Soft set theory is a general mathematical framework for dealing with uncertainty. In this regard, soft set operations can be regarded as crucial concepts in soft set theory, since they offer new perspectives for dealing with issues containing parametric information. In this paper, we give a theoretical study on AND-product (∧-product), which is an essential concept in decision making problems, by investigating its whole algebraic properties in detail regarding soft F-subsets and soft M- equality, the strictest type of soft equality. Moreover, in order to complete some incomplete results concerning AND-product in the literature, we compare our properties by the formerly obtained properties regarding soft L-equality and soft J-equality. Furthermore, we handle the whole relations between AND-product and OR-product, the other keystone in decision making. Besides, by establishing some new results on distributive properties of AND-product over restricted, extended, and soft binary piecewise soft set operations, we prove that the set of all the soft sets over U together with restricted/extended union and AND-product is a commutative hemiring with identity as the set of all the soft sets over U together with restricted/extended symmetric difference and AND-product forms a commutative hemiring with identity in the sense of soft L-equality. As analyzing the algebraic structure of soft sets from the standpoint of operations gives profound insight into the potential uses of soft sets in classical and nonclassical logic and since theoretical foundations of soft computing approaches are derived from purely mathematical principles, this research will pave the way for a wide range of applications, including new decision-making approaches and innovative cryptography techniques based on soft sets.