Abbreviation: AbLGrp

An \emph{abelian lattice-ordered group} (or abelian $\ell$\emph{-group}) is a lattice-ordered group $\mathbf{L}=\langle L, \vee, \wedge, \cdot, ^{-1}, e\rangle$ such that

$\cdot$ is commutative: $x\cdot y=y\cdot x$

Let $\mathbf{L}$ and $\mathbf{M}$ be $\ell$-groups. A morphism from $\mathbf{L}$ to $\mathbf{M}$ is a function $f:L\rightarrow M$ that is a homomorphism: $f(x\vee y)=f(x)\vee f(y)$ and $f(x\cdot y)=f(x)\cdot f(y)$.

Remark: It follows that $f(x\wedge y)=f(x)\wedge f(y)$, $f(x^{-1})=f(x)^{-1}$, and $f(e)=e$

An \emph{abelian lattice-ordered group} (or \emph{abelian $\ell$-group}) is a commutative residuated lattice $\mathbf{L}=\langle L, \vee, \wedge, \cdot, \to, e\rangle$ that satisfies the identity $x\cdot(x\to e)=e$.

Remark: $x^{-1}=x\to e$ and $x\to y=x^{-1}y$

$\langle\mathbb{Z}, \mbox{max}, \mbox{min}, +, -, 0\rangle$, the integers with maximum, minimum, addition, unary subtraction and zero. The variety of abelian $\ell$-groups is generated by this algebra.

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

None

Totally ordered abelian groups

Representable lattice-ordered groups