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A semidomain is an additive submonoid of an integral domain that is closed under multiplication and contains the identity element. Although factorizations and divisibility in atomic domains have been systematically investigated for more than 30 years, the same aspects in the more general context of atomic semidomains have been considered just recently. Here we study subatomicity in the context of semidomains; that is, we study semidomains satisfying divisibility properties weaker than atomicity. We mostly focus on the Furstenberg property, which is due to P. Clark and motivated by the work of H. Furstenberg on the infinitude of primes, and the almost atomic and quasi-atomic properties, introduced by J. G. Boynton and J. Coykendall in the context of divisibility in integral domains. We investigate these three properties in the context of semidomains, paying special attention to whether they ascend from a semidomain to its polynomial and Laurent polynomial extensions.