  We study the categorical-algebraic properties of the semi-abelian variety
$\ell \mathbb{G}rp$ of lattice-ordered groups. In particular, we show that this
category is fiber-wise algebraically cartesian closed, arithmetical, and
strongly protomodular. Moreover, we observe that $\ell \mathbb{G}rp$ is not
action accessible, despite the good behaviour of centralizers of internal
equivalence relations. Finally, we restrict our attention to the subvariety
$\ell \mathbb{A}b$ of lattice-ordered abelian groups, showing that it is
algebraically coherent; this provides an example of an algebraically coherent
category which is not action accessible.
