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Commutativity conditions for truncated Brown–Peterson spectra of height 2

An algebraic criterion, in terms of closure under power operations, is determined for the existence and uniqueness of generalized truncated Brown–Peterson spectra of height 2 as E∞‐ring spectra. The criterion is checked for an example at the prime 2 derived from the universal elliptic curve equipped with a level Γ1(3) structure.