The Belluce lattice associated with a bounded BCK -algebra

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 ⋏,⋎ and ⊔ 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 τA,Max(A) on M ax(A). We prove (M ax(A), τA,Max(A) ) is a compact topological space if A has Glivenko property. Using the open and the closed sets of M ax(A), we define a congruence relation on a bounded BCK-algebra A and we show LA, the quotient set, is a bounded distributive lattice. We call this lattice the Belluce lattice associated with A. Finally, we show (LA, pA) is a reticulation of A (in the sense of Definition 5.1) and the lattices LA and SA are isomorphic.