The aim of the talk is to study the notions of lax protomodular category [1] and \( \mathsf{Ord} \)-Mal'tsev category [2] in the context of (coherent) varieties of (pre)ordered algebras and to compare them, as has been done in the non-ordered context. We characterise varieties of ordered algebras which are (co)lax protomodular and those which are \( \mathsf{Ord} \)-Mal'tsev, in terms of operations of arities given by ordered sets and inequalities involving them. This study allowed us to obtain examples of (co)lax protomodular non-degenerate \( \mathsf{Ord} \)-categories (which had been an open question in [1]). For those which are \( \mathsf{Ord} \)-Mal'tsev categories, the order of their algebras is degenerate (i.e. is symmetric). Consequently, the implication "protomodular \( \Rightarrow \) Mal'tsev" cannot be carried out to our context. In the context of non-coherent ordered varieties which are \( \mathsf{Ord} \)-Mal'tsev categories, we give exampes of algebras with non-degenerate order.

References:
[1] M. M. Clementino, A. Montoli, D. Rodelo, On lax protomodularity for Ord-enriched categories, J. Pure Appl. Algebra 227(8) (2023) 107348.
[2] M. M. Clementino, D. Rodelo, Enriched aspects of calculus of relations and 2-permutability, J. Algebra Appl, 2025, 2650233.
[3] M. M. Clementino, D. Rodelo, A note on varieties of ordered algebras, DMUC 26-06 Preprint.