Definition:Ringoid
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Definition
A ringoid $\left({S, *, \circ}\right)$ is a set $S$ on which is defined two binary operations, here denoted $\circ$ and $*$, defined on all the elements of $S \times S$, where one operation $\circ$ distributes over the other $*$.
That is:
- $\forall a, b, c \in S: a \circ \left({b * c}\right) = \left({a \circ b}\right) * \left({a \circ c}\right)$
- $\forall a, b, c \in S: \left({a * b}\right) \circ c = \left({a \circ c}\right) * \left({b \circ c}\right)$
Note that in a ringoid, there is no insistence that $S$ is closed under either of these operations, neither does $\circ$ have to be associative.
In the denotation of this structure, $\left({S, *, \circ}\right)$, the distributor is shown after the distributand.
In the context of a ringoid, the fact that $\circ$ distributes over $*$ is known as the distributive law.
