P. V. Ramana Murty and M. Krishna Murty

Copyright © 1982 P. V. Ramana Murty and M. Krishna Murty. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

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.