An abelian lattice-ordered group (or abelian ℓ-group) is a lattice-ordered group L = (L, ∨, ∧, ·, −1, e) such that · is commutative: x·y = y·x.
Let L and M be ℓ-groups. A morphism from L to M is a function f : L→M that is a homomorphism: f(x∨y) = f(x)∨f(y) and f(x·y) = f(x)·f(y).
Remark: It follows that f(x∧y) = f(x)∧f(y), f(x−1) = f(x)−1, and f(e) = e
An abelian lattice-ordered group (or abelian ℓ-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
(Z, max, min, + , −, 0), the integers with maximum, minimum, addtion, unary subtraction and zero. The variety of abelian ℓ-groups is generated by this algebra.
The lattice reducts of (abelian) ℓ-groups are distributive lattices.
|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]
|Congruence distributive||yes (see lattices)|
|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|
|Strong amalgamation property|
|Epimorphisms are surjective|