Bender-Knuth involutions on linear extensions of posets

We study the permutation group $\mathcal{BK}_P$ generated by Bender-Knuth moves on linear extensions of a poset $P$, an analog of the Berenstein-Kirillov group on column-strict tableaux. We explore the group relations, with an emphasis on identifying posets $P$ for which the cactus relations hold in $\mathcal{BK}_P$. We also examine $\mathcal{BK}_P$ as a subgroup of the symmetric group $\mathfrak{S}_{\mathcal{L}(P)}$ on the set of linear extensions of $P$ with the focus on analyzing posets $P$ for which $\mathcal{BK}_P = \mathfrak{S}_{\mathcal{L}(P)}$.