Suppose that $A$ and $B$ are subsets of a group or semigroup. We call $A$ left upper [resp. lower] $B$-periodic if $BA\subseteq A$ [resp. $A\subseteq BA$]. If $A$ is both left upper and lower $B$-periodic (i.e. $BA=A$ ), then we call it left $B$-periodic. It is a generalization of the conceptions ideals, sub-semigroups and subgroups (for more information, one can see http://www.worldscientific.com/doi/abs/10.1142/S1005386711000332?journalCode=ac) .
Now, we are looking for some (finite, infinite) groups/semigroups containing a non-singleton subset $B\neq \emptyset$ such that for every subset $A$ the following property holds $$ A\subseteq BA\Rightarrow A\subseteq bA\; \mbox{; for all } b\in B $$
(Note that in groups it is equivalent to "$A\subseteq BA\Rightarrow B^{-1}A\subseteq A$". Also, if $B^{-1}\subseteq B$ then $A=BA$ implies $A=bA$, for all $b\in B$)