# Math Help - a "ring" whose addition lacks inverses, terminology

1. ## a "ring" whose addition lacks inverses, terminology

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 $\cup$ and $\cap$, but also the set of all binary relations on a set with the operations $\cup$ and $\circ$, the latter being the composition of relations.

2. ## Re: a "ring" whose addition lacks inverses, terminology

the name is semiring.

3. ## Re: a "ring" whose addition lacks inverses, terminology

Thanks. I checked "quasi-" and "pseudo-", but forgot "semi-". :-)

4. ## Re: a "ring" whose addition lacks inverses, terminology

Originally Posted by ymar
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 $\cup$ and $\cap$, but also the set of all binary relations on a set with the operations $\cup$ and $\circ$, the latter being the composition of relations.
You also get semifields (a semiring, but multiplication forms a group). For example, the tropical semifield. Your set is the integers, multiplication is addition, a.b=a+b, while addition is taken to be max, a+b=max(a, b). Trippy...