THE PROBLEM ON THE POWER OF LATTICES OF SUBVARIETIES THE MONOID WREATH PRODUCT OF SEMIGROUP VARIETIES AND VARIETIES GENERATED BY SMALL VARIETIES
At present time the problem of finding of the power of the lattice of the subvarieties of monoid wreath product of semigroup varieties which are simultaneously atoms of the lattice of all semigroup varieties is not solved. If this lattice is infinite then to find its power is a simple problem exept the case of monoid wreath product of the variety of all semilattices with the same variety. In our paper we find these powers for infinite powers and show the connection this problem with open problem of the lattice of the variety generated by the known five-element semigroup. This semigroup has been indicated in the paper by Edmond Lee in 2013 year as the only five-element semigroup which may be not hereditarily finite based.