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Semi-ring

From Encyclopedia of Mathematics
Revision as of 17:13, 27 December 2015 by Richard Pinch (talk | contribs) (→‎Comments: Exotic semirings)
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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.

Comments

The term "exotic" semi-rings has been used to describe subsets of the real numbers with $\min$ or $\max$ as ${+}$ and addition as ${\star}$. These are thus idempotent semi-rings. Examples include the tropical semiring on $\mathbf{N} \cup \{\infty\}$ with operations ${\min},\, +$.

References

  • K. Glazek, A Guide to the Literature on Semirings and their Applications in Mathematics and Information Sciences: With Complete Bibliography Springer (2013) ISBN 9401599645
How to Cite This Entry:
Semi-ring. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Semi-ring&oldid=37114
This article was adapted from an original article by L.A. Skornyakov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article