Categorical-Algebraic Properties of Lattice-ordered Groups

We study the categorical-algebraic properties of the semi-abelian variety ℓGrp 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 ℓGrp is not action accessible, despite the good behaviour of centralizers of internal equivalence relations. Finally, we restrict our attention to the subvariety ℓAb 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.