A note on varieties of ordered algebras

TL;DR

This paper investigates the properties of ordered algebra varieties, characterizing lax protomodular and Ord-Mal'tsev categories, and explores their relationships and examples within the context of ordered and preordered algebras.

Contribution

It provides a characterization of ordered algebra varieties that are lax protomodular or Ord-Mal'tsev, including new examples and analysis of their order properties.

Findings

Ordered algebra varieties can be characterized by operations and inequalities.

Examples of non-degenerate Ord-categories that are lax protomodular are provided.

In Ord-Mal'tsev categories, the algebra's order is degenerate (symmetric).

Abstract

The aim of this work is to study the notions of lax protomodular and Ord-Mal'tsev category at the level of (coherent) varieties of (pre)ordered algebras and to further 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. We exhibit examples of (co)lax protomodular non-degenerate Ord-categories, which were unknown. We prove that, for varieties of ordered algebras which are Ord-Mal'tsev categories, the order of their algebras is degenerate (i.e. is symmetric). As a consequence, the implication "protomodular => Mal'tsev" cannot be carried out to our context. The case of non-coherent ordered varieties which are Ord-Mal'tsev categories is also addressed, where we show the existence of algebras with non-degenerate order.