Local biquandles and Niebrzydowski's tribracket theory

We introduce a new algebraic structure called \textit{local biquandles} and show how colorings of oriented classical link diagrams and of broken surface diagrams are related to tribracket colorings. We define a (co)homology theory for local biquandles and show that it is isomorphic to Niebrzydowski's tribracket (co)homology. This implies that Niebrzydowski's (co)homology theory can be interpreted similary as biqandle (co)homology theory. Moreover through the isomorphism between two cohomology groups, we show that Niebrzydowski's cocycle invariants and local biquandle cocycle invariants are the same.