Abstract
An ordered regular semigroup S is E-special if for every x ∈ S there is a biggest x + ∈ 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 → 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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