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Embedding Finitely Generated Abelian Lattice‐Ordered Groups: Higman's Theorem and a Realisation of π
Author(s) -
Glass A. M. W.,
Marra Vincenzo
Publication year - 2003
Publication title -
journal of the london mathematical society
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.441
H-Index - 62
eISSN - 1469-7750
pISSN - 0024-6107
DOI - 10.1112/s002461070300468x
Subject(s) - realisation , abelian group , embedding , mathematics , pi , finitely generated abelian group , discrete mathematics , lattice (music) , pure mathematics , combinatorics , physics , computer science , quantum mechanics , geometry , artificial intelligence , acoustics
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 ℓ‐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. Theorem. Every finitely generated Abelian lattice‐ordered group that has finite rank and a recursively enumerable set of defining relations can be ℓ‐ embedded in a finitely presented lattice‐ordered group . If ξ is a real number, let D (ξ) be the Abelian rank 2 group Z 2 with order ( m, n )>0 if and only if m + n ξ>0. Corollary. D (ξ) can be ℓ‐ embedded in a finitely presented lattice‐ordered group if and only if ξ is a recursive real number . Thus an algebraic characterisation of recursive real numbers is obtained. In particular, π is ‘ℓ‐algebraic’ in that it can be captured by finitely many relations in this language.