Definition:Ringoid (Abstract Algebra)

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Definition

A ringoid is a triple $\left({S, *, \circ}\right)$ where:

$S$ is a set
$*$ and $\circ$ are binary operations on $S$
the operation $\circ$ distributes over $*$.

That is:

$\forall a, b, c \in S: a \circ \left({b * c}\right) = \left({a \circ b}\right) * \left({a \circ c}\right)$
$\forall a, b, c \in S: \left({a * b}\right) \circ c = \left({a \circ c}\right) * \left({b \circ c}\right)$


Closedness

For the expression $a \circ \left({b * c}\right)$ to make sense, we require that $S$ is closed under $*$.

Similarly, for $\left({a \circ b}\right) * \left({a \circ c}\right)$ to make sense, we require that $S$ is closed under $\circ$.

Note that $\circ$ does not have to be associative.


Note on order of operations

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.


Also see

Stronger properties


Sources