Regular semigroups with a left ideal inverse transversal (Q1964678)

Let \(S\) be a regular semigroup. An inverse subsemigroup \(S^0\) of \(S\) which contains a unique inverse \(x^0\) for each element \(x\in S\) is called an inverse transversal of \(S\). \(S^0\) is a quasi-ideal inverse transversal when \(S^0\) is a quasi-ideal (i.e. \(S^0SS^0\subseteq S^0\)). Similarly, if \(S^0\) is a left ideal (i.e. \(SS^0\subseteq S^0\)), then \(S^0\) is called a left ideal inverse transversal. The author describes the smallest inverse semigroup congruence, the minimum group congruence, and the maximum idempotent-separating congruence on a regular semigroup \(S\) with a left ideal inverse transversal \(S^0\).