\mathop{\ll}_{\Phi} = \left\{(A,B)\in 2^X\times 2^X: \mathcal{Q}(A)\subset \mathcal{Q}(\mbox{int}B)\right\}.
That is, A\ \mathop{\ll}_{\Phi}\ B, provided the description of each a\in A is contained in the set of descriptions of the points b\in \mbox{int}B.