Distributive property

Given two binary operations, $\times$ and $+$ , acting on a set $S$ , we say that $\times$ has the distributive property over $+$ (or $\times$ distributes over $+$ ) if, for all $a, b, c \in S$ we have

$a\times(b + c) = (a\times b) + (a \times c)$ and $(a + b) \times c = (a \times c) + (b \times c)$ .

Note that if the operation $\times$ is commutative, these two conditions are the same, but if $\times$ does not commute then we could have operations which left-distribute but do not right-distribute, or vice-versa.

Also note that there is no particular reason that distributivity should be one-way, as it is with conventional multiplication and addition. For example, in a distributive lattice, each of the operations meet and join distributes over the other.

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Distributive property