Difference between revisions of "Semi-ring"
From Encyclopedia of Mathematics
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a \cdot (b+c) = a\cdot b + a\cdot c \ . | a \cdot (b+c) = a\cdot b + a\cdot c \ . | ||
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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 | + | 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 [[unit element]] 1, the two classes are combined by the condition |
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\forall x \, \exists y \ x+y=1 \ . | \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 18:32, 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 unit element 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=35628
Semi-ring. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Semi-ring&oldid=35628
This article was adapted from an original article by L.A. Skornyakov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article