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

Re: a "ring" whose addition lacks inverses, terminology

Re: a "ring" whose addition lacks inverses, terminology

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

Re: a "ring" whose addition lacks inverses, terminology

Quote:

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