Congruences on regular semigroups
Francis J. Pastijn, Mario Petrich · Transactions of the American Mathematical Society · 1986
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