Abstract
Let τ be a congruence on a full regular subsemigroup R of a regular semigroup S. The least congruence on S that contains τ is described. The description is in closed form modulo τ when R is self conjugate and τ extends to a congruence on S. Important congruences on S such as the least inverse and least orthodox congruences can be explicitly described by this approach in terms of extendable band congruences on the idempotent generated subsemigroup. Conditions for a congruence on R to extend to S are given. For the special case where S is E-solid, the least inverse congruence on S has an especially simple description.
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