A note on the congruence lattice of a finitely generated algebra

The study of finitely generated algebras is important to the theory of algebraic systems. Problems concerning free algebras and varieties of algebras often turn on this theme. At the same time properties of the lattice of congruence relations of an algebra often bear immediate consequences for the structure theory of an algebraic system. Let 9t be an algebra, that is, a pair , where A is a set and F is a family of operations on A. We say that 9t is of finite type if I FI is finite. Let Con(W) denote the lattice of all congruence relations of 9t partially ordered by set inclusion. For elements x and y of A let 0 (x, y) denote the principal congruence relation generated by identifying x and y; that is, 9 (x, y) = 0OIO E Con(%) and x _ y(9)). For a general reference to terminology we refer the reader to [1] or [3].