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Countable lattice-ordered groups
| Content Provider | Scilit |
|---|---|
| Author | Glass, A. M. W. |
| Copyright Year | 1983 |
| Abstract | A lattice-ordered group is a group and a lattice such that the group operation distributes through the lattice operations (i.e. f(g ∨ h)k = fgk ∨ fhk and dually). Lattice-ordered groups are torsion-free groups and distributive lattices. They further satisfy f ∧ g = $(f^{−1}$ ∨ $g^{−1})^{−1}$ and f ∨ g = $(f^{−1}$ ∧ $g^{−1})^{−1}$. Since the lattice is distributive, each lattice-ordered group word can be written in the form $∨_{A}$ $∧_{B}$ $ω_{αβ}$ where A and B are finite and each $ω_{αβ}$ is a group word in ${x_{i}$: i ∈ I}. Unfortunately, even for free lattice-ordered groups, this form is not unique. We will use the prefix l- for maps between lattice-ordered groups that preserve both the group and lattice operations, and e for the identity element. A presentation $(x_{i};r_{j}$(x) = $e)_{i∈I}$,$ _{j∈J}$ is the quotient of the free lattice-ordered group F on ${x_{i}$: i∈I} by the l-ideal (convex normal sublattice subgroup) generated by its subset ${r_{j}$(x): j ∈ J}. ${x_{i}$: i ∈ I} is called a generating set and ${r_{i}$(x):j∈J} a defining set of relations. If I and J are finite we have a finitely presented lattice-ordered group. If we can effectively enumerate all lattice-ordered group words $r_{1}$(x), $r_{2}$(x),… in $x_{i}$; i∈I}. If I is finite and J (for this enumeration) is a recursively enumerable set, we say that we have a recursively presented lattice-ordered group. Throughout Z denotes the group of integers and ℝ the real line.Our purpose in this paper is to prove the natural analogues of three theorems from combinatorial group theory (5), chapter IV, theorems 4·9, 3·1 and 3·5-in particular, theorem C is a natural analogue of an unpublished theorem of Philip Hall (4). |
| Related Links | https://www.cambridge.org/core/services/aop-cambridge-core/content/view/E0E3FE136CEAC42596633596429F0C1D/S0305004100060898a.pdf/div-class-title-countable-lattice-ordered-groups-div.pdf |
| Ending Page | 33 |
| Page Count | 5 |
| Starting Page | 29 |
| ISSN | 03050041 |
| e-ISSN | 14698064 |
| DOI | 10.1017/s0305004100060898 |
| Journal | Mathematical Proceedings of the Cambridge Philosophical Society |
| Issue Number | 1 |
| Volume Number | 94 |
| Language | English |
| Publisher | Cambridge University Press (CUP) |
| Publisher Date | 1983-07-01 |
| Access Restriction | Open |
| Subject Keyword | Mathematical Proceedings of the Cambridge Philosophical Society Group and Lattice |
| Content Type | Text |
| Resource Type | Article |
| Subject | Mathematics |
