The Belluce lattice associated with a bounded BCK -algebra


Faculty of Sciences, Department of Mathematics, University of Craiova, Craiova, Romania


In this paper, we introduce the notions of Belluce lattice associated with a bounded $BCK$-algebra and reticulation of a bounded $BCK$-algebra. To do this, first, we define the operations  $\curlywedge ,$ $\curlyvee $ and $\sqcup $ on $BCK$-algebras and we study some algebraic properties of them. Also, for a bounded $BCK$-algebra $A$ we define the Zariski topology on $\ Spec(A)$ and the induced topology $\tau _{A,Max(A)}$ on $Max(A)$. We prove $(Max(A),\tau_{A,Max(A)})$ is a compact topological space if $A$ has Glivenko property. Using the open and the closed sets of $Max(A)$, we define a congruence relation on a bounded $BCK$-algebra $A$ and we show $L_{A}$, the quotient set, is a bounded distributive lattice. We call this lattice the Belluce lattice associated with $A.$ Finally, we show $(L_{A},p_{A})$ is a reticulation of $A$ (in the sense of Definition \ref{d7}) and the lattices $L_{A}$ and $S_{A}$ are isomorphic.