Using filters to describe congruences and band congruences of semigroups

It is well known that the smallest semilattice congruence can be described via filters. We generalise this result to the smallest left (right) normal band congruences and also to arbitrary semilattice (left normal band, right normal band) congruences, describing them all via filters. To achieve this, we introduce filters relative to arbitrary quasiorders on a semigroup (traditional filters are filters relative to the smallest negative operation-compatible quasiorder). We study congruences which can be described via filters. We show that the lattice of semilattice (left normal band, right normal band) congruences is a homomorphic image of the lattice of negative (right negative, left negative) operation-compatible quasiorders. © 2011 Springer Science+Business Media, LLC.