An ordered semigroup $S$ is called a ${\it dual ordered semigroup}$ if $l(r(L)) = L$ for every left ideal $L$ of $S$ and
$r(l(R)) = R$ for every right ideal $R$ of $S$ where $r(A)$ and $l(A)$ denoted the ${\it right annihilator}$ and
the ${\it left annihilator}$ of a nonempty subset $A$ of $S$, respectively.
The main result of this paper is to show the existence of 0-minimal ideals of a dual ordered semigroup.
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