  Subresiduated lattices were introduced during the decade of 1970 by Epstein
and Horn as an algebraic counterpart of some logics with strong implication
previously studied by Lewy and Hacking. These logics are examples of
subuintuitionistic logics, i.e., logics in the language of intuitionistic logic
that are defined semantically by using Kripke models, in the same way as
intuitionistic logic is defined, but without requiring of the models some of
the properties required in the intuitionistic case. Also in relation with the
study of subintuitionistic logics, Celani and Jansana get these algebras as the
elements of a subvariety of that of weak Heyting algebras.
  Here, we study both the implicative and the implicative-infimum subreducts of
subresiduated lattices. Besides, we propose a calculus whose algebraic
semantics is given by these classes of algebras. Several expansions of this
calculi are also studied together to some interesting properties of them.
