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

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

    the name is semiring.
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    Re: a "ring" whose addition lacks inverses, terminology

    Thanks. I checked "quasi-" and "pseudo-", but forgot "semi-". :-)
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    MHF Contributor Swlabr's Avatar
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    Re: a "ring" whose addition lacks inverses, terminology

    Quote Originally Posted by ymar View Post
    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...
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