Lattice Fully Orderable Groups

Let $\Omega$ be a linearly ordered set, $A(\Omega)$ be the group of all order automorphisms of $\Omega$, and $L(\Omega)$ be a normal subgroup of $A(\Omega)$ consisting of all automorphisms whose support is bounded above. We argue to show that, for every linearly ordered set $\Omega$ such that: (1) $A(\Omega)$ is an $o$-2-transitive group, and (2) $\Omega$ contains a countable unbounded sequence of elements, the simple group $A(\Omega)/L(\Omega)$ has exactly two maximal and two minimal non-trivial (mutually inverse) partial orders, and that every partial order of $A(\Omega)/L(\Omega)$ extends to a lattice one. It is proved that every lattice-orderable group is isomorphically embeddable in a simple lattice fully orderable group. We also state that some quotient groups of Dlab groups of the real line and unit interval are lattice fully orderable.