# rig

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.

Title rig Rig 2013-03-22 14:44:29 2013-03-22 14:44:29 HkBst (6197) HkBst (6197) 8 HkBst (6197) Definition msc 20M99 semigroup ring