On the \(E\)-disjunctive inverse semigroups (Q1281270)

Let \(S\) be an inverse semigroup. The symbol \(E(S)\) denotes the set of all idempotents in \(S\). A congruence \(\rho\) on \(S\) is called an idempotent pure congruence if for any \(a\in S\), \(e\in E(S)\), \(\langle a,e\rangle\in\rho\) implies \(a\in E(S)\). If the identity congruence \(\{\langle s,s\rangle;\;s\in S\}\) is the only idempotent pure congruence on \(S\), then \(S\) is called an \(E\)-disjunctive inverse semigroup. \(E\)-disjunctive inverse semigroups were already studied by M. Petrich in 1984. The present paper gives some new characterizations of \(E\)-disjunctive inverse semigroups.