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For example, let's say we're using the operators +, and *, and the set {0,1,2}

The Cayley tables look like this:

* 0 1 2    + 0 1 2
0 0 0 1    0 1 2 0
1 1 2 1    1 0 1 0
2 0 0 2    2 1 2 2

These Cayley tables are totally random, but the point is that the algebraic structure isn't necessarily like any other common type of algebraic structure with two binary operators (e.a. field, ring, boolean algebra). The two operators just obey closure, so it's basically an abstraction of a magma to more than one operator.

Is there a specific, agreed upon name for this in mathematics yet? The most obvious thing to me would be to call this a bimagma, then to call something similar with three binary operators a trimagma, then in general a n-magma. Do these structures have a common, agreed upon name?

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    $\begingroup$ If the two operations do not talk to each other, via something like distributivity or Jacobi's identity, then they're just that, two separate operations. $\endgroup$ – lhf Jun 10 '14 at 1:36
  • $\begingroup$ Is that true though? There may be properties like distributivity or Jacobi's identity that the two operators obey, or there may not be, that's the point. Shouldn't all that information be derivable from the Cayley tables anyway? $\endgroup$ – Nathan BeDell Jun 10 '14 at 1:41
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    $\begingroup$ I haven't seen a term for this sort of structure before, but I do like the "$n$-magma" idea. $\endgroup$ – Malice Vidrine Jun 10 '14 at 1:55
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    $\begingroup$ As @lhf suggests, there's really no point in making a specific term for a set with multiple operations unless they are related somehow. Just like if a set is a group under two different operations, you merely speak of it as two separate groups, not as one double-group. There must be some relation between the two operations in order for it to make sense to bunch them together. $\endgroup$ – 6005 Jun 10 '14 at 14:31
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    $\begingroup$ @Sintrastes: Hi! Yeah I had seen that page...but I don't know much of it indeed. Bigroupoid is the word: I would swear some authors were using it decades ago (R.H. Bruck, 1958? I should check again and again...) while I never heard/read bimagma. But this is the way to follow ;) $\endgroup$ – MattAllegro Jun 13 '14 at 13:00
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From Burris, Sankappanavar A Course in Universal Algebra page 26 (42 of the pdf): "An algebra A is unary if all of its operations are unary, and it is mono-unary if it has just one unary operation."

Although from what I read it is not clear whether or not in practice this terminology has been extended before, an algebra with two binary operators could be called di-binary.

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