Varieties of algebras arising from K-perfect m-cycle systems

A class of algebras forms a variety if it is characterised by a collection of identities. There is a well-known method, often called the standard construction, which gives rise to algebras from m  -cycle systems. It is known that the algebras arising from {1}{1}-perfect m  -cycle systems form a variety for m∈{3,5}m∈{3,5} only, and that the algebras arising from {1,2}{1,2}-perfect m  -cycle systems form a variety for m∈{3,5,7}m∈{3,5,7} only. Here we give, for any set K of positive integers, necessary and sufficient conditions under which the algebras arising from K-perfect m-cycle systems form a variety.