Algebras with Boolean and Stonean congruence lattices. (English) Zbl 0618.08001
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.
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.
Reviewer: O.Frink
MSC:
| 08A30 | Subalgebras, congruence relations |
| 06B15 | Representation theory of lattices |
| 06C15 | Complemented lattices, orthocomplemented lattices and posets |
| 06B10 | Lattice ideals, congruence relations |
Keywords:
lattice of congruence relations; finitary algebra; Boolean algebra; Stone lattice; distributive pseudocomplemented lattice; strong center; complemented, distributive elements; atomic completely Stonean latticeReferences:
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