$b_\infty$-algebra structure in homology of a homotopy Gerstenhaber algebra

The minimality theorem states, in particular, that on cohomology $H(A)$ of a dg algebra there exists sequence of operations $m_i:H(A)^{\otimes i}\to H(A)$, $i=2,3,\dots$, which form a minimal $A_\infty$-algebra $(H(A),\{m_i\})$. This structure defines on the bar construction $BH(A)$ a correct differential $d_m$ so that the bar constructions $(BH(A),d_m)$ and $BA$ have isomorphic homology modules. It is known that if $A$ is equipped additionally with a structure of homotopy Gerstenhaber algebra, then on $BA$ there is a multiplication which turns it into a dg bialgebra. In this paper, we construct algebraic operations $E_{p,q}:H(A)^{\otimes p}\otimes H(A)^{\otimes q}\to H(A)$, $p,q=0,1,2,\dots$, which turn $(H(A),\{m_i\},\{E_{p,q}\})$ into a $B_\infty$-algebra. These operations determine on $BH(A)$ correct multiplication, so that $(BH(A),d_m)$ and $BA$ have isomorphic homology algebras.