@ShayBenMoshe i thought that there was a choice involved

afaik, a form of E(n) is a complex oriented p-local associative and commutative ring spectrum for which there is a choice of indecomposables v_1, ..., v_n such that the composite Z_(p)[v_1, ..., v_n] -> BP_* -> E(n)_* sends v_n to something invertible, and such that the map induced by inverting v_n in the source is an isomorphism

@ShayBenMoshe at least in the case of Morava E-theory, there is a uniqueness result for E_1 and E_oo-structures, so the one defined in that way (which corresponds to the honda fgl) is unique as an E_oo-ring up to galois twisting

my question was motivated by the angeltveit-lind uniqueness result for BP<n> as a spectrum

@skd I see, but we have a canonical such spectrum, Johnson-Wilson. Also, if I'm not mistaken, Lubin-Tate are the ones for which there is a (unique) E_oo structure, and not Johnson-Wilson, and they have different homotopy groups. Are you sure then that that's the definition you wanted?

Btw, there is some uniqueness for K(n) as you said if you also require it to be E_1 and complex oriented of height n, if that's what you were looking for.

@ShayBenMoshe sorry, by "Morava E-theory" i mean Lubin-Tate theory, and by E(n) i mean Johnson-Wilson theory. what i'm curious about is whether different choices for the v_i's give homotopy equivalent E(n)'s

Aha, I see. Don't you get this by the fact that they are Landweber flat?

yes, i think it does

i also found ams.org/journals/tran/1999-351-07/S0002-9947-99-02436-8/…, and i think it's shown there that E(n) is unique up to non-canonical isomorphism under MU

In his book "Geometry of iterated loop spaces" May covers the connected case of the Recognition Principle in detail. In the final chapter he very briefly discusses how to extend it to the group-like case, and even then mostly for $n=\infty$.

He says that you should replace the little n-cubes monad C_n with the monad \Omega C_{n-1} \Sigma, with some claim that group-like little n-cube algebras are the same as algebras over the latter monad

This is essentially Theorem 2 in Segal's "configuration spaces and iterated loop spaces"