On ranks of the planarity of varieties of all idempotent semigroups, nilsemigroups, and semigroups with the permutation identity
We prove that the rank of the planarity of the variety of idempotent semigroupsis equal to 3; the rank of the planarity of the variety var { xw = w, wx = w } of nilsemigroups for any word w that does not contain the variable x is equal to infinity; the rank of planarity of the permutation variety of semigroups is equal to 1 or 2.