Semilattice congruences viewed from quasi-orders
Takayuki Tamura · Proceedings of the American Mathematical Society · 1973
Let $S$ be a semigroup. This paper proves that the smallest semilattice congruence ${\rho _0}$ containing a compatible binary relation $\xi$ on $S$ equals the natural equivalence of the smallest lower-potent positive quasi-order ${\sigma _0}$ containing $\xi$.