Special classes of positive implication algebras

We construct a positive implication algebra PI(A) inherited from a chain. For a non-empty subset ∇ of a positive implication algebra, using the special set ∇(a,b), we give a necessary and sufficient condition for ∇ to be an implicative filter. We also introduce the notion of a ∇-type positive implication algebra, and prove the followings: (1) every ∇-type positive implication algebra is a commutative monoid under the operation ȯ, but not a group; (2) every ∇-type positive implication algebra is a lower semilattice; (3) in a ∇-type positive implication algebra the class of implicative filters coincides with the class of convex subsemigroups with V. Finally we show that every ∇-type positive implication algebra is a semi-Brouwerian algebra, and consider the direct product of implicative algebras.