Boolean Properties Of Normal Completions Of Lattices

In this paper,as some generalizations of complete Heyting algebras,the con-cepts of E-Heyting posets,Heyting semilattices and pseudocomplement semilattices are introduced.We prove that(1)a poset P is a Heyting poset if and only if its Dedekind-MacNeille completion is a complete Heyting algebra;(2)a Heyting semi-lattice is an E-Heyting poset;and(3)for a lattice B=(B,∨.∧,0,1)with 0 and 1,the following three conditions are equivalent:(i)B is a Boolean algebra;(ii)B is a Heyting and its normal completion is a Boolean algebra;and(iii)B is a pseudocomplement semilattice and its normal completion is a Boolean algebra.