On different definitions of wreath product of semigroup varieties

There are considered three different definitions of wreath product of semigroup varieties, namely: the general, monoidal and standard ones. It is shown that these are three different operations. An algorithm is found which gives us a possibility to decide if a given identity in a wreath product of semigroups is true provided such algorithms exist for initial semigroups. As a consequence of this result we get algorithms which give the answer to such question in monoidal, general and standard wreath product of semigroup varieties. It is known that the monoidal and general wreath products of semigroup varieties are associative. It is proved that standard wreath product of semigroup varieties is not associative even if the initial varieties are the atoms in the lattice of all semigroup varieties. The known variety generated by five-element completely $0$-simple semigroup $A_2=\langle a,b\mid a^2=a,\ b^2=0,\ aba=a,\ bab=b\rangle$ is represented as the monoidal wreath product of the variety of semilattices and the variety of right bands. The general wreath product of varieties coincides with the monoidal one if the second semigroup variety is not a periodic group variety.