# Group laws on elliptic curves and varieties

To what extent does the existence of a group-law for adding points depend on the binary operations in the polynomial? For example what if the addition in the polynomial y^2 = ax^3+bx+c is a semigroup/monoid/quasigroup/loop and the multiplication isn't necessarily associative. Will you instead end up with a monoid-law or loop-law ?

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What do you mean, binary operations in the polynomial? –  Simon Rose Sep 26 '11 at 17:49
The symbol"+" replaced with a non-group addition and the "." in b.x replaced with non-ring multiplication (but still distributive over "+") –  asri Sep 26 '11 at 17:52
The group law on the elliptic curve assumes that the underlying algebraic structure is a field. Even just having a commutative ring creates problems and the sum of two points may not always be defined. I can't imagine you will get anything reasonable if the underlying algebraic structure is a semi-ring or worse. –  Felipe Voloch Sep 26 '11 at 19:33