Planar varieties of semigroups

We study the property of planarity of Cayley graphs for varieties of semigroups. It is proved that the variety of semigroups with zero multiplication $\mathrm{ var}\{xy=zt\}$, the variety of left zero semigroups $\mathrm{ var}\{xy=x\}$ and the variety $\mathrm{ var}\{xy=xz\}$ are the only non-trivial planar variety of semigroups. We find the ranks of planarity for some important series of varieties of semigroups.