On congruence lattices
AbstractInvestigations of the lattice of congruences on a semigroup have taken two different directions. One approach is to study special congruences on a semigroup, and describe their relative positions within the lattice of congruences. For some classes C1 and C2, it will happen that the intersection σ is, of course, the minimum C1 congruence on S, and S/σ is a maximal homomorphic image of S of type C1. For instance, it is easily seen that the intersection of all commutative congruences on any semigroup is a commutative congruence, and so every semigroup has a minimum commutative congruence. Similarly, every semigroup has a minimum band congruence (denoted β) and a minimum semilattice congruence (denoted η). We outline some results dealing with the lattice of congruences of a semigroup. It is clear that a modular lattices is a semimodular, but the converse, however, is not true.
Journal of the Nigerian Association of Mathematical Physics, Volume 15 (November, 2009), pp 281 - 286