  We study axiomatic foundations for different classes of constant-function
automated market makers (CFMMs). We focus particularly on separability and on
different invariance properties under scaling. Our main results are an
axiomatic characterization of a natural generalization of constant product
market makers (CPMMs), popular in decentralized finance, on the one hand, and a
characterization of the Logarithmic Scoring Rule Market Makers (LMSR), popular
in prediction markets, on the other hand. The first class is characterized by
the combination of independence and scale invariance, whereas the second is
characterized by the combination of independence and translation invariance.
The two classes are therefore distinguished by a different invariance property
that is motivated by different interpretations of the num\'eraire in the two
applications.
  However, both are pinned down by the same separability property.
  Moreover, we characterize the CPMM as an extremal point within the class of
scale invariant, independent, symmetric AMMs with non-concentrated liquidity
provision. Our results add to a formal analysis of mechanisms that are
currently used for decentralized exchanges and connect the most popular class
of DeFi AMMs to the most popular class of prediction market AMMs.
