On \(E_ k\)-semigroups (Q1091479)

Let S be a semigroup and E the set of idempotents of S. Let k be a positive integer. If \(S\neq E\), \(| E| \geq k\) and every nonidempotent subsemigroup of S, containing k idempotents, contains the whole E, then S is said to be an \(E_ k\)-semigroup. If \(S\neq E\) and \(| E| =k\) then S is obviously an \(E_ k\)-semigroup which is called trivial. The authors study \(E_ k\)-semigroups for \(k=1,2\). It is shown that there are no nontrivial \(E_ 1\)-semigroups. A characterization of nontrivial \(E_ 2\)-semigroups is given which is used for obtaining a stucture theorem for these semigroups.