MONOTONE CLONES AND CONGRUENCE MODULARITY

We investigate the relationship between the local shape of an ordered set P=(P; ≤) and the congruence-modularity of the variety V generated by an algebra A=(P; F) each of whose operations is order-preserving with respect to P. For example, if V is k-permutable (k≥2) then P is an antichain; if P is both up and down directed and V is congruence-modular, then V is congruence-distributive; if A is a dual discriminator algebra, then either P is an antichain or a two-element chain. We also give a useful necessary condition on P for V to be congruence-modular. Finally a class of ordered sets called braids is introduced and it is shown that if P is a braid of length 1, in particular if P is a crown, then the variety V is not congruence-modular. © 1990 Kluwer Academic Publishers.