A Cacti theoretical interpretation of the axioms of bialgebras and   H-module algebras
  We establish a dictionary between the Cacti algebra axioms on a Cacti algebra structure with underlying free associative algebra, under suitable good behavior with degrees. Using these ideas, for an associative algebra $A$ and a bialgebra $H$, we also translate Cacti algebra maps $\Omega(H)\to C^\bullet(A)$ (where $\Omega(H)$ stands for the cobar construction on $H$ and $C^\bullet(A)$ is the Hochschild cohomology complex) with $H$-module algebra structures on $A$, and illustrate with examples of applications. 
Marco Farinati and Leandro Lombardi
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