In this paper we study the Arens triadjoints of some bilinear maps on vector lattices. In particular, we prove that, for Archimedean vector lattices $A$ and $B$, the Arens triadjoint i) $T^{***}:A"\times A"\to B"$ of a positive orthosymmetric bilinear map $T:A\times A\to B$ is positive orthosymmetric, and ii) $T^{***}:A"\times A"\to A"$ of a bi-orthomorphism $T:A\times A\to A$ is a bi-orthomorphism. These generalize results on the order bidual of $f$-algebras and almost $f$-algebras in [4].