The aim of this work is to explore the 1-dimensional algebraic Mal'tsev property from an Ord-enriched point of view. A 1-dimensional (regular) Mal'tsev category [2, 1] may be characterised through nice properties on (internal) relations such as:
- every reflexive relation \( R \rightarrowtail X \times X \) is an equivalence relation;
- any relation \( D \rightarrowtail X \times Y \) is difunctional, meaning that \( DD^o D \leq D \).
The proof of such characterisations are easily obtained through the calculus of relations, which has been well established for regular categories for several years (see [1]).

In order to explore the Mal'tsev property in an Ord-enriched context we have to develop the calculus of relations for regular Ord-categories. To capture the enriched features of a regular Ord-category and obtain a good calculus,
the relations we work with are precisely the ideals. We introduce the notion of Ord-Mal'tsev category and show that these may be characterised through enriched versions of the above mentioned properties adapted to ideals. Any Ord-enrichment of a 1-dimensional Mal'tsev category is necessarily an Ord-Mal'tsev category. We also give some examples of categories which are not Mal'tsev categories, but are Ord-Mal'tsev categories.

* Joint work with Maria Manuel Clementino.

[1] A. Carboni, G. M. Kelly, M. C. Pedicchio, Some remarks on Maltsev and Goursat categories, Appl. Categ. Struct. 14 (1993) 385-421.

[2] A. Carboni, J. Lambek, M. C. Pedicchio, Diagram chasing in Mal'cev categories, J. Pure Appl. Algebra 69 (1991) 271-284.

[3] M.M. Clementino, D. Rodelo, Enriched aspects of calculus of relations and 2-permutability, DMUC preprint 24-33 (2024) 24 pgs.