Abstract
Motivated by interior operators and by analogous generalisations of closure operators to semigroups in general, we define left, right and two-sided interior operations on a semigroup S in terms of a distinguished set of idempotents E using the natural order. In the left-sided case, we require that for all sin S, there is a largest ein E under the natural order such that es=e. We generalise them using suitable Green-like relations, and characterise the resulting classes of unary and biunary semigroups as quasivarieties or varieties.
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