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Is there an algebraic structure over an infinite set $S$, composed by $3$ different binary operators $*$, $+$,$×$ such that :

  1. $*$ form an abelian group over $S$ and let's call $e$ its identity element
  2. $*$ and $+$ form a field over $S$
  3. $+$ and $×$ form a field over $S\setminus\{e\}$

If this structure exist what is its name? Can you give an example? If it does not exist, why? What is "wrong" in this structure?

Arnaud D.
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user100415
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    A field has two operations - what do you mean by '+' forms a field over S? – Guy Paterson-Jones Oct 30 '16 at 17:12
  • There seemed to be a problem with the formatting; I tried to correct it but can you confirm that I got it right? – Arnaud D. Oct 30 '16 at 17:17
  • For items 2 and 3, can you specify which of the two operations is the multiplication and which the addition of the field? The notation would suggest + is addition, but the fact that 3 excludes e suggests that * is addition and + is multiplication in (2). Please clarify. – Caleb Stanford Oct 30 '16 at 17:21
  • @ArnaudD. : thanks, your formatting is OK – user100415 Oct 30 '16 at 19:26
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    @GuyPaterson-Jones: probably before ArnaudD formatting it wasn't clear, but it is the couple of binary operation $*$ and $+$ that form the first field, while the second field is formed by the couple $+$ and $×$. – user100415 Oct 30 '16 at 19:55
  • @6005 : to mantain an internal coherency between the three item, since item 1 state that $$ form a group over S, in item 2 $$ must assume the 'additive' role and $+$ the 'multiplicative' role. From this descent that $+$ form a group over $S\setminus{e}$, so in item 3 $+$ must assume the 'additive' role and $x$ the 'multiplicative' role. For sake of notation we can use the 0 symbol for the identity element of $+$ operation and 1 for the identity element of $×$ – user100415 Oct 30 '16 at 20:03

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I doubt that there is a name for such things, since they are rare, but they do exist. You are simply looking for a pair of fields $(S, *, +)$ and $(S-\{e\}, +, \times)$ where the multiplicative group of nonzero elements of the first field equals the additive group of the second field. Such a pair of fields can be constructed using this answer.

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Keith Kearnes
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