Publication Date:
2018-03-06

Description:
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.

Print ISSN:
1367-0751

Electronic ISSN:
1368-9894

Topics:
Mathematics

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