Following the study of the categorical features of preordered groups of [1], in this talk we study the category of rightpreordered groups and monotone homomorphisms. We start by showing that this category has exactly the same exactness properties as the category of preordered groups. Regarding the behaviour of split extensions, guided by results of [1] and [2], we analyse the existence of compatible rightpreorders, showing that, for semidirect products, it is more natural to ask for a compatible rightpreorder than for a compatible preorder  this is not surprising due to the different roles played by the left and right summands in the addition of the semidirect product. This conclusion leads to interesting examples of rightpreordered groups which cannot be preordered.
This talk is based on joint work with Andrea Montoli [3].
References: [1] M.M. Clementino, N. MartinsFerreira, A. Montoli, On the categorical behaviour of preordered groups. J. Pure Appl. Algebra 223 (2019) 42264245. [2] M.M. Clementino, C. Ruivo, On split extensions of preordered groups. Port. Math. 80 (2023) 327341. [3] M.M. Clementino, A. Montoli, Rightpreordered groups from a categorical perspective. DMUC preprint 2432, arXiv 2406.10071
