ACTA MATHEMATICA UNIVERSITATIS COMENIANAE

Vol. 68,   1   (1999)
pp.   153-170

WEAK CONGRUENCE SEMIDISTRIBUTIVITY LAWS AND THEIR CONJUGATES
G. CZEDLI

Abstract.  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 $SD_\meet$. 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\in \V\$ is characterized by a Mal'cev condition. Using this Mal'cev condition it is shown that $C(n,2)\cimply SD(n,2)$, which means that, for every variety $\V$, whenever $C(n,2)$ holds in $\\Con(A): A\in \V\$ then so does $SD(n,2)$. In particular, $C(2,2)\cimply SD(2,2)$, which is a stronger statement than $SD_\join \cimply SD_\meet$, the only previously known $\cimply$ result between lattice Horn sentences not below congruence modularity''. Some other $\cimply$ statements are also presented.

AMS subject classification.  08B10; Secondary 08B05
Keywords.  Mal'cev condition, congruence variety, lattice, semidistributivity, Horn sentence