Categorical-Algebraic Properties of Lattice-ordered Groups

Andrea Cappelletti

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.

Keywords: lattice-ordered group, semi-abelian category, algebraically coherent category

2020 MSC: 06F15, 18E13

Theory and Applications of Categories, Vol. 39, 2023, No. 31, pp 916-948.

Published 2023-12-01.

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