on heyting algebras and dual bck-algebras

a heyting algebra is a distributive lattice with implication and a dual $bck$-algebra is an algebraic system having as models logical systems equipped with implication. the aim of this paper is to investigate the relation of heyting algebras between dual $bck$-algebras. we define notions of $i$-invariant and $m$-invariant on dual $bck$-semilattices and prove that a heyting semilattice is equivalent to an $i$-invariant and $m$-invariant dual $bck$-semilattices, and show that a commutative heyting algebra is equivalent to a bounded implicative dual $bck$-algebra.