Let \(S\) be a semigroup. Let \(\leq\) be the standard partial order on the set \(E(S)\) of all idempotents of \(S\): \(e\leq f\Leftrightarrow ef=fe=e\). Let \(U\subseteq E(S)\) be such that for all \(a\in S\) there is a smallest \(e\in U\) (denoted by \(D(a)\)) for which \(ea=a\). In this case, the semigroup \(S\) is equipped with the unary operation \(D\colon a\mapsto D(a)\) and is called a \(D\)-semigroup. Properties of \(D\)-semigroups are studied and many examples of such semigroups are given. The author introduces notions of \(D\)-abundant, \(D\)-adequate \(D\)-semigroups, strong \(D\)-semigroups, involuted \(D\)-semigroups and so on, and studies them. The same is done for \(D\)-rings. A ring is called a \(D\)-ring if it is multiplicatively a \(D\)-semigroup. Important examples of \(D\)-rings are provided by Rickart \(*\)-rings.