We introduce the concepts of an annihilator and a relative annihilator of a given subset of a BCK-algebra . We prove that annihilators of deductive systems of BCK-algebras are again deductive systems and moreover pseudocomplements in the lattice of all deductive systems on . Moreover, relative annihilators of with respect to are introduced and serve as relative pseudocomplements of w.r.t. in .
Given a groupoid , and , we say that is antiassociative if an only if for all , and are never equal. Generalizing this, is -antiassociative if and only if for all , any two distinct expressions made by putting parentheses in are never equal. We prove that for every , there exist finite groupoids that are -antiassociative. We then generalize this, investigating when other pairs of groupoid terms can be made never equal.
A tolerance on an algebra is defined similarly to a congruence, only the requirement of transitivity is omitted. The paper studies a special type of tolerance, namely atomary tolerances. They exist on every finite algebra.