MATHEMATICAL LOGIC IN LATIN AMERICA A.I. Arruda, R. Chuaqui, N.C.A. da Costa (eds.) © NQrth-Holland Publishing Company, 1980 A CHARACTERIZATION OF PRINCIPAL CONGRUENCES OF DE MORGAN ALGEBRAS AND ITS APPLICATIONS H.P. Sankappanavall. ABSTRACT. In this paper a characterization of prin- cipal congruences of De Morgan algebras is given and from it we derive that the variety of De Morgan alge- bras has DPC and CEP. The characterization is then appl ied to give a new proof of Kalman's characteriza- tion of subdirectly irreducibles in this variety and thus to obtain the representation theorem for De Morgan algebras first proved by Kalman and independently, usingtopological methods, by 8ialynicki- Bi rula and Rasiowa. From this representation it is deduced that finite De Morgan algebras are not the only ones wi th Boolean congruence lattices. Finally it is shown that the compact elements in the congruence lattice of a De Morgan algebra form a Boolean sublattice. §1. INTRODUCTION, Principal congruences have turned out to be a useful tool, especially after the publication of Day 1971, in obtaining some interesting information about a given variety (equational class) of algebras. For example, the "equational" char- acterization of principal congruences of distributive lattices - first given by Gratzer and Schmidt (see Gratzer 1971, Theorem 3, p. 87) - has been used, among other things, to characterize subdirect irreducibility (Example 3 in Gratzer 1971); and it also implies that the variety of distributive lattices has definable prin- cipal congruences(DPC) (see § 4 below for definition) and has congruence extension property (CEP). (This is immediate via Day 1971.) Lasker 1973 found a similar characterization in the case of distributive pseudo-complemented lattices and used it to deduce CEP and OPC. Recently we observed in Sankappanavar 197+ b that his characterization could also be used to give a very simple proof of the characteri- zation by Lakser (see Gratzer 1971, Theorem 7, p. 178) of subdirectly irreducible distributive pseudocomplemented lattices. To cite still another example, princi- ple congruences of pseudo-complemented semilattices are characterized in Sankappa- navar 197+ a and DPC and CEP are deduced; moreover, this characterization is applied in Sankappanavar 197+ b to give a new direct proof of a result, due 'to Jones, characterizing subdirectly irreducible ones. In this paper a characterization of principal of De Morgan alge- bras is given. From this it follows that the variety of De Morgan algebras has DPe and CEP. This characterization is then applied to give a new proof of Kal- man's characterization of subdirectly irreducibles in this variety and thus to ob- tain the representation theorem for De Morgan algebras first proved by Kalman and later, using topological methods, by Bialynicki-Birula and Rasiowa. From this it 341