Congruences on regular semigroups

Abstract

Let S S be a regular semigroup and let ρ \rho be a congruence relation on S S . The kernel of ρ \rho , in notation ker ⁡ ρ \ker \rho , is the union of the idempotent ρ \rho -classes. The trace of ρ \rho , in notation tr ρ \operatorname {tr}\,\rho , is the restriction of ρ \rho to the set of idempotents of S S . The pair ( ker ⁡ ρ , tr ρ ) (\ker \rho ,\operatorname {tr}\,\rho ) is said to be the congruence pair associated with ρ \rho . Congruence pairs can be characterized abstractly, and it turns out that a congruence is uniquely determined by its associated congruence pair. The triple ( ( ρ ∨ L ) / L , ker ⁡ ρ , ( ρ ∨ R ) / R ) ((\rho \vee \mathcal {L})/\mathcal {L},\ker \rho ,(\rho \vee \mathcal {R})/\mathcal {R}) is said to be the congruence triple associated with ρ \rho . Congruence triples can be characterized abstractly and again a congruence relation is uniquely determined by its associated triple. The consideration of the parameters which appear in the above-mentioned representations of congruence relations gives insight into the structure of the congruence lattice of S S . For congruence relations ρ \rho and θ \theta , put ρ T l θ [ ρ T r θ , ρ T θ ] \rho {T_l}\theta \;[\rho {T_r}\theta ,\rho T\theta ] if and only if ρ ∨ L = θ ∨ L [ ρ ∨ R = θ ∨ R , tr ⁡ ρ = tr ⁡ θ ] \rho \vee \mathcal {L} = \theta \vee \mathcal {L}\;[\rho \vee \mathcal {R} = \theta \vee \mathcal {R},\operatorname {tr}\rho = \operatorname {tr}\theta ] . Then T l , T r {T_l},{T_r} and T T are complete congruences on the congruence lattice of S S and T = T l ∩ T r T = {T_l} \cap {T_r} .