# a "ring" whose addition lacks inverses, terminology

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• Jul 22nd 2011, 04:06 PM
ymar
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.
• Jul 22nd 2011, 05:48 PM
NonCommAlg
Re: a "ring" whose addition lacks inverses, terminology
the name is semiring.
• Jul 23rd 2011, 04:50 AM
ymar
Re: a "ring" whose addition lacks inverses, terminology
Thanks. I checked "quasi-" and "pseudo-", but forgot "semi-". :-)
• Jul 23rd 2011, 04:54 AM
Swlabr
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 $\cup$ 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...