  We equip the space of lattice cones with a coproduct which makes it a
connected cograded colagebra. The exponential sum and exponential integral on
lattice cones can be viewed as linear maps on this space with values in the
space of meromorphic germs with linear poles at zero. We investigate the
subdivision properties-- reminiscent of the inclusion-exclusion principle for
the cardinal on finite sets-- of such linear maps and establish a compatibility
of these properties with respect to the convolution quotient of the coalgebra.
Implementing the Algebraic Birkhoff Factorization procedure on the linear maps
under consideration, we factorize the exponential sum as a convolution quotient
of two maps, with each of the maps in the factorization satisfying a
subdivision property. Consequently, the Algebraic Birkhoff Factorization
specializes to the Euler-Maclaurin formula on lattice cones and provides a
simple formula for the interpolating factor by means of a projection map.
