A Characterization of Principal Congruences of De Morgan Algebras and its Applications

Abstract

In this paper a characterization of principal congruences of De Morgan algebras is given and from it we derive that the variety of De Morgan algebras has DPC and CEP. The characterization is then applied to give a new proof of Kalman's characterization of subdirectly irreducibles in this variety and thus to obtain the representation theorem for DeMorgan algebras first proved by Kalman and independently, using topological methods, by Bialynicki-Birula and Rasiowa. From this representation it is deduced that finite De Morgan algebras are not the only ones with Boolean congruence lattices. Finally it is shown that the compact elements in the congruence lattice of a De Morgan algebra form a Boolean sublattice.