Sublattice subgroups of finitely presented lattice-ordered groups

Abstract

Graham Higman proved that the finitely generated groups that occur as subgroups of finitely presented groups are precisely those that can be defined by recursively enumerable sets of relations.We prove the analogue for lattice-ordered groups: Theorem. A finitely generated lattice-ordered group is a sublattice subgroup of some finitely presented lattice-ordered group if and only if it can be defined by a recursively enumerable set of relations.Consequently, there is a universal finitely presented lattice-ordered group.

Finitely generated lattice-ordered groups with soluble word problem

“…In Section 3, we summarise the construction and formal proof of Theorem D from [5], and in Section 4, we outline the permutation construction used there and provide a modification. In Section 5, we use this modification to consider the solubility of the word problem for a given recursively generated lattice-ordered group defined by a recursively enumerable set of relations.…”

“…Then there is an algorithm that constructs a 2-generator ℓ-groupH and an explicit ℓ-embedding of H intoH such that (the image of) every element of H is equal to a group term in the generators ofH andH is definable by a recursively enumerable set of ℓ-group words; the defining relations forH are group terms or finite meets of group terms and are explicitly obtainable from the defining relations of H (see the proof of Theorem E in [5], Section 6). Moreover, the proof in [5] shows that this set of defining relations forH is recursive if the set of defining relations for H is, andH has soluble word problem whenever H does.…”

“…In [5], Section 3, we called these ℓ-group words meet strings and gave an explicit recursive Gödel numbering for the set of all meet strings occurring in the free ℓ-group on the n free generators y 1 , . .…”

“…. , y n ( [5], Section 3); each natural number was a pseudo-Gödel number of a unique meet string and each non-empty meet string had an infinite recursive set of pseudo-Gödel numbers.…”

“…, y n that hold in H. In [5], Section 5.1, we constructed from H (and X) a finitely presented ℓ-group L(X), and provided an explicit map ϕ of H into L(X). In Section 5.2 of [5], we proved that ϕ was a well-defined ℓ-homomorphism.…”