Some remarks on certain classes of semilattices

In this paper the concept of a ∗ -semilattice is introduced as a generalization to distributive ∗ -lattice first introduced by Speed [1]. It is shown that almost all the results of Speed can be extended to a more eneral class of distributive ∗ -semilattices. In pseudocomplemented semilattices and distributive semilattices the set of annihilators of an element is an ideal in the sense of Grätzer [2]. But it is not so in general and thus we are led to the definition of a weakly distributive semilattice. In § 2 we actually obtain the interesting corollary that a modular ∗ -semilattice is weakly distributive if and only if its dense filter is neutral. In § 3 the concept of a sectionally pseudocomplemented semilattice is introduced in a natural way. It is proved that given a sectionally pseudocomplemented semilattice there is a smallest quotient of it which is a sectionally Boolean algebra. Further as a corollary to one of the theorems it is obtained that a sectionally pseudocomplemented semilattice with a dense element becomes a ∗ -semilattice. Finally a necessary and sufficient condition for a ∗ -semilattice to be a pseudocomplemented semilattice is obtained.