Vol. 43, No. 1, 1972

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Notes on related structures of a universal algebra

William A. Lampe

Vol. 43 (1972), No. 1, 189–205

The related structures of a universal algebra A that are studied here are the subalgebra lattice of A, the congruence lattice of A, the automorphism group of A, and the endomorphism semigroup of A. Characterizations of these structures known, and E. T. Schmidt proved the independence of the automorphism group and the subalgebra lattice. It has been conjectured that the first three of the structures listed above are independent, i.e., that the congruence lattice, subalgebra lattice, and automorphism group are independent. One result in this paper is a proof of a special case of this conjecture. Various observations concerning the relationship between the endomorphism semigroup and the congruence lattice are also in this paper. In the last section a problem of G. Grätzer is solved, namely that of characterizing the endomorphism semigroups of simple unary algebras. (An algebra is simple when the only congruences are the trivial ones.)

Mathematical Subject Classification
Primary: 08A25
Received: 10 June 1971
Published: 1 October 1972
William A. Lampe