  A semidomain is an additive submonoid of an integral domain that is closed
under multiplication and contains the identity element. Although atomicity and
divisibility in integral domains have been systematically investigated for more
than thirty years, the same aspects in the more general context of semidomains
have been considered just recently. Here we study subatomicity in the context
of semidomains, focusing on whether certain subatomic properties ascend from a
semidomain to its polynomial extension and its Laurent polynomial extension. We
investigate factorization and divisibility notions generalizing that of
atomicity. First, we consider the Furstenberg property, which is due to P.
Clark and motivated by the work of H. Furstenberg on the infinitude of primes.
Then we consider the almost atomic and quasi-atomic properties, both introduced
by J. G. Boynton and J. Coykendall in their study of divisibility in integral
domains.
