Congruences on ample semigroups
Abdulsalam El-Qallali · Semigroup Forum · 2019
Congruences on an ample semigroup S are investigated. For any admissible congruence \( \rho \) on S , the minimum \(\sigma _{\rho }\) (the maximum \( \mu _{\rho }\) ) admissible congruence on S whose restriction to E ( S ) is the trace of \(\rho \) is obtained. An expression for a congruence which is contained in (contains) any admissible congruence of the same kernel is given. The concept of congruence pairs for S is introduced. It is shown that for any admissible congruence \(\rho \) of S ; ( \({{\,\mathrm{tr}\,}}\rho , \ker \rho \) ) is a congruence pair for S . Conversely, the congruence of S which contains (is contained in) any admissible congruence associated with a given congruence pair for S is formulated.