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float: left;" onfocus="this.value == 'Search' ? this.value = '' : true" onblur="this.value == '' ? this.value = 'Search' : true" /></fieldset> </form> <span id='navEnd'></span> </div> <div id="revision"> <html xmlns="http://www.w3.org/1999/xhtml" xmlns:svg="http://www.w3.org/2000/svg" xml:lang="en" lang="en"> <head><meta http-equiv="Content-type" content="application/xhtml+xml;charset=utf-8" /><title>Contents</title></head> <body> <div class="rightHandSide"> <div class="toc clickDown" tabindex="0"> <h3 id="context">Context</h3> <h4 id="algebra">Algebra</h4> <div class="hide"><div> <ul> <li><a class="existingWikiWord" href="/nlab/show/algebra">algebra</a>, <a class="existingWikiWord" href="/nlab/show/higher+algebra">higher algebra</a></li> <li><a class="existingWikiWord" href="/nlab/show/universal+algebra">universal algebra</a></li> <li><a class="existingWikiWord" href="/nlab/show/monoid">monoid</a>, <a class="existingWikiWord" href="/nlab/show/semigroup">semigroup</a>, <a class="existingWikiWord" href="/nlab/show/quasigroup">quasigroup</a></li> <li><a class="existingWikiWord" href="/nlab/show/nonassociative+algebra">nonassociative algebra</a></li> <li><a class="existingWikiWord" href="/nlab/show/associative+unital+algebra">associative unital algebra</a></li> <li><a class="existingWikiWord" href="/nlab/show/commutative+algebra">commutative algebra</a></li> <li><a class="existingWikiWord" href="/nlab/show/Lie+algebra">Lie algebra</a>, <a class="existingWikiWord" href="/nlab/show/Jordan+algebra">Jordan algebra</a></li> <li><a class="existingWikiWord" href="/nlab/show/Leibniz+algebra">Leibniz algebra</a>, <a class="existingWikiWord" href="/nlab/show/pre-Lie+algebra">pre-Lie algebra</a></li> <li><a class="existingWikiWord" href="/nlab/show/Poisson+algebra">Poisson algebra</a>, <a class="existingWikiWord" href="/nlab/show/Frobenius+algebra">Frobenius algebra</a></li> <li><a class="existingWikiWord" href="/nlab/show/lattice">lattice</a>, <a class="existingWikiWord" href="/nlab/show/frame">frame</a>, <a class="existingWikiWord" href="/nlab/show/quantale">quantale</a></li> <li><a class="existingWikiWord" href="/nlab/show/Boolean+ring">Boolean ring</a>, <a class="existingWikiWord" href="/nlab/show/Heyting+algebra">Heyting algebra</a></li> <li><a class="existingWikiWord" href="/nlab/show/commutator">commutator</a>, <a class="existingWikiWord" href="/nlab/show/center">center</a></li> <li><a class="existingWikiWord" href="/nlab/show/monad">monad</a>, <a class="existingWikiWord" href="/nlab/show/comonad">comonad</a></li> <li><a class="existingWikiWord" href="/nlab/show/distributive+law">distributive law</a></li> </ul> <h2 id="group_theory">Group theory</h2> <ul> <li><a class="existingWikiWord" href="/nlab/show/group">group</a>, <a class="existingWikiWord" href="/nlab/show/normal+subgroup">normal subgroup</a></li> <li><a class="existingWikiWord" href="/nlab/show/action">action</a>, <a class="existingWikiWord" href="/nlab/show/Cayley%27s+theorem">Cayley's theorem</a></li> <li><a class="existingWikiWord" href="/nlab/show/centralizer">centralizer</a>, <a class="existingWikiWord" href="/nlab/show/normalizer">normalizer</a></li> <li><a class="existingWikiWord" href="/nlab/show/abelian+group">abelian group</a>, <a class="existingWikiWord" href="/nlab/show/cyclic+group">cyclic group</a></li> <li><a class="existingWikiWord" href="/nlab/show/group+extension">group extension</a>, <a class="existingWikiWord" href="/nlab/show/Galois+extension">Galois extension</a></li> <li><a class="existingWikiWord" href="/nlab/show/algebraic+group">algebraic group</a>, <a class="existingWikiWord" href="/nlab/show/formal+group">formal group</a></li> <li><a class="existingWikiWord" href="/nlab/show/Lie+group">Lie group</a>, <a class="existingWikiWord" href="/nlab/show/quantum+group">quantum group</a></li> </ul> <h2 id="ring_theory">Ring theory</h2> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/ring">ring</a>, <a class="existingWikiWord" href="/nlab/show/commutative+ring">commutative ring</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/local+ring">local ring</a>, <a class="existingWikiWord" href="/nlab/show/Artinian+ring">Artinian ring</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Noetherian+ring">Noetherian ring</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/skewfield">skewfield</a>, <a class="existingWikiWord" href="/nlab/show/field">field</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/integral+domain">integral domain</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/ideal">ideal</a>, <a class="existingWikiWord" href="/nlab/show/prime+ideal">prime ideal</a>, <a class="existingWikiWord" href="/nlab/show/maximal+ideal">maximal ideal</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Ore+localization">Ore localization</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/group+extension">group extension</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/central+simple+algebra">central simple algebra</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/derivation">derivation</a>, <a class="existingWikiWord" href="/nlab/show/Ore+extension">Ore extension</a></p> </li> </ul> <h2 id="module_theory">Module theory</h2> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/module">module</a>, <a class="existingWikiWord" href="/nlab/show/bimodule">bimodule</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/vector+space">vector space</a>, <a class="existingWikiWord" href="/nlab/show/linear+algebra">linear algebra</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/matrix">matrix</a>, <a class="existingWikiWord" href="/nlab/show/eigenvalue">eigenvalue</a>, <a class="existingWikiWord" href="/nlab/show/trace">trace</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/determinant">determinant</a>, <a class="existingWikiWord" href="/nlab/show/quasideterminant">quasideterminant</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/representation+theory">representation theory</a>, <a class="existingWikiWord" href="/nlab/show/Schur+lemma">Schur lemma</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/extension+of+scalars">extension of scalars</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/restriction+of+scalars">restriction of scalars</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Frobenius+reciprocity">Frobenius reciprocity</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Morita+equivalence">Morita equivalence</a>, <a class="existingWikiWord" href="/nlab/show/Morita+context">Morita context</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Wedderburn-Artin+theorem">Wedderburn-Artin theorem</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/abelian+category">abelian category</a>, <a class="existingWikiWord" href="/nlab/show/additive+category">additive category</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/monoidal+category">monoidal category</a>, <a class="existingWikiWord" href="/nlab/show/Tannaka+duality">Tannaka duality</a></p> </li> </ul> <h2 id=""><a class="existingWikiWord" href="/nlab/show/gebra+theory">Gebras</a></h2> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/coalgebra">coalgebra</a>, <a class="existingWikiWord" href="/nlab/show/coring">coring</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/bialgebra">bialgebra</a>, <a class="existingWikiWord" href="/nlab/show/Hopf+algebra">Hopf algebra</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/comodule">comodule</a>, <a class="existingWikiWord" href="/nlab/show/Hopf+module">Hopf module</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Yetter-Drinfeld+module">Yetter-Drinfeld module</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/associative+bialgebroid">associative bialgebroid</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/dual+gebra">dual gebra</a>, <a class="existingWikiWord" href="/nlab/show/cotensor+product">cotensor product</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Hopf-Galois+extension">Hopf-Galois extension</a></p> </li> </ul> </div></div> </div> </div> <h1 id="contents">Contents</h1> <div class='maruku_toc'> <ul> <li><a href='#idea'>Idea</a></li> <li><a href='#Definition'>Definition</a></li> <ul> <li><a href='#further_weakening'>Further weakening</a></li> </ul> <li><a href='#properties'>Properties</a></li> <li><a href='#examples'>Examples</a></li> <li><a href='#related_concepts'>Related concepts</a></li> <li><a href='#references'>References</a></li> </ul> </div> <h2 id="idea">Idea</h2> <p>In <a class="existingWikiWord" href="/nlab/show/algebra">algebra</a>, by a <em>rig</em> one means a <a class="existingWikiWord" href="/nlab/show/mathematical+structure">mathematical structure</a> much like a <a class="existingWikiWord" href="/nlab/show/ring">ring</a> but without the assumption that every element has an additive inverse, hence without the assumption of <em>negatives</em> (whence the omission of “n” from “ring” [<a href="#Schanuel91">Schanuel 1991 p. 379</a>, <a href="#Lawvere92">Lawvere 1992 p. 2</a>])</p> <p> <div class='num_remark'> <h6>Remark</h6> <p><strong>(Terminology: Rigs and semirings)</strong> Rigs are commonly also called <em>semirings</em>, but the term ‘semiring’ is overloaded in the mathematics literature, with different authors each defining a semiring to be different algebraic structures from each other. See <em><a class="existingWikiWord" href="/nlab/show/semiring">semiring</a></em> for a discussion about the various definitions of semirings; only one of the proposed definitions is the same as the one of <em>rigs</em> as considered here.</p> </div> </p> <h2 id="Definition">Definition</h2> <p>A <em>rig</em> is a <a class="existingWikiWord" href="/nlab/show/set">set</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>R</mi></mrow><annotation encoding="application/x-tex">R</annotation></semantics></math> with binary operations of addition and multiplication, such that</p> <ul> <li><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>R</mi></mrow><annotation encoding="application/x-tex">R</annotation></semantics></math> is a <a class="existingWikiWord" href="/nlab/show/monoid">monoid</a> under multiplication;</li> <li><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>R</mi></mrow><annotation encoding="application/x-tex">R</annotation></semantics></math> is a <a class="existingWikiWord" href="/nlab/show/commutative+monoid">commutative monoid</a> under addition;</li> <li>multiplication distributes over addition, i.e. the distributivity laws hold:<div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>x</mi><mo>⋅</mo><mo stretchy="false">(</mo><mi>y</mi><mo>+</mo><mi>z</mi><mo stretchy="false">)</mo><mo>=</mo><mo stretchy="false">(</mo><mi>x</mi><mo>⋅</mo><mi>y</mi><mo stretchy="false">)</mo><mo>+</mo><mo stretchy="false">(</mo><mi>x</mi><mo>⋅</mo><mi>z</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">x\cdot (y+z) = (x\cdot y) + (x\cdot z)</annotation></semantics></math></div><div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>y</mi><mo>+</mo><mi>z</mi><mo stretchy="false">)</mo><mo>⋅</mo><mi>x</mi><mo>=</mo><mo stretchy="false">(</mo><mi>y</mi><mo>⋅</mo><mi>x</mi><mo stretchy="false">)</mo><mo>+</mo><mo stretchy="false">(</mo><mi>z</mi><mo>⋅</mo><mi>x</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(y+z)\cdot x = (y\cdot x) + (z\cdot x)</annotation></semantics></math></div> <p>and also the absorption laws, which are the nullary version of the distributive laws:</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mn>0</mn><mo>⋅</mo><mi>x</mi><mo>=</mo><mn>0</mn><mo>=</mo><mi>x</mi><mo>⋅</mo><mn>0</mn><mo>.</mo></mrow><annotation encoding="application/x-tex">0\cdot x = 0 = x\cdot 0.</annotation></semantics></math></div></li> </ul> <p>In a ring, the absorption laws follow from distributivity, since for example <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mn>0</mn><mo>⋅</mo><mi>x</mi><mo>+</mo><mn>0</mn><mo>⋅</mo><mi>x</mi><mo>=</mo><mo stretchy="false">(</mo><mn>0</mn><mo>+</mo><mn>0</mn><mo stretchy="false">)</mo><mo>⋅</mo><mi>x</mi><mo>=</mo><mn>0</mn><mo>⋅</mo><mi>x</mi></mrow><annotation encoding="application/x-tex">0\cdot x + 0\cdot x = (0+0)\cdot x = 0\cdot x</annotation></semantics></math> and we can cancel one copy to obtain <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mn>0</mn><mo>⋅</mo><mi>x</mi><mo>=</mo><mn>0</mn></mrow><annotation encoding="application/x-tex">0\cdot x = 0</annotation></semantics></math>. In a <a class="existingWikiWord" href="/nlab/show/rig">rig</a>, however, we have to assert the absorption laws separately.</p> <p>More sophisticatedly, we can say that just as a ring is a <a class="existingWikiWord" href="/nlab/show/monoid+object">monoid object</a> in <a class="existingWikiWord" href="/nlab/show/abelian+group">abelian group</a>s, so a rig is a monoid object in <a class="existingWikiWord" href="/nlab/show/commutative+monoid">commutative monoid</a>s. Here the categories of abelian groups and commutative monoids must be given suitable monoidal structures: not the cartesian product, but the tensor product <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo>⊗</mo></mrow><annotation encoding="application/x-tex">\otimes</annotation></semantics></math> that has a universal property for bilinear maps.</p> <p>Equivalently, a rig is the <a class="existingWikiWord" href="/nlab/show/hom-set">hom-set</a> of a category with a single object that is <a class="existingWikiWord" href="/nlab/show/enriched+category">enriched</a> in the category of <a class="existingWikiWord" href="/nlab/show/commutative+monoids">commutative monoids</a>.</p> <p>Rigs and rig <a class="existingWikiWord" href="/nlab/show/homomorphisms">homomorphisms</a> form the category <a class="existingWikiWord" href="/nlab/show/Rig">Rig</a>.</p> <h3 id="further_weakening">Further weakening</h3> <p>As with rings, one sometimes considers non-associative or non-unital versions (where multiplication may not be <a class="existingWikiWord" href="/nlab/show/associative">associative</a> or may have no <a class="existingWikiWord" href="/nlab/show/identity">identity</a>). It is rarer to remove requirements from addition as we have done here. But notice that while <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>R</mi></mrow><annotation encoding="application/x-tex">R</annotation></semantics></math> can be proved (from the other axioms) to be an <a class="existingWikiWord" href="/nlab/show/abelian+group">abelian group</a> under addition (and therefore a ring) as long as it is a <a class="existingWikiWord" href="/nlab/show/group">group</a>, this argument does not go through if it is only a monoid. If we assert only distributivity on one side, however, then we can have a noncommutative addition; see <a class="existingWikiWord" href="/nlab/show/near-ring">near-ring</a>.</p> <h2 id="properties">Properties</h2> <p>Many rigs are either <a class="existingWikiWord" href="/nlab/show/ring">ring</a>s or <a class="existingWikiWord" href="/nlab/show/distributive+lattice">distributive lattice</a>s. Indeed, a ring is precisely a rig that forms a group under addition, while a distributive lattice is precisely a commutative, simple rig in which both operations are idempotent (see (<a href="#Golan2003">Golan 2003, Proposition 2.25</a>)). Note that a <a class="existingWikiWord" href="/nlab/show/Boolean+algebra">Boolean algebra</a> is a rig in both ways: as a lattice and as a <a class="existingWikiWord" href="/nlab/show/Boolean+ring">Boolean ring</a>.</p> <p>Any rig can be “completed” to a ring by adding negatives, in generalization of how the <a class="existingWikiWord" href="/nlab/show/natural+numbers">natural numbers</a> are completed to the <a class="existingWikiWord" href="/nlab/show/integers">integers</a>. When applied to the <a class="existingWikiWord" href="/nlab/show/set">set</a> of <a class="existingWikiWord" href="/nlab/show/isomorphism+classes">isomorphism classes</a> of objects in a <a class="existingWikiWord" href="/nlab/show/rig+category">rig category</a>, the result is part of <a class="existingWikiWord" href="/nlab/show/algebraic+K-theory">algebraic K-theory</a>.</p> <p>More formally, the ring completion of a rig <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>R</mi></mrow><annotation encoding="application/x-tex">R</annotation></semantics></math> is obtained by applying the <a class="existingWikiWord" href="/nlab/show/group+completion">group completion</a> functor to the underlying additive monoid of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>R</mi></mrow><annotation encoding="application/x-tex">R</annotation></semantics></math>, and extending the rig multiplication to a ring multiplication by exploiting distributivity; this gives the <a class="existingWikiWord" href="/nlab/show/left+adjoint">left adjoint</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>F</mi><mo>:</mo><mi>Rig</mi><mo>→</mo><mi>Ring</mi></mrow><annotation encoding="application/x-tex">F: Rig \to Ring</annotation></semantics></math> to the forgetful functor <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>U</mi><mo>:</mo><mi>Ring</mi><mo>→</mo><mi>Rig</mi></mrow><annotation encoding="application/x-tex">U: Ring \to Rig</annotation></semantics></math>. Note however that the unit of the <a class="existingWikiWord" href="/nlab/show/adjunction">adjunction</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>R</mi><mo>→</mo><mi>U</mi><mi>F</mi><mo stretchy="false">(</mo><mi>R</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">R \to U F(R)</annotation></semantics></math> is not <a class="existingWikiWord" href="/nlab/show/monomorphism">monic</a> if the additive monoid of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>R</mi></mrow><annotation encoding="application/x-tex">R</annotation></semantics></math> is not <span class="newWikiWord">cancellative<a href="/nlab/new/cancellation+monoid">?</a></span>, despite an informal convention that “<a class="existingWikiWord" href="/nlab/show/completion">completion</a>” should usually mean a <a class="existingWikiWord" href="/nlab/show/monad">monad</a> where the unit <em>is</em> monic.</p> <p>Matrices of rigs can be used to formulate versions of <a class="existingWikiWord" href="/nlab/show/matrix+mechanics">matrix mechanics</a>.</p> <p>Every rig with <a class="existingWikiWord" href="/nlab/show/positive+characteristic">positive characteristic</a> is a <a class="existingWikiWord" href="/nlab/show/ring">ring</a>.</p> <h2 id="examples">Examples</h2> <p>Some rigs which are neither rings nor distributive lattices include:</p> <ul> <li> <p>The <a class="existingWikiWord" href="/nlab/show/natural+numbers">natural numbers</a>.</p> </li> <li> <p>The nonnegative <a class="existingWikiWord" href="/nlab/show/rational+number">rational number</a>s and the nonnegative <a class="existingWikiWord" href="/nlab/show/real+numbers">real numbers</a>.</p> </li> <li> <p>Polynomials with coefficients in any rig.</p> </li> <li> <p>The set of isomorphism classes of objects in any <a class="existingWikiWord" href="/nlab/show/distributive+category">distributive category</a>, or more generally in any <a class="existingWikiWord" href="/nlab/show/rig+category">rig category</a>.</p> </li> <li> <p>The <a class="existingWikiWord" href="/nlab/show/tropical+rig">tropical rig</a>, which is <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ℝ</mi><mo>∪</mo><mo stretchy="false">{</mo><mn>∞</mn><mo stretchy="false">}</mo></mrow><annotation encoding="application/x-tex">\mathbb{R}\cup \{\infty\}</annotation></semantics></math> with addition <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>x</mi><mo>⊕</mo><mi>y</mi><mo>=</mo><mi>min</mi><mo stretchy="false">(</mo><mi>x</mi><mo>,</mo><mi>y</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">x\oplus y = min(x,y)</annotation></semantics></math> and multiplication <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>x</mi><mo>⊗</mo><mi>y</mi><mo>=</mo><mi>x</mi><mo>+</mo><mi>y</mi></mrow><annotation encoding="application/x-tex">x\otimes y = x+y</annotation></semantics></math>.</p> <p>Tropical rigs are among the important class of <em><a class="existingWikiWord" href="/nlab/show/idempotent+semirings">idempotent semirings</a></em>.</p> </li> <li> <p>The <a class="existingWikiWord" href="/nlab/show/ideals">ideals</a> of a <a class="existingWikiWord" href="/nlab/show/commutative+ring">commutative ring</a> form a rig under ideal addition and multiplication, where the unit and zero ideals are the unit and zero elements of the rig, respectively. They also form a <a class="existingWikiWord" href="/nlab/show/distributive+lattice">distributive lattice</a> and therefore a rig in another way; note that the addition operation is the same in both rigs but the multiplication operation is different (being intersection in the lattice).</p> </li> </ul> <h2 id="related_concepts">Related concepts</h2> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/semiring">semiring</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/rng">rng</a> (<a class="existingWikiWord" href="/nlab/show/nonunital+ring">nonunital ring</a>)</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/ring">ring</a>, <a class="existingWikiWord" href="/nlab/show/near-ring">near-ring</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/tropical+geometry">tropical geometry</a>, <a class="existingWikiWord" href="/nlab/show/tropical+semiring">tropical semiring</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/idempotent+semiring">idempotent semiring</a></p> </li> <li> <p>A <a class="existingWikiWord" href="/nlab/show/categorification">categorification</a> of the notion of rig is the notion of <a class="existingWikiWord" href="/nlab/show/rig+category">rig category</a>, or more generally <a class="existingWikiWord" href="/nlab/show/colax-distributive+rig+category">colax-distributive rig category</a>. See also <a class="existingWikiWord" href="/nlab/show/2-rig">2-rig</a> and <a class="existingWikiWord" href="/nlab/show/distributivity+for+monoidal+structures">distributivity for monoidal structures</a>.</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/semi-monoid">semi-monoid</a>/<a class="existingWikiWord" href="/nlab/show/semi-group">semi-group</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/semi-category">semi-category</a>, <a class="existingWikiWord" href="/nlab/show/semi-Segal+space">semi-Segal space</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/semi-simplicial+set">semi-simplicial set</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Burnside+rig">Burnside rig</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/division+rig">division rig</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/multiplicatively+cancellable+rig">multiplicatively cancellable rig</a></p> </li> </ul> <h2 id="references">References</h2> <p>The terminology “rig” is due to:</p> <ul> <li id="Schanuel91"> <p><a class="existingWikiWord" href="/nlab/show/Stephen+H.+Schanuel">Stephen H. Schanuel</a>, p. 379 of: <em>Negative sets have Euler characteristic and dimension</em>, in: <em>Category Theory</em>, Lecture Notes in Mathematics <strong>1488</strong> (1991) 379–385 [<a href="https://doi.org/10.1007/BFb0084232">doi:10.1007/BFb0084232</a>]</p> </li> <li id="Lawvere92"> <p><a class="existingWikiWord" href="/nlab/show/William+Lawvere">William Lawvere</a>, pp. 1 of: <em>Introduction to Linear Categories and Applications</em>, course lecture notes (1992) [<a href="https://github.com/mattearnshaw/lawvere/blob/192dac273e8bf352f307f87b9ec4fe8ef7dc85b9/pdfs/1992-introduction-to-linear-categories-and-applications.pdf">pdf</a>, <a class="existingWikiWord" href="/nlab/files/Lawvere-LinearCategories.pdf" title="pdf">pdf</a>]</p> </li> </ul> <p>as recalled in:</p> <ul> <li><a class="existingWikiWord" href="/nlab/show/F.+William+Lawvere">F. William Lawvere</a>: <em>The legacy of Steve Schanuel!</em> (2015) [<a class="existingWikiWord" href="/nlab/files/Lawvere-LegacyOfSchanuel.pdf" title="pdf">pdf</a>, <a href="https://web.archive.org/web/20230311022135/https://www.acsu.buffalo.edu/~wlawvere/Schanuel%20Memorial%20posting.htm">archive</a>]</li> </ul> <blockquote> <p>“We were amused when we finally revealed to each other that we had each independently come up with the term ‘rig’.”</p> </blockquote> <p>Discussion under the name <em>semirings</em>:</p> <ul> <li id="Golan1999"> <p>Jonathan S. Golan, <em>Semirings and their applications</em>. Updated and expanded version of The theory of semirings, with applications to mathematics and theoretical computer science, Longman Sci. Tech., Harlow, 1992, MR1163371. Kluwer Academic Publishers, Dordrecht, 1999. xii+381 pp.</p> </li> <li id="Golan2003"> <p>Jonathan S. Golan, <em>Semirings and affine equations over them: theory and applications</em> (Vol. 556). Springer Science & Business Media, 2003.</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/M.+Marcolli">M. Marcolli</a>, R. Thomgren, <em>Thermodynamical semirings</em>, <a href="http://arxiv.org/abs/1108.2874">arXiv/1108.2874</a></p> </li> <li> <p>wikipedia <a href="http://en.wikipedia.org/wiki/Semiring">semiring</a></p> </li> <li> <p>J. Jun, S. Ray, J. Tolliver, <em>Lattices, spectral spaces, and closure operations on idempotent semirings</em>, <a href="https://arxiv.org/abs/2001.00808">arxiv/2001.00808</a></p> </li> </ul> </body></html> </div> <div class="revisedby"> <p> Last revised on November 10, 2024 at 21:44:31. See the <a href="/nlab/history/rig" style="color: #005c19">history</a> of this page for a list of all contributions to it. </p> </div> <div class="navigation navfoot"> <a href="/nlab/edit/rig" accesskey="E" class="navlink" id="edit" rel="nofollow">Edit</a><a href="https://nforum.ncatlab.org/discussion/14877/#Item_15">Discuss</a><span class="backintime"><a href="/nlab/revision/rig/47" accesskey="B" class="navlinkbackintime" id="to_previous_revision" rel="nofollow">Previous revision</a></span><a href="/nlab/show/diff/rig" accesskey="C" class="navlink" id="see_changes" rel="nofollow">Changes from previous revision</a><a href="/nlab/history/rig" accesskey="S" class="navlink" id="history" rel="nofollow">History (47 revisions)</a> <a href="/nlab/show/rig/cite" style="color: black">Cite</a> <a href="/nlab/print/rig" accesskey="p" id="view_print" rel="nofollow">Print</a> <a href="/nlab/source/rig" id="view_source" rel="nofollow">Source</a> </div> </div>  </div>  </body> </html>