ACTAS DEL VIII CONGRESO DR. ANTONIO A. R. MONTEIRO (2005), Páginas 25–32 SOME REMARKS ON OCKHAM CONGRUENCES LEONARDO CABRER AND SERGIO CELANI ABSTRACT. In this work we shall describe the lattice of congruences of an Ockham alge- bra whose quotient algebras are in the Urquhart clases P m,n . This description is obtained using the Duality for Ockham algebras given by Urquhart (see [3]). This work is a natural generalization for some of the results obtained by Rodriguez and Silva in [5]. 1. PRELIMINARIES In [5] Rodriguez and Silva describe the lattice of congruences of an Ockham algebra whose quotient algebras are Boolean. Given an Ockham algebra, they characterize them in two different ways, one by means of pro-boolean ideals and the other using the set of fixed points of the dual space. Here we will give a generalization of this results describing the lattice of congruences whose quotient algebras belong to the subvarieties of Ockham algebras defined by Urquhart (see [3]). We will see that this congruences do not admit a description by means of ideals, but they can be described by means of some subsets of the dual space. In this section we will recall the definitions, results and notations that will be needed in the rest of the paper. In section 2 we will introduce the set Con m,n (O) for every Ockham algebra and develop the main results of this paper. Given 〈X , ≤〉 a poset, we will say that a subset Y ⊆ X is increasing if for every y ∈ Y and for every x ∈ X such that y ≤ x, then x ∈ Y . A map g : X −→ X is an order reversing map if for every x, y ∈ X such that x ≤ y , g (y) ≤ g (x). If X is a set and Y ⊆ X , when there is no risk of misunderstanding, we will note Y c = X \ Y . Given a lattice L we will note the set of atoms of L by At (L), and with CoAt (L) the set of co-atoms of L. Definition 1. An algebra O = 〈O, ∧, ∨, f , 0, 1〉 of type (2, 2, 1, 0, 0) is an Ockham algebra if it verifies the following conditions: O1 〈O, ∧, ∨, 0, 1〉 is a bounded distributive lattice. O2 f (0)= 1, f (1)= 0. O3 f (a ∧ b) ≈ f (a) ∨ f (b). O4 f (a ∨ b) ≈ f (a) ∧ f (b). For the rest of the paper O = 〈O, ∧, ∨, f , 0, 1〉 will be an arbitrary Ockham algebra. 25