On definable sets in some definably complete locally o-minimal structure

In this paper, we show that the Grothendieck ring of a definably complete locally o-minimal expansion of the set (not the field) of real numbers $\mathbb R$ is trivial. Afterwards, we will give a sufficient condition for which a definably complete locally o-minimal expansion of an ordered group has no nontrivial definable subgroups. In the last section, we study some sets that are definable in a definably complete locally o-minimal expansion of an ordered field. Finally, a decomposition theorem for a definable set into finite union of $\pi_L$-quasi-special $\mathcal{C}^r$ submanifolds is demonstrated.