Shadow biquandles and local biquandles

Given a shadow biquandle $(B,X)$ composed of a biquandle $B$ and a strongly connected $B$-set $X$, we have a local biquandle structure on $X$. The (co)homology groups of such shadow biquandles are isomorphic to those of the corresponding local biquandles. Moreover, cocycle invariants, of oriented links and oriented surface-links, using such shadow biquandles coincide with those using the corresponding local biquandles. These results imply that for some cases, the Niebrzydowski's theory in [14, 15, 16] for knot-theoretic ternary quasigroups is the same as shadow biquandle theory. We also show that some local biquandle $2$- or $3$-cocycles and some $1$- or $2$-cocycles of the Niebrzydowski's (co)homology theory can be induced from Mochizuki's cocycles.