Lattices of blocks of regular algebras and solvable groups (Q804702)

Let A be a universal algebra. B is called a block of A if and only if B is an equivalence class of \(\theta\) for some congruence \(\theta\) on A. The set of all blocks with partial ordering \(\subseteq\) forms a lattice which is denoted by C(A). An algebra A is called regular if different congruences on A have no common equivalence classes. The author studies the lattice C(A). He gives a test of simplicity of the lattice C(A). He proves that if A is a regular algebra of order \(\geq 3\) then the lattice C(A) is either simple or contains a unique non-trivial congruence. The test of simplicity is applied for the study of lattices of blocks of solvable groups. The author proves for a solvable group G of order \(\geq 3\) that the lattice C(G) has a non-trivial congruence if and only if the factor group \(G/G'\) \((G'\) is the commutator subgroup of G) is a quasi- cyclic group. There exists a metabelian group G such that C(G) has a non- trivial congruence. For a nilpotent group G the author shows that if C(G) is not a simple lattice then G is an Abelian group.