Does anyone know what would it mean that the merge operation is asymmetrical? Would this mean that the order of merged objects is important?


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Not precisely (for the order implications see Kayne 1994 - The antisymmetry of syntax - but note that the work is not uniformly accepted within Minimalism).

Merge is asymmetrical because one of the merged constituents (commonly called head though the term can be misleading) projects all its features to the resulting constituent - hence "David's brothers" is a plural constituent despite "David" being singular (and, vice versa, "Davids' brother" is singular despite "Davids" being plural) - whereas the other usually projects ("percolates") none of them (or, in some versions, only those which do not have features of their type in the "head" constituent - see, e.g., my term paper if you can read Russian).

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