Algebras with Boolean and Stonean congruence lattices (Q1089019)

There is a large literature on when the lattice con(A) of congruence relations on a finitary algebra A is atomic, a Boolean algebra, or a Stone lattice. The lattice con(A) is always algebraic, that is, complete and compactly generated. The authors consider cases where it is also distributive and hence pseudocomplemented. A Stone lattice is a distributive pseudocomplemented lattice satisfying the identity \((a\wedge b)^*=a^*\vee b^*.\) In 11 theorems the authors extend many of the known results on con(A) to larger classes of algebras. Their chief new concept is that of an algebra with a strong center. The center C(L) of a lattice L consists of the complemented, distributive elements of L, and is a Boolean algebra. An algebra A which is a lattice has a strong center if all the operations of A are center-preserving. They give various conditions on an algebra A that con(A) be atomic, a Boolean algebra, or a Stone lattice. An example of the authors' theorems is: If con(A) is distributive, then it is a Boolean lattice if and only if (i) A has a subdirect factorization with simple factors, and (ii) con(A) is an atomic completely Stonean lattice.