\(E\)-minimal semigroups (Q1343221)

The concept of an \(E\)-minimal universal algebra comes from the tame congruence theory. The \(E\)-minimal modules, groups, rings, and loops have all been characterized. These classes of algebras are congruence modular. A finite semigroup \(S\) is said to be \(E\)-minimal if the only idempotents of the transformation semigroup \(\text{Pol}_ 1 S\) are the constant functions and the identity function. The author gives a complete characterization of \(E\)-minimal semigroups. There are four classes of \(E\)-minimal semigroups: \(p\)-groups, left (right) zero semigroups, nilpotent semigroups, and the two element semilattice. Consequently, there are \(E\)-minimal semigroups which do not generate congruence modular varieties.