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If $\mathbb{R}$ is the real numbers and x#y=max{x,y} then ($\mathbb{R}$,#,+) is a semiring where ($\mathbb{R}$,#) is a semigroup and + distributes over #.

If you have a set R with three distinct binary operations *, +, # such that * distributes over + , and + distributes over # then must # be an idempotent operation? (i.e. x#x=x)

Does it make any difference if (R,+,*) is a ring and (R,#,+) is a semiring or does the double distributivity on its own force the idempotency?

Whether or not it turns out to be the case that # must be idempotent, does the double distributivity imply any other restrictions on the characteristics of *,+ or #?

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It's a good idea to provide some context for your questions: is this just something you were wondering about (say because you were reading a book on semirings - in that case mention the book), or is it homework? It's considered acceptable to ask about homework on the site, but you should mention it is homework and use the 'homework' tag. –  Tara B Feb 11 '13 at 10:30

1 Answer 1

Just to get you started, try making $*$ something silly, say for example $R = \mathbb{R}$ and $x*y = 0$ for all $x,y\in R$. Then you should be able to choose + and # to be some standard operations such that all the required distributivity laws hold and # is not idempotent.

Of course in this case $(R,*,+)$ is not a ring. How might it help force # to be idempotent if it were a ring?

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