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The semigroup inclusion class I = [ x y x y = x y ; x y z ∈ { x y w z , x u y z } ] is the union of two maximal subvarieties of GRB = [ x y z x y = x y ] . Monzo (A lattice of semigroup inclusion classes consisting of unions of varieties of generalised inflations of rectangular bands, arXiv:1411.4860) described the lattice of semigroup inclusion classes below I and asked if I is covered by GRB . Our main result is a characterization of I which makes it easy to answer Monzo's question in the negative. [ABSTRACT FROM AUTHOR]

We determine the structure of semigroups that satisfy xyzw∈{ xy, xw, zy, zw}. These semigroups are precisely those whose power semigroup is a generalised inflation of a band. The structure of generalised inflations of the following types of semigroups is determined: the direct product of a group and a band, a completely simple semigroup and a free semigroup F( X) on a set X. In the latter case the semigroup must be an inflation of F( X). We also prove that in any semigroup that equals its square, the power semigroup is a generalised inflation of a band if and only if it is an inflation of a band. [ABSTRACT FROM AUTHOR]