Monotone clones and congruence modularity (Q922571)

The scope of this paper is to investigate the relationship between the local shape of an ordered set \({\mathbb{P}}=(P;\leq)\) and the congruence- modularity of the variety \({\mathcal V}\) generated by an algebra \({\mathbb{A}}=(P;F)\) each of whose operations is order-preserving with respect to \({\mathbb{P}}\). Finally a class of ordered sets called braids is introduced and it is shown that if \({\mathbb{P}}\) is a braid of length 1, in particular if \({\mathbb{P}}\) is a crown, then the variety \({\mathcal V}\) is not congruence- modular.