Aeppli and Bott-Chern cohomology for bi-generalized Hermitian manifolds and $d^{\mathrm{\prime }}{d}^{\mathrm{\prime }\mathrm{\prime }}$-lemma

We define Aeppli and Bott-Chern cohomology for bi-generalized complex manifolds and show that they are finite dimensional for compact bi-generalized Hermitian manifolds. For totally bounded double complexes $(A,d^{\mathrm{\prime }},d^{\mathrm{\prime }\mathrm{\prime }})$, we show that the validity of $d^{\mathrm{\prime }}{d}^{\mathrm{\prime }\mathrm{\prime }}$-lemma is equivalent to having the same dimension of several cohomology groups. Some calculations of Bott-Chern cohomology groups of some bi-generalized Hermitian manifolds are given.