We describe a theory of logarithmic Chow rings and tautological subrings for logarithmically smooth algebraic stacks, via generalisation the notion piecewise-polynomial functions. Using this machinery we prove that double-double ramification cycle lies in subring (classical) ring moduli space curves, double is divisorial (as conjectured by Molcho, Pandharipande, Schmitt).

1. Let T be a ternary ring with the ternary operation T(a, b, c) and the distinguished elements 0, 1 (see [4]). On T, two loop structures can be defined by means of the binary operations a+b = T(a, 1, b) and ab=T(a, b, 0). The resulting loops are called the additive and the multiplicative loop of Ty respectively. Together with a0 = 0a = 0 they define the structure of a double loop on T, which s...

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