  Suppose $X$ is a locally solid vector lattice. In this paper, we introduce
the notion "$AM$-property" in $X$ as an extension for $AM$-spaces in the
category of all Banach lattices. With the aid of this concept, we characterize
spaces in which bounded sets and order bounded sets agree. This, in turn,
characterizes conditions under which each class of bounded operators on $X$ is
order bounded and vice versa. Also, we show that under some natural
assumptions, different types of bounded order bounded operators on $X$ have the
Lebesgue or Levi property if and only if so is $X$.
