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.)
