Seminário de Álgebra, Lógica e Topologia: On the protomodular and Mal'tsev properties for varieties of ordered algebras

The aim of the talk is to study the notions of lax protomodular category [1] and 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 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 Ord-categories (which had been an open question in [1]). For those which are Ord-Mal'tsev categories, the order of their algebras is degenerate (i.e. is symmetric). Consequently, the implication "protomodular implies Mal'tsev" cannot be carried out to our context. In the context of non-coherent ordered varieties which are 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.