On subsemigroup lattices of aperiodic groups

This note is inspired by some results and questions from the survey [9] of Shevrin and Ovsyannikov and by Shevrin personally. The author wishes to thank Professor Shevrin for fruitful discussions. In the note some problems related to determinability of aperiodic (=non-torsion) groups by their subsemigroup lattices are settled in the negative. It is worth pointing out that the groups constructed below in the negative. It is worth pointing out that the groups that satisfy the maximal condition for subsemigroups.