  Tropical ideals are a class of ideals in the tropical polynomial semiring
that combinatorially abstracts the possible collections of supports of all
polynomials in an ideal over a field. We study zero-dimensional tropical ideals
I with Boolean coefficients in which all underlying matroids are paving
matroids, or equivalently, in which all polynomials of minimal support have
support of size deg(I) or deg(I)+1 -- we call them paving tropical ideals. We
show that paving tropical ideals of degree d+1 are in bijection with $\mathbb
Z^n$-invariant d-partitions of $\mathbb Z^n$. This implies that
zero-dimensional tropical ideals of degree 3 with Boolean coefficients are in
bijection with $\mathbb Z^n$-invariant 2-partitions of quotient groups of the
form $\mathbb Z^n/L$. We provide several applications of these techniques,
including a construction of uncountably many zero-dimensional degree-3 tropical
ideals in one variable with Boolean coefficients, and new examples of
non-realizable zero-dimensional tropical ideals.
