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rig (Definition)

A rig $ (R, +, \cdot)$ is a set $ R$ together with two binary operations $ +:R^2 \to R:(a, b) \mapsto a + b$ and $ \cdot:R^2 \to R:(a, b) \mapsto ab$, such that both $ (R, +)$ and $ (R, \cdot)$ are monoids, where $ \cdot$ distributes over $ +$. That is if $ \{a, b, c, d\} \subset R$ then $ (a+b)(c+d) = ac + ad + bc +bd$. The natural numbers with ordinary addition and multiplication $ (\mathbf{N}, +, \cdot)$ is a rig.

A rig $ (R, +, \cdot)$ is a ring if $ (R, +)$ is a group. The integers with ordinary addition and multiplication $ (\mathbf{Z}, +, \cdot)$ is a ring.



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"rig" is owned by HkBst.
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See Also: semigroup, ring
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Cross-references: integers, group, ring, multiplication, addition, natural numbers, distributes over, monoids, binary operations
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This is version 5 of rig, born on 2004-10-16, modified 2006-09-15.
Object id is 6378, canonical name is Rig.
Accessed 2099 times total.

Classification:
AMS MSC: 20M99 (Group theory and generalizations :: Semigroups :: Miscellaneous)
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