The purpose of this appendix is to establish Holland's theorem and some immediate consequences of it.

THEOREM (Holland). Every ℓ-group can be ℓ-embedded in A(Ω) for some chain Ω. Indeed, if G is an ℓ-group, then (G,Ω) is an ℓ-permutation group for some chain Ω with |Ω| ≤ |G|.

This is the natural analogue of Cayley's theorem for groups. In order to prove the theorem, we first need to establish some lemmas. If C is a subgroup of G, write R(C) for the set of right cosets of C in G.

LEMMA1. Let C be a convex subgroup of a p.o. group G. Then R(C) is a.p.o. set if we define Cg ≤ Cf if and only if cg ≤ f for some c ∈ C. If G is an ℓ-group and C is a convex ℓ-subgroup of G, then R(C) becomes a lattice with Cg ∨ Ch = C(g ∨ h) and Cg ∧ Ch = C(g ∧ h) (g,h ∈ G).

Proof: By routine verification.

A convex ℓ-subgroup C of an ℓ-group G is said to be prime if f,g ∈ G and f ∧ g = e imply f ∈ C or g ∈ C.

LEMMA2. Let C be a convex ℓ-subgroup of an ℓ-group G. Then the following are equivalent:

1. C is prime.

2. R(C) is a chain.

3. The set of convex ℓ-subgroups of G that contain C forms a chain under inclusion.

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