The lattice of prefilters of an EQ-algebra

In this paper, the notion of a prefilter generated by a nonempty subset of an EQ-algebra is introduced and a characterization of it is obtained. It is proved that the set of all prefilters of an EQ-algebra is an algebraic lattice and it is a Brouwerian lattice for an ℓEQ-algebra. Furthermore, it is shown that the set of all principal prefilters of an ℓEQ-algebra is a sublattice of the lattice of prefilters. Then by defining an implication between two prefilters, it is determined that the lattice of prefilters is a Heyting algebra, for an ℓEQ-algebra. Finally, the EQ-algebras for which the lattice of prefilters is a Boolean algebra are given.