On Kalman’s functor for bounded hemi-implicative semilattices and hemi-implicative lattices
Ramón Jansana, Hernán Javier San Martín · Logic Journal of IGPL · 2017
Hemi-implicative semilattices (lattices), originally defined under the name of weak implicative semilattices (lattices), were introduced by the second author of the present article. 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 article, 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.