to $A$ we get a differential on $A$.

The cochain complex (\[Hom-A-A\]) with the differential (\[Hoch\]) is called the Hochschild cochain complex of the $A_{\infty}$-algebra $(A,m)$. In this paper we denote this cochain complex by ${C^{\bullet}}(A,A)$.

The operad ${{\bf Ger}}^+_{\infty}$ {#sect:Ger-plus}
===================================

Let us start the construction of the operad ${{\bf Ger}}^+_{\infty}$ with the following definition

\[dfn:S-coalg\] An ${{\bf S}}$-coalgebra is a pair of ${{\bf Ger}}^{\vee}$-coalgebra $V$ and a ${{\Lambda}}{{\bf coassoc}}$-coalgebra $A$ together with a unary operation $$\rho : A \to V$$ of degree[^1] $-1$ such that the composition of the cobracket ${{\delta}}_{{{\mathfrak{c}}}}$ with $\rho$ is zero: $$\label{de-rho}
{{\delta}}_{{{\mathfrak{c}}}} \circ  \rho (a) = 0\,,$$ and the following diagrams commute: $$\label{rho-map-coalg}
\xymatrix@M=0.4pc{
A\ar[r]^{\rho} \ar[d]_{{{\Delta}}_{{{\mathfrak{o}}}}}  &  V \ar[d]^{{{\Delta}}_{{{\mathfrak{c}}}}}  \\
A\otimes A \ar[r]^{\rho \otimes \rho} & V \otimes V
}$$ $$\label{left-right-via-rho}
\xymatrix@M=0.4pc{
~ & A \otimes A \ar[r]^{\rho \otimes {{\rm id}}} & V\otimes A\\
A \ar[ur]^{{{\Delta}}_{{{\mathfrak{o}}}}} \ar[dr]_{{{\Delta}}_{{{\mathfrak{o}}}}} & ~ & ~ \\
~ & A \otimes A \ar[r]^{{{\rm id}}\otimes \rho} & A\otimes V \ar[uu]^{{{\sigma}}}
}$$ where ${{\Delta}}_{{{\mathfrak{c}}}}$ (resp. ${{\Delta}}_{{{\mathfrak{o}}}}$) is the comultiplication in $V$ (resp. in $A$) and $${{\sigma}}: A \otimes V \to V \otimes A$$ is the mapping which switches tensor components.

Diagram (\[rho-map-coalg\]) says that $\rho$ is a map of coassociative 