Abstract
In this paper we, contribute the notation of natural epimorphism of a semilattice on the quotient semilattice and subsemilattice. If S is distributive semilattice and F is a filter of S, then we demonstrate that θF is the smallest congruence on S containing F in a single equivalence class and that S/θF is distributive. In addition, the author proved that map FθF is an isomorphism from the lattice of F0(S) all non-empty filters of S into a permutable sublattice of the lattice C(S) of all congruencies on S.
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