Name this algebraic structure

Question

I found an algebraic structure $(P(S), \cup, \cap)$, where $S$ is a set of some elements and $P(S)$ is its power set such that the given algebraic structure satisfies all the properties of a semiring except distributivity. When it comes to distributivity, it is found that $A\cap(B\cup C)\subset(A\cap B)\cup (A\cap C)$ , where $A, B, C\in P(S)$. Similarly, $\subset$ holds for right distributivity also. Now i wish such structure to be assigned a suitable name which is almost a semiring. Pliz suggest it a name. Note that $(S, \cup, \cap)$ is a semiring.