  We define transalgebraic functions on a compact Riemann surface as
meromorphic functions except at a finite number of punctures where they have
finite order exponential singularities. This transalgebraic class is a
topological multiplicative group. We extend the action of the e\~ne product to
the transcendental class on the Riemann sphere. This transalgebraic class,
modulo constant functions, is a commutative ring for the multiplication, as the
additive structure, and the e\~ne product, as the multiplicative structure. In
particular, the divisor action of the e\~ne product by multiplicative
convolution extends to these transalgebraic divisors. The polylogarithm
hierarchy appears related to transalgebraic e\~ne poles of higher order.
