Part II Invited Speech

In 2001, an open question that was posed by Blyth, Silva and Varlet in [2] is the following: for an Ockham algebra L determine the congruences on L that are kernels of endomorphisms on L. Whereas a general solution to this is stll an open problem, there has been some progress in investigating kernels of endomorphisms in various lattice-ordered algebras. In this connection, an algebra A is said to have the endomorphism kernel property if every congruence on A, other than the universal congruence, is a kernel of an endomorphism on A. A strengthening of this notion was given by Blyth and Silva [6] in the context of Ockham algebras. Specically, if A is an algebra and # is a congruence on A then an endomorphism e on A is said to be compatible with # if (8x; y 2 A) (x; y) 2 # ) (e(x); e(y)) 2 #: If such an endomorphism e is compatible with every congruence on A then it is said to be strong. Then A is said to have the strong endomorphism kernel property if every congruence on A, other than the universal congruence, is the kernel of a strong endomorphism on A. In [6], Blyth and Silva investigated Ockham algebras with the strong endomorphism kernel property by way of Priestley duality. They particularly showed that if L is an MS-algebra with dual space X that has the strong endomorphism kernel property then X can be characterized in terms of 1point compactications of discrete spaces. A similar approach adopted in some lattice-ordered algebras such as distributive p-algebras, distributive double p-algebras and double MS-algebras. By a direct algebraic way, the class of Heyting algebras and class of semilattices with this property also can be characterised. We recall that a Priestley space is a compact totally order-disconnected topological space. For a Priestley space X, we shall denote by O(X) the set of all clopen down-sets of X. Then O(X) is a distributive lattice. Conversely, if L is a distributive lattice, then (X; ; ) is a Priestley space, where X = Ip(L) is the lattice of prime ideals of L and the topology has as a base the sets fx 2 Ip(L) j x 3 ag and fx 2 Ip(L) j x 63 ag. These constructions give that L ' O(Ip(L)) and X ' � Ip(O(X) ; ; ). The power of duality theory is particular evident in the study of congruence relations. If L is a distributive lattice and X is the dual space of L then for every closed subset Q of X the relation Q dened on O(X) by (A;B) 2 Q () A \ Q = B \ Q is a congruence. Moreover, the lattice of congruences on L is dually