Bender-Knuth involutions on linear extensions of posets

We study the permutation group ${\mathcal{B}\mathcal{K}}_P$ 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 ${\mathcal{B}\mathcal{K}}_P$. We also examine ${\mathcal{B}\mathcal{K}}_P$ 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 ${\mathcal{B}\mathcal{K}}_P={\mathfrak{S}}_{\mathcal{L}(P)}$.