A. M. W. Glass

Partially Ordered Groups

Definitions and examples basic properties values, primes and polars Abelian and normal-valued lattice-ordered groups Archimediean function groups soluble right partially ordered groups and generalizations permutations applications completions varieties of lattice-ordered groups unsolved problems.

Rigid homogeneous chains

Classifying (unordered) sets by the elementary (first order) properties of their automorphism groups was undertaken in (7), (9) and (11). For example, if Ω is a set whose automorphism group, S (Ω), satisfies then Ω has cardinality at most ℵ 0 and conversely (see (7)). We are interested in classifying homogeneous totally ordered sets (homogeneous chains, for short) by the elementary properties of their automorphism groups. (Note that we use ‘homogeneous’ here to mean that the automorphism group is transitive.) This study was begun in (4) and (5). For any set Ω, S (Ω) is primitive (i.e. has no congruences). However, the automorphism group of a homogeneous chain need not be o -primitive (i.e. it may have convex congruences). Fortunately, ‘ o -primitive’ is a property that can be captured by a first order sentence for automorphisms of homogeneous chains. Hence our general problem falls naturally into two parts. The first is to classify (first order) the homogeneous chains whose automorphism groups are o -primitive; the second is to determine how the o -primitive components are related for arbitrary homogeneous chains whose automorphism groups are elementarily equivalent.

Highly transitive representations of free groups and free products

A permutation group is highly transitive if it is n –transitive for every positive integer n . A group G of order-preserving permutations of the rational line Q is highly order-transitive if for every α 1 α n and β 1 β n in Q there exists g ∈ G such that α ig = β i , i = 1, …, n . The free group F n (2 ≤ η ≤ א o ) can be faithfully represented as a highly order-transitive group of order-preserving permutations of Q, and also (reproving a theorem of McDonough) as a highly transitive group on the natural numbers N. If G and H are nontrivial countable groups having faithful representations as groups of order-preserving permutations of Q, then their free product G * H has such a representation which in addition is highly order-transitive. If G and H are nontrivial finite or countable groups and if H has an element of infinite order, then G * H can be faithfully represented as a highly transitive group on N. Some of the representations of F η on Q can be extended to faithful representations of the free lattice-ordered group L η .

Free abelian lattice-ordered groups

Abstract Let n be a positive integer and F A l ( n ) be the free abelian lattice-ordered group on n generators. We prove that F A l ( m ) and F A l ( n ) do not satisfy the same first-order sentences in the language L = { + , − , 0 , ∧ , ∨ } if m ≠ n . We also show that Th ( F A l ( n ) ) is decidable iff n ∈ { 1 , 2 } . Finally, we apply a similar analysis and get analogous results for the free finitely generated vector lattices.

Finitely generated lattice-ordered groups with soluble word problem

Abstract William W. Boone and Graham Higman proved that a finitely generated group has soluble word problem if and only if it can be embedded in a simple group that can be embedded in a finitely presented group. We prove the exact analogue for lattice-ordered groups: Theorem. A finitely generated lattice-ordered group has soluble word problem if and only if it can be ℓ-embedded in an ℓ-simple lattice-ordered group that can be ℓ-embedded in a finitely presented lattice-ordered group. The proof uses permutation groups, a technique of Holland and McCleary, and the ideas used to prove the lattice-ordered group analogue of Higmans embedding theorem.

Embedding finitely generated Abelian lattice-ordered groups : Higman's theorem and a realisation of \pi

Graham Higman proved that a finitely generated group can be embedded in a finitely presented group if and only if it has a recursively enumerable set of defining relations. The analogue for lattice-ordered groups is considered here. Clearly, the finitely generated lattice-ordered groups that can be

FREE GROUPS FROM FIELDS

\ell

Right orders and amalgamation for lattice-ordered groups

-embedded in finitely presented lattice-ordered groups must have recursively enumerable sets of defining relations. The converse direction is proved for a special class of lattice-ordered groups. T HEOREM . Every finitely generated Abelian lattice-ordered group that has finite rank and a recursively enumerable set of defining relations can be

Finitely presented abelian lattice-ordered groups

\ell

Composites of translations and odd rational powers act freely

- embedded in a finitely presented lattice-ordered group . If