The ranks of planarity for varieties of commutative semigroups
We study the concept of the planarity rank suggested by L.M. Martynov for semigroup varieties. Let V be a variety of semigroups. If there is a natural number r  1 that all V-free semigroups of ranks  r allow planar Cayley graphs and the V-free semigroup of a rank r + 1 doesn't allow planar Cayley graph, then this number r is called the planarity rank for variety V. If such a number r doesn't exist, then we say that the variety V has the infinite planarity rank. We prove that a non-trivial variety of commutative semigroups either has the infinite planarity rank and coincides with the variety of semigroups with the zero multiplication or has a planarity rank 1, 2 or 3. These estimates of planarity ranks for varieties of commutative semigroups are achievable.