# Abelian lattice-ordered groups

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### Definition

An abelian lattice-ordered group (or abelian -group) is a lattice-ordered group L = (L, ∨, ∧, ·, −1, e) such that · is commutative:   x·y = y·x.

### Morphisms

Let L and M be -groups. A morphism from L to M is a function f : LM that is a homomorphism: f(xy) = f(x)∨f(y) and f(x·y) = f(xf(y).

Remark: It follows that f(xy) = f(x)∧f(y), f(x−1) = f(x)−1, and f(e) = e

### Definition

An abelian lattice-ordered group (or abelian -group) is a commutative residuated lattice L = (L, ∨, ∧, ·,  → , e) that satisfies the identity x·(x → e) = e.

Remark: x−1 = x → e and x → y = x−1y

### Examples

(Z, max, min, + , −, 0), the integers with maximum, minimum, addtion, unary subtraction and zero. The variety of abelian -groups is generated by this algebra.

### Some results

The lattice reducts of (abelian) -groups are distributive lattices.

### Properties

 Classtype variety Equational theory decidable Quasiequational theory decidable First-order theory hereditarily undecidable [Yuri Gurevic, Hereditary undecidability of a class of lattice-ordered Abelian groups, Algebra i Logika Sem. 6 (1967) 45--62 MRreview] [Stanley Burris, A simple proof of the hereditary undecidability of the theory of lattice-ordered abelian groups, Algebra Universalis 20 (1985) 400--401 MRreview] Locally finite No Residual size Congruence distributive yes (see lattices) Congruence modular yes Congruence n-permutable yes, n = 2 (see groups) Congruence regular yes, (see groups) Congruence uniform yes, (see groups) Congruence extension property Definable principal congruences Equationally definable principal congruences Amalgamation property yes Strong amalgamation property Epimorphisms are surjective

None

### Subclasses

[Totally ordered abelian groups]?

### Superclasses

[Representable lattice-ordered groups]?

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