Difference between revisions of "Semi-ring"
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A non-empty set $S$ with two associative binary operations $+$ and $\cdot$, satisfying the [[distributive law]]s | A non-empty set $S$ with two associative binary operations $+$ and $\cdot$, satisfying the [[distributive law]]s | ||
Revision as of 16:38, 27 December 2015
2020 Mathematics Subject Classification: Primary: 16Y60 [MSN][ZBL]
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.
Semi-ring. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Semi-ring&oldid=37111