On the relation between left thickness and topological left thickness in semigroups

In this paper, we establish an interesting relation between left thickness and topological left thickness in semigroups by showing that a Borel subset $T$ of a locally compact semigroup $S$ is topological left thick in $S$ iff a certain subset ${M_T}$ associated with $T$ is left thick in a semigroup ${S_1}$ containing $S$, or equivalent, iff ${M_T}$ contains a left ideal of ${S_1}$. Our results contain a topological analogue of a result of H. Junghenn in [Amenability of function spaces on thick subsemigroups, Proc. Amer. Math. Soc. 75 (1979), 37-41]. However, even in the case of discrete semigroups, our results are more general and in a way more natural than those of Junghenn's. Furthermore, the fact that ${M_T}$ is left thick iff it contains a left ideal in ${S_1}$ is quite surprising, since in general, a left thick subset need not contain a left ideal although the converse is always true.