2017/08/30 by Jansana, Ramon, Martín, Hernán Javier San · 1 citation
#FOS: Mathematics #Logic (math.LO)
paper · doi:10.48550/arxiv.1708.09490
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,→,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→ a = 1 for every a∈ H and for every a, b, c∈ H, if a≤ b→ c then a\wedge b ≤ c. A bounded hemi-implicative semilattice is an algebra (H,\wedge,→,0,1) of type (2,2,0,0) such that (H,\wedge,→,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,→,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,→,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.