Lattice-Ordered Permutation Groups

Abstract

Although most of the elementary theorems about o-groups are as easy to prove in the general case as in the commutative case, there are no natural examples of non-commutative o-groups. Of course, it is easy to construct examples of non-commutative o-groups, but all of these are artificial and without much interest outside of the context of ordered groups. The situation is quite different for general lattice-ordered groups, however, because there is a class of non-commutative examples of great intrinsic interest, independent of the fact that they are lattice-ordered. These are the groups A(Ω) of order-preserving permutations of totally ordered sets Ω, endowed with the pointwise order. This means that for f,g ∈ A(Ω), we declare that f ≤ g iff for all α ∈ Ω, αf≤ αg. It is easily checked that this makes A(Ω) a lattice-ordered group in which α(f∧g) = αf∧αg. Only rarely is A(Ω) commutative.