The Endomorphism Kernel Property in Finite Distributive Lattices and de Morgan Algebras

Abstract

Abstract An algebra 𝒜 has the endomorphism kernel property if every congruence on 𝒜 different from the universal congruence is the kernel of an endomorphism on 𝒜. We first consider this property when 𝒜 is a finite distributive lattice, and show that it holds if and only if 𝒜 is a cartesian product of chains. We then consider the case where 𝒜 is an Ockham algebra, and describe in particular the structure of the finite de Morgan algebras that have this property. Key Words: Endomorphism kernelde Morgan algebraKleene algebra1991 Mathematics Subject Classification: 06D30 Acknowledgments The authors are indebted to Professor Brian Davey who, on reading an earlier version of this paper, made valuable suggestions which have acted as a catalyst in the evolution of Theorem 3. The second author expresses his gratitude to the Centro de Matemática e Aplicações, F.C.T., Universidade Nova de Lisboa where part of this research was carried out. Notes #Communicated by P. Higgins.