Is there a unique factorization commutative semiring such that every element except zero does not have additive inverse in the semiring and each element can be decomposed into unique finite sum decomposition?
That is, for every $x$ in the element, it has unique sum decomposition into $k \geq 1$ elements, $x = a_1 + ... + a_k$ and each $a_i$ can be uniquely factorizable into prime elements. There does not necessarily need to be multiplicative identity.