Abstract. A semigroup $S$ is called $E$-($0$-)inversive if for every $a\in S(a\not=0)$ there exists $x\in S$ such that $(ax) ^2=ax(\not=0)$. (For example, every finite and every regular semigroup is $E$-inversive.) Imposing different restrictions on the ordering of the idempotents of such a semigroup $S$ the impact of these conditions on the structure of all of $S$ is investigated. First, those $E$-($0$-)inversive semigroups are characterized which admit a unique idempotent $(\not=0)$. Next, the case when there exists a least idempotent $(\not=0 )$ is described. Also primitive $E$-($0$-)inversive semigroups, that is, all of whose idempotents $(\not=0)$ are incomparable in the natural order, are dealt with. Finally, all $E$-($0$-)inversive semigroups whose (nonzero) idempotents form an $\omega $-chain, that is, which are ordered dually to the natural numbers, are characterized.

AMS Subject Classification (1991): 20M10

Received May 8, 2000, and in final form March 22, 2001. (Registered under 2804/2009.)