  In "Singularities on Normal Varieties", de Fernex and Hacon started the study
of singularities on non-Q-Gorenstein varieties using pullbacks of Weil
divisors. In "Log Terminal Singularities", the author of this paper and
Urbinati introduce a new class of singularities, called log terminal+, or
simply lt+, which they prove is rather well behaved. In this paper we will
continue the study of lt+ singularities, and we will show that they satisfy a
Bertini type result, inversion of adjunction and small deformation invariance,
and that they are naturally related to rational singularities. Finally, we will
provide a list of example (all of them with lt+ singularities) of the
pathologies that can occur in the study of non-Q-Gorenstein singularities.
