Topological Duality for Boolean Algebras with a Normal n-ary Monotonic Operator

Abstract

In this paper we shall give a topological duality for Boolean algebras endowed with an n-ary monotonic operator (BAMOs). The dual spaces of BAMOs are structures of the form $\left\langle X,R,\tau\right\rangle$, such that $\left\langle X,\tau\right\rangle$ is a Boolean space, and R is a relation between X and a finite sequences of non-empty closed subsets of X. By means of this duality we shall characterize the equivalence relations of the dual space of a BAMO A that correspond biunivocally to subalgebras of A. We shall prove that there exist bijective correspondences between the lattice of congruences, the lattice of closed filters, and the lattice of certain closed subsets of the dual space of a BAMO. These correspondences are used to study the simple and the subdirectly irreducible algebras.