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Hemi-implicative semilattices (lattices), originally defined under the nameof weak implicative semilattices (lattices), were introduced by the secondauthor 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 meetsemilattice, $1$ 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-implicativesemilattice is an algebra $(H,\wedge,\rightarrow,0,1)$ of type $(2,2,0,0)$ suchthat $(H,\wedge,\rightarrow,1)$ is a hemi-implicative semilattice and $0$ isthe first element with respect to the order. A hemi-implicative lattice is analgebra $(H,\wedge,\vee,\rightarrow,0,1)$ 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 boundedhemi-implicative semilattices and hemi-implicative lattices, respectively,which is motivated by an old construction due J. Kalman that relates boundeddistributive lattices and Kleene algebras.

On Kalman's functor for bounded hemi-implicative semilattices and hemi-implicative lattices