introduced in [@OCHAStasheff] open-closed homotopy algebras (OCHA). These homotopy algebras were studied in papers [@OCHA], [@OCHA1], [@OCHAStasheff1], and [@OCHAStasheff11]. In this subsection we show that the operad ${{\bf Ger}}^+_{\infty}$ is a natural extension of the operad ${{\mathcal{O C}}}_{\infty}$.

To recall the definition of OCHA we introduce ${{\mathcal{O C}}}^{\vee}$-coalgebras.

\[dfn:OC-vee\] An ${{\mathcal{O C}}}^{\vee}$-coalgebra is a pair of a ${{\Lambda}}^2{{\bf cocomm}}$-coalgebra $V$ and a ${{\Lambda}}{{\bf coassoc}}$-coalgebra $A$ together with a unary operation $$\rho : A \to V$$ of degree $-1$ such that the following diagrams commute: $$\label{rho-map-coalg-OC}
\xymatrix@M=0.4pc{
A\ar[r]^{\rho} \ar[d]_{{{\Delta}}_{{{\mathfrak{o}}}}}  &  V \ar[d]^{{{\Delta}}_{{{\mathfrak{c}}}}}  \\
A\otimes A \ar[r]^{\rho \otimes \rho} & V \otimes V
}$$ $$\label{left-right-via-rho-OC}
\xymatrix@M=0.4pc{
~ & A \otimes A \ar[r]^{\rho \otimes {{\rm id}}} & V\otimes A\\
A \ar[ur]^{{{\Delta}}_{{{\mathfrak{o}}}}} \ar[dr]_{{{\Delta}}_{{{\mathfrak{o}}}}} & ~ & ~ \\
~ & A \otimes A \ar[r]^{{{\rm id}}\otimes \rho} & A\otimes V \ar[uu]^{{{\sigma}}}
}$$ 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.

We denote by ${{\mathcal{O C}}}^{\vee}$ the cooperad which governs ${{\mathcal{O C}}}^{\vee}$-coalgebras.

The operad ${{\mathcal{O C}}}_{\infty}$ governing OCHA algebras is by definition the cobar construction of the cooperad ${{\mathcal{O C}}}^{\vee}$. $$\label{OC-infinity}
{{\mathcal{O C}}}_{\infty} = {{\rm Cobar}}({{\mathcal{O 