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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?

Nathan BeDell
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    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. – lhf Jun 10 '14 at 01:36
  • 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? – Nathan BeDell Jun 10 '14 at 01:41
  • I'm not sure if a formal proof would be possible or not, but given two binary operators like this, it should be possible to prove some kind of identity for any two given Cayley tables, don't you think? – Nathan BeDell Jun 10 '14 at 01:43
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    I haven't seen a term for this sort of structure before, but I do like the "$n$-magma" idea. – Malice Vidrine Jun 10 '14 at 01:55
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    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. – Caleb Stanford Jun 10 '14 at 14:31
  • I don't see why there's no point. In fact, let me state my use case for why a term like this would be useful: I am developing a python library that allows users to analyze properties of magmas, as well as properties of (insert word for algebraic structures with two binary operators here). This generic term would include rings, fields, and any other type of algebraic structure which relates two binary operators with some identity. – Nathan BeDell Jun 10 '14 at 16:17
  • The reason this is futile is that a set with two magma structures, is just like two magmas. You don't need to program anything new. If you have a library for handling magmas, and now you have a single set with two magma structures, then just treat it as two magmas. – Ittay Weiss Jun 11 '14 at 00:01
  • Bigroupoid! http://books.google.it/books?id=udIEPHzSuuQC&pg=PA127&lpg=PA127&dq=bigroupoid&source=bl&ots=xRtdfFvexO&sig=0XP7NU3WZiNL-dTx7tWi6pfvTds&hl=it&sa=X&ei=niOaU4eJC-G_0QXH0oDoAg&ved=0CGEQ6AEwBg#v=onepage&q=bigroupoid&f=false but this term was used much before, I'm sure. – MattAllegro Jun 12 '14 at 22:01
  • @MattAllegro, Interesting, I think the term bimagma might be more appropriate though, since "groupoid" has two common definitions, whereas magma has one, making "bimagma" less ambiguous. The term bigroupoid also apparently has other definitions. – Nathan BeDell Jun 13 '14 at 00:44
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    @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 ;) – MattAllegro Jun 13 '14 at 13:00
  • Yeah, I looked up "bimagma", and found it in a few places on Google books, but the actual content wasn't available to browse, so I couldn't get the definition. Here is the book I found. The exact thing it picked up was actually "Bimagmatic bialgebra", and I found out that bialgebra is actually related to vector spaces. However, I also found this "...This defines a bimagma (a set with two binary operations)" (pg 1) – Nathan BeDell Jun 13 '14 at 15:40
  • Unless what you say is true, and there are a decent number of authors that have used the term decades ago, it doesn't really seem that there is really a settled upon, often used term for this. I'd be interested to see whether or not the usage I discussed in my answer to this (from A Course in Universal Algebra) has been used before or not though. – Nathan BeDell Jun 13 '14 at 15:43

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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.

MattAllegro
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Nathan BeDell
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