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Difference between revisions of "Semi-ring"

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A non-empty set with two associative binary operations <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084320/s0843201.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084320/s0843202.png" />, satisfying the distributive laws
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{{TEX|done}}
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<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084320/s0843203.png" /></td> </tr></table>
 
  
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A non-empty set $S$ with two associative binary operations $+$ and $\cdot$, satisfying the [[distributive law]]s
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$$
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(a+b) \cdot c = a\cdot c + b \cdot c
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$$
 
and
 
and
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$$
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<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084320/s0843204.png" /></td> </tr></table>
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a \cdot (b+c) = a\cdot b + a\cdot c \ .
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$$
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In most cases one also assumes that the addition is commutative and that there exists a zero <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084320/s0843205.png" /> such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084320/s0843206.png" /> for every <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084320/s0843207.png" />. The most important examples of semi-rings are rings and distributive lattices (cf. [[Ring|Ring]]; [[Distributive lattice|Distributive lattice]]). If there is a multiplicative identity 1, the two classes are combined by the condition
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In most cases one also assumes that the addition is commutative and that there exists a zero element $0$ such that $a + 0 = a$ for every $a \in S$. The most important classes of semi-rings are [[ring]]s and [[distributive lattice]]s. If there is a multiplicative identity 1, the two classes are combined by the condition
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$$
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<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084320/s0843208.png" /></td> </tr></table>
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\forall x \, \exists y \  x+y=1 \ .
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The non-negative integers with the usual operations provide an example of a semi-ring that does not satisfy this condition.
 
The non-negative integers with the usual operations provide an example of a semi-ring that does not satisfy this condition.

Revision as of 18:16, 13 December 2014


A non-empty set $S$ with two associative binary operations $+$ and $\cdot$, satisfying the distributive laws $$ (a+b) \cdot c = a\cdot c + b \cdot c $$ and $$ a \cdot (b+c) = a\cdot b + a\cdot c \ . $$ In most cases one also assumes that the addition is commutative and that there exists a zero element $0$ such that $a + 0 = a$ for every $a \in S$. The most important classes of semi-rings are rings and distributive lattices. If there is a multiplicative identity 1, the two classes are combined by the condition $$ \forall x \, \exists y \ x+y=1 \ . $$ The non-negative integers with the usual operations provide an example of a semi-ring that does not satisfy this condition.

How to Cite This Entry:
Semi-ring. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Semi-ring&oldid=16685
This article was adapted from an original article by L.A. Skornyakov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article
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