the name is semiring.
Is there a name for an algrebraic structure with two operations, addition and multiplication, such that
1) the additive structure is that of a commutative monoid;
2) the multiplicative structue is that of a monoid;
3) multiplication distributes over addition?
A simple example would be any power set with the operations and , but also the set of all binary relations on a set with the operations and , the latter being the composition of relations.