If $(X,m)$ is a median algebra, then for each $x\in X$, define an operation $\wedge_{x}$ by letting $y\wedge_{x}z=m(x,y,z)$. Then $(X,\wedge_{x})$ is a meet-semilattice with least element $x$. Define the ordering $\leq_{x}$ on $X$ where $y\leq_{x}z$ iff $y=y\wedge_{x}z=m(x,y,z)$. An unrooted lobster is a median algebra $(X,m)$ where if $x,y\in X$, then the interval $([x,y],\leq_{x})$ is a linear ordering. We say that a median algebra $(X,m)$ is non-linear if there does not exist a linear ordering $\leq$ where $m(x,y,z)=(x\wedge y)\vee(x\wedge z)\vee(y\wedge z)$ for all $x,y,z$ and $\wedge,\vee$ are the lattice operations on $(X,\leq)$.
A left lobster group is a structure $(G,m)$ where $G$ is a group, $m$ is a median algebra operation on $G$ that makes $(G,m)$ into a lobster, and where $r\cdot m(x,y,z)=m(r\cdot x,r\cdot y,r\cdot z)$ for all $r,x,y,z\in G$. A two-sided lobster group is a left-lobster group $(G,m)$ where $m(x,y,z)\cdot r=m(x\cdot r,y\cdot r,z\cdot r)$ for all $r,x,y,z\in G$.
Can the braid group $B_{n}$ for $n>2,n\in\mathbb{N}\cup\{\infty\}$ be endowed with a ternary operation $m$ so that $(B_{n},m)$ is a non-linear left-lobster group or perhaps even a two-sided lobster group? Can the pure braid group $PB_{n}$ for $n>2,n\in\mathbb{N}\cup\{\infty\}$ be endowed with a ternary operation $m$ so that $(PB_{n},m)$ is a non-linear left-lobster group or perhaps even a non-linear two-sided lobster group?
The braid group $B_{n}$ can be endowed with a left linear ordering (the Dehornoy ordering) and $PB_{n}$ can be endowed with a two-sided linear ordering (the Magnus ordering). Furthermore, the free group $F_{n}$ can be endowed with a non-linear left lobster ordering and the pure braid group $PB_{n}$ can be factored as a semidirect product $F_{n-1}\rtimes PB_{n-1}$. It therefore seems like it may be possible to make $B_{n}$ or $PB_{n}$ into at least a non-linear left lobster group.
