Using filters to describe congruences and band congruences of semigroups. (Q766193)

It is well known that the smallest semilattice congruence of a semigroup can be described via filters. The authors generalize 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, they introduce filters relative to arbitrary quasiorders on a semigroup (traditional filters are filters relative to the smallest negative operation-compatible quasiorder). Congruences which can be described via filters are studied. It is shown 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.