  Hemi-implicative semilattices (lattices), originally defined under the name
of weak implicative semilattices (lattices), were introduced by the second
author of the present paper. A hemi-implicative semilattice is an algebra
$(H,\wedge,\rightarrow,1)$ of type $(2,2,0)$ such that $(H,\wedge)$ is a meet
semilattice, $1$ is the greatest element with respect to the order,
$a\rightarrow a = 1$ for every $a\in H$ and for every $a$, $b$, $c\in H$, if
$a\leq b\rightarrow c$ then $a\wedge b \leq c$. A bounded hemi-implicative
semilattice is an algebra $(H,\wedge,\rightarrow,0,1)$ of type $(2,2,0,0)$ such
that $(H,\wedge,\rightarrow,1)$ is a hemi-implicative semilattice and $0$ is
the first element with respect to the order. A hemi-implicative lattice is an
algebra $(H,\wedge,\vee,\rightarrow,0,1)$ of type $(2,2,2,0,0)$ such that
$(H,\wedge,\vee,0,1)$ is a bounded distributive lattice and the reduct algebra
$(H,\wedge,\rightarrow,1)$ is a hemi-implicative semilattice.
  In this paper we introduce an equivalence for the categories of bounded
hemi-implicative semilattices and hemi-implicative lattices, respectively,
which is motivated by an old construction due J. Kalman that relates bounded
distributive lattices and Kleene algebras.

Logic