Ideals of quasi-absorbing elements in semigroups

Motivated by situations in which the removal of a zero (a.k.a., an absorbing element) from a semigroup yields a subsemigroup with another zero, sets of quasi-zeros (a.k.a., quasi-absorbing elements) are introduced as well as primitive elements, minimal ideals and simple semigroups all with respect to a given ideal. An application to set optimization is discussed.