Fuzzy Ideals and Fuzzy Filters of Pseudocomplemented Semilattices

In this paper, we introduce the concept of kernel fuzzy ideals and ⁎-fuzzy filters of a pseudocomplemented semilattice and investigate some of their properties. We observe that every fuzzy ideal cannot be a kernel of a ⁎-fuzzy congruence and we give necessary and sufficient conditions for a fuzzy ideal to be a kernel of a ⁎-fuzzy congruence. On the other hand, we show that every fuzzy filter is the cokernel of a ⁎-fuzzy congruence. Finally, we prove that the class of ⁎-fuzzy filters forms a complete lattice that is isomorphic to the lattice of kernel fuzzy ideals.

Finite semilattices whose non-invertible endomorphisms are products of idempotents

For a finite semilattice S, is is proved that if every noninvertible endomorphism is a product of idempotents, then S is a chain; the converse was proved, independently, by A. Ya. Aĭzenštat and J. M. Howie. For a finite pseudocomplemented semilattice S, with pseudocomplementation regarded as a unary operation, it is proved that all noninvertible endomorphisms are products of idempotents if and only if S is Boolean or a chain.

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§2we actually obtain the interesting corollary that a modular∗-semilattice is weakly distributive if and only if its dense filter is neutral. In§3the 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.