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Abelian lattice-ordered groupsAbelian lattice-ordered groups
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.
| 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 |