Uniqueness of the group operation on the lattice of order-automorphisms of the real line

A(R) is the lattice-ordered group (l-group) of all order-automorphisms of the real lineR, with the usual pointwise order and “of course” with composition as the group operation. In fact, what other choices are there for a group operation having the same identity that would give anl-group? Composition in the reverse order would work. But there are no other choices — the group operation can be recognized in the lattice. Several classes of abelianl-groups having a unique group operation have been found by Conrad and Darnel, but this is the first non-abelian example having the minimum of two group operations. “Conversely”, Holland has shown that for the groupA(R) under composition, the only lattice orderings yielding anl-group are the pointwise order and its dual. These results also hold for the rational lineQ.