  A subset $S$ of an integral domain $R$ is called a semidomain if the pairs
$(S,+)$ and $(S, \cdot)$ are semigroups with identities; additionally, we say
that $S$ is additively reduced provided that $S$ contains no additive inverses.
Given an additively reduced semidomain $S$ and a torsion-free monoid $M$, we
denote by $S[M]$ the semidomain consisting of polynomial expressions with
coefficients in $S$ and exponents in $M$; we refer to these objects as
additively reduced monoid semidomains. We study the factorization properties of
additively reduced monoid semidomains. Specifically, we determine necessary and
sufficient conditions for an additively reduced monoid semidomain to be a
bounded factorization semidomain, a finite factorization semidomain, and a
unique factorization semidomain. We also provide large classes of semidomains
with full and infinity elasticity. Throughout the paper we provide examples
aiming to shed some light upon the arithmetic of additively reduced
semidomains.
