Sa-small subacts in the category of S-acts with zero

In this paper, we introduce the concept of semiannihilator small (sa-small) subacts within the framework of [Formula: see text]-acts with zero. The definition builds on the classical notion of small submodules. A subact is called sa-small if, whenever its union with another subact covers the entire act, the annihilator of the complementary subact forms a small ideal in the underlying monoid. We study various characterizations of sa-small subacts, analyze their behavior under [Formula: see text]-act homomorphisms, and examine their stability in standard algebraic constructions such as direct sums, products, pushouts, and tensor products. Additionally, we establish important structural theorems that help clarify their significance in advancing our knowledge of monoid act theory.