Larger locally compact monothetic semigroups

We show that for every k. N, there is a locally compact noncompact monothetic semigroup S with identity such that S is homeomorphic to a closed nowhere dense subset of Rk+1 and the connected components of S are homeomorphic to Rk+, so dim S = k. The semigroup S can be algebraically embedded in R and cannot be topologically and algebraically embedded in a topological group. Finally, we show that for every second countablemonothetic semigroup M with identity, the semigroup MxTc is also monothetic. Consequently, there are locally compact noncompact monothetic semigroups with identity which have infinite dimension and cardinality 2c.