An ordered regular semigroup S is E-special if for every x is an element of S there is a biggest x(+) is an element of S such that both xx(+) and x(+)x are idempotent. Every regular strong Dubreil-Jacotin semigroup is E-special, as is every ordered completely simple semigroup with biggest inverses. In an E-special ordered regular semigroup S in which the unary operation x bar right arrow x(+) is antitone the subset P of perfect elements is a regular ideal, the biggest inverses in which form an inverse transversal of P if and only if S has a biggest idempotent. If S+ is a subsemigroup and S does not have a biggest idempotent, then P contains a copy of the crown bootlace semigroup.
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