  The aim of this work is to further develop the calculus of (internal)
relations for a regular Ord-category C. To capture the enriched features of a
regular Ord-category and obtain a good calculus, the relations we work with are
precisely the ideals in C. We then focus on an enriched version of the
1-dimensional algebraic 2-permutable (also called Mal'tsev) property and its
well-known equivalent characterisations expressed through properties on
ordinary relations. 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.
