Lattice Fully Orderable Groups

Abstract

Let Ω be a linearly ordered set, A(Ω) be the group of all order automorphisms of Ω, and L(Ω) be a normal subgroup of A(Ω) consisting of all automorphisms whose support is bounded above. We argue to show that, for every linearly ordered set Ω such that: (1) A(Ω) is an o-2-transitive group, and (2) Ω contains a countable unbounded sequence of elements, the simple group A(Ω)/L(Ω) has exactly two maximal and two minimal non-trivial (mutually inverse) partial orders, and that every partial order of A(Ω)/L(Ω) extends to a lattice one (Thm. 2.1). It is proved that every lattice-orderable group is isomorphically embeddable in a simple lattice fully orderable group (Thm. 2.2). We also state that some quotient groups of Dlab groups of the real line and unit interval are lattice fully orderable (Thms. 3.1 and 3.2).