The aim of this work is to explore the 1dimensional algebraic Mal'tsev property from an Ordenriched point of view. A 1dimensional (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 Ordenriched context we have to develop the calculus of relations for regular Ordcategories. To capture the enriched features of a regular Ordcategory and obtain a good calculus, the relations we work with are precisely the ideals. We introduce the notion of OrdMal'tsev category and show that these may be characterised through enriched versions of the above mentioned properties adapted to ideals. Any Ordenrichment of a 1dimensional Mal'tsev category is necessarily an OrdMal'tsev category. We also give some examples of categories which are not Mal'tsev categories, but are OrdMal'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) 385421.
[2] A. Carboni, J. Lambek, M. C. Pedicchio, Diagram chasing in Mal'cev categories, J. Pure Appl. Algebra 69 (1991) 271284.
[3] M.M. Clementino, D. Rodelo, Enriched aspects of calculus of relations and 2permutability, DMUC preprint 2433 (2024) 24 pgs.
