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

HERNÁN JAVIER SAN MARTÍN; RAMON JANSANA

LOGIC JOURNAL OF THE IGPL (PRINT)

OXFORD UNIV PRESS

Lugar: Oxford; Año: 2018 vol. 26 p. 47 - 82

1367-0751

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 $(H,wedge,ightarrow,1)$ of type $(2,2,0)$ such that $(H,wedge)$ is a meet semilattice, $1$ is the greatest element with respect to the order, $aightarrow a = 1$ for every $ain H$ and for every $a$, $b$, $cin H$, if $aleq bightarrow c$ then $awedge b leq c$. A bounded hemi-implicative semilattice is an algebra $(H,wedge,ightarrow,0,1)$ of type $(2,2,0,0)$ such that $(H,wedge,ightarrow,1)$ is a hemi-implicative semilattice and $0$ is thefirst element with respect to the order. A hemi-implicative lattice is an algebra $(H,wedge,ee,ightarrow,0,1)$ of type $(2,2,2,0,0)$ such that $(H,wedge,ee,0,1)$ is a bounded distributive lattice and the reduct algebra $(H,wedge,ightarrow,1)$ 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.