Weak Congruence Semidistributivity Laws and Their Conjugates

Lattice Horn sentences including Geyer’s SD(n, 2) and their conjugates C(n, 2) are considered. SD(2, 2) is the meet semidistributivity law SD∧. Both SD(n, 2) and C(n, 2) become strictly weaker when n grows. For varieties V the satisfaction of SD(n, 2) in {Con(A) : A ∈ V} is characterized by a Mal’cev condition. Using this Mal’cev condition it is shown that C(n, 2) |=con SD(n, 2), which means that, for every variety V , whenever C(n, 2) holds in {Con(A) : A ∈ V} then so does SD(n, 2). In particular, C(2, 2) |=con SD(2, 2), which is a stronger statement than SD∨ |=con SD∧, the only previously known |=con result between lattice Horn sentences “not below congruence modularity”. Some other |=con statements are also presented.