Categorical-Algebraic Properties of Lattice-ordered Groups
Andrea Cappelletti · Theory and applications of categories · 2023
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.The author would like to thank the referee for the useful comments and interesting suggestions, which have led to the present improved version of the paper.Gratitude is also extended to Andrea Montoli, the author's PhD supervisor, for his valuable insights and guidance, which were fundamental in the development of this work.