian</a></td><td class='docblock-short'><p>[Alias] Algebraic structure specialized for one kind of operation.</p>
</td></tr><tr class='module-item'><td><a class="trait" href="trait.AdditiveLoop.html" title='alga::general::AdditiveLoop trait'>AdditiveLoop</a></td><td class='docblock-short'><p>[Alias] Algebraic structure specialized for one kind of operation.</p>
</td></tr><tr class='module-item'><td><a class="trait" href="trait.AdditiveMagma.html" title='alga::general::AdditiveMagma trait'>AdditiveMagma</a></td><td class='docblock-short'><p>[Alias] Algebraic structure specialized for one kind of operation.</p>
</td></tr><tr class='module-item'><td><a class="trait" href="trait.AdditiveMonoid.html" title='alga::general::AdditiveMonoid trait'>AdditiveMonoid</a></td><td class='docblock-short'><p>[Alias] Algebraic structure specialized for one kind of operation.</p>
</td></tr><tr class='module-item'><td><a class="trait" href="trait.AdditiveQuasigroup.html" title='alga::general::AdditiveQuasigroup trait'>AdditiveQuasigroup</a></td><td class='docblock-short'><p>[Alias] Algebraic structure specialized for one kind of operation.</p>
</td></tr><tr class='module-item'><td><a class="trait" href="trait.AdditiveSemigroup.html" title='alga::general::AdditiveSemigroup trait'>AdditiveSemigroup</a></td><td class='docblock-short'><p>[Alias] Algebraic structure specialized for one kind of operation.</p>
</td></tr><tr class='module-item'><td><a class="trait" href="trait.ClosedAdd.html" title='alga::general::ClosedAdd trait'>ClosedAdd</a></td><td class='docblock-short'><p>[Alias] Trait alias for <code>Add</code> and <code>AddAssign</code> with result of type <code>Self</code>.</p>
</td></tr><tr class='module-item'><td><a class="trait" href="trait.ClosedDiv.html" title='alga::general::ClosedDiv trait'>ClosedDiv</a></td><td class='docblock-short'><p>[Alias] Trait alias for <code>Div</code> and <code>DivAssign</code> with result of type <code>Self</code>.</p>
</td></tr><tr class='module-item'><td><a class="trait" href="trait.ClosedMul.html" title='alga::general::ClosedMul trait'>ClosedMul</a></td><td class='docblock-short'><p>[Alias] Trait alias for <code>Mul</code> and <code>MulAssign</code> with result of type <code>Self</code>.</p>
</td></tr><tr class='module-item'><td><a class="trait" href="trait.ClosedNeg.html" title='alga::general::ClosedNeg trait'>ClosedNeg</a></td><td class='docblock-short'><p>[Alias] Trait alias for <code>Neg</code> with result of type <code>Self</code>.</p>
</td></tr><tr class='module-item'><td><a class="trait" href="trait.ClosedSub.html" title='alga::general::ClosedSub trait'>ClosedSub</a></td><td class='docblock-short'><p>[Alias] Trait alias for <code>Sub</code> and <code>SubAssign</code> with result of type <code>Self</code>.</p>
</td></tr><tr class='module-item'><td><a class="trait" href="trait.ComplexField.html" title='alga::general::ComplexField trait'>ComplexField</a></td><td class='docblock-short'><p>Trait shared by all complex fields and its subfields (like real numbers).</p>
</td></tr><tr class='module-item'><td><a class="trait" href="trait.Field.html" title='alga::general::Field trait'>Field</a></td><td class='docblock-short'><p>[Alias] Algebraic structure specialized for one kind of operation.</p>
</td></tr><tr class='module-item'><td><a class="trait" href="trait.Identity.html" title='alga::general::Identity trait'>Identity</a></td><td class='docblock-short'><p>A type that is equipped with identity.</p>
</td></tr><tr class='module-item'><td><a class="trait" href="trait.JoinSemilattice.html" title='alga::general::JoinSemilattice trait'>JoinSemilattice</a></td><td class='docblock-short'><p>A set where every two elements have a supremum (i.e. smallest upper bound).</p>
</td></tr><tr class='module-item'><td><a class="trait" href="trait.Lattice.html" title='alga::general::Lattice trait'>Lattice</a></td><td class='docblock-short'><p>Partially orderable sets where every two elements have a supremum and infimum.</p>
</td></tr><tr class='module-item'><td><a class="trait" href="trait.MeetSemilattice.html" title='alga::general::MeetSemilattice trait'>MeetSemilattice</a></td><td class='docblock-short'><p>A set where every two elements have an infimum (i.e. greatest lower bound).</p>
</td></tr><tr class='module-item'><td><a class="trait" href="trait.Module.html" title='alga::general::Module trait'>Module</a></td><td class='docblock-short'><p>A module which overloads the <code>*</code> and <code>+</code> operators.</p>
</td></tr><tr class='module-item'><td><a class="trait" href="trait.MultiplicativeGroup.html" title='alga::general::MultiplicativeGroup trait'>MultiplicativeGroup</a></td><td class='docblock-short'><p>[Alias] Algebraic structure specialized for one kind of operation.</p>
</td></tr><tr class='module-item'><td><a class="trait" href="trait.MultiplicativeGroupAbelian.html" title='alga::general::MultiplicativeGroupAbelian trait'>MultiplicativeGroupAbelian</a></td><td class='docblock-short'><p>[Alias] Algebraic structure specialized for one kind of operation.</p>
</td></tr><tr class='module-item'><td><a class="trait" href="trait.MultiplicativeLoop.html" title='alga::general::MultiplicativeLoop trait'>MultiplicativeLoop</a></td><td class='docblock-short'><p>[Alias] Algebraic structure specialized for one kind of operation.</p>
</td></tr><tr class='module-item'><td><a class="trait" href="trait.MultiplicativeMagma.html" title='alga::general::MultiplicativeMagma trait'>MultiplicativeMagma</a></td><td class='docblock-short'><p>[Alias] Algebraic structure specialized for one kind of operation.</p>
</td></tr><tr class='module-item'><td><a class="trait" href="trait.MultiplicativeMonoid.html" title='alga::general::MultiplicativeMonoid trait'>MultiplicativeMonoid</a></td><td class='docblock-short'><p>[Alias] Algebraic structure specialized for one kind of operation.</p>
</td></tr><tr class='module-item'><td><a class="trait" href="trait.MultiplicativeQuasigroup.html" title='alga::general::MultiplicativeQuasigroup trait'>MultiplicativeQuasigroup</a></td><td class='docblock-short'><p>[Alias] Algebraic structure specialized for one kind of operation.</p>
</td></tr><tr class='module-item'><td><a class="trait" href="trait.MultiplicativeSemigroup.html" title='alga::general::MultiplicativeSemigroup trait'>MultiplicativeSemigroup</a></td><td class='docblock-short'><p>[Alias] Algebraic structure specialized for one kind of operation.</p>
</td></tr><tr class='module-item'><td><a class="trait" href="trait.Operator.html" title='alga::general::Operator trait'>Operator</a></td><td class='docblock-short'><p>Trait implemented by types representing abstract operators.</p>
</td></tr><tr class='module-item'><td><a class="trait" href="trait.Real.html" title='alga::general::Real trait'>Real</a></td><td class='docblock-short'><span class="stab deprecated">Deprecated</span><p>The field of reals. This has been renamed to <code>RealField</code>.</p>
</td></tr><tr class='module-item'><td><a class="trait" href="trait.RealField.html" title='alga::general::RealField trait'>RealField</a></td><td class='docblock-short'><p>Trait shared by all reals.</p>
</td></tr><tr class='module-item'><td><a class="trait" href="trait.Ring.html" title='alga::general::Ring trait'>Ring</a></td><td class='docblock-short'><p>[Alias] Algebraic structure specialized for one kind of operation.</p>
</td></tr><tr class='module-item'><td><a class="trait" href="trait.RingCommutative.html" title='alga::general::RingCommutative trait'>RingCommutative</a></td><td class='docblock-short'><p>[Alias] Algebraic structure specialized for one kind of operation.</p>
</td></tr><tr class='module-item'><td><a class="trait" href="trait.SubsetOf.html" title='alga::general::SubsetOf trait'>SubsetOf</a></td><td class='docblock-short'><p>Nested sets and conversions between them (using an injective mapping). Useful to work with
substructures. In generic code, it is preferable to use <code>SupersetOf</code> as trait bound whenever
possible instead of <code>SubsetOf</code> (because SupersetOf is automatically implemented whenever
<code>SubsetOf</code> is).</p>
</td></tr><tr class='module-item'><td><a class="trait" href="trait.SupersetOf.html" title='alga::general::SupersetOf trait'>SupersetOf</a></td><td class='docblock-short'><p>Nested sets and conversions between them. Useful to work with substructures. It is preferable
to implement the <code>SupersetOf</code> trait instead of <code>SubsetOf</code> whenever possible (because
<code>SupersetOf</code> is automatically implemented whenever <code>SubsetOf</code> is.</p>
</td></tr><tr class='module-item'><td><a class="trait" href="trait.TwoSidedInverse.html" title='alga::general::TwoSidedInverse trait'>TwoSidedInverse</a></td><td class='docblock-short'><p>Trait used to define the two_sided_inverse element relative to the given operator.</p>
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