Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. Join them; it only takes a minute:

Sign up
Here's how it works:
  1. Anybody can ask a question
  2. Anybody can answer
  3. The best answers are voted up and rise to the top

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 #?

share|cite|improve this question

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?

share|cite|improve this answer

Your Answer


By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.