Semigroups with the ideal retraction property.
A semigroup \(S\) has the (ideal) retraction property (RP) if \(S\) has at least one proper ideal and if any ideal \(I\) of \(S\) is a retract of \(S\) (i.e., there exists a homomorphism \(f\colon S\to I\) whose restriction to \(I\) is the identity function). This property is studied for some classes of semigroups; for example: (1) A semigroup \(S\) with zero has RP if for each ideal \(I\) of \(S\) there is a dual ideal \(J\) in \(S\) (i.e., \(I\cap J=\{0\}\) and \(I\cup J=S\)). (2) A semigroup \(S\) with zero, for which each congruence is a Rees congruence, has RP if and only if for each ideal of \(S\) there exists a dual ideal in \(S\) (that is, the congruence lattice of \(S\) is complemented). (3) A cyclic semigroup \(S\) has RP if and only if some generator of \(S\) has index two. Examples are given which show that RP is neither preserved by subsemigroups nor by direct products. Some classes of semigroups are indicated, for which RP is preserved by homomorphisms. Remark: A construction of all semigroups having the retraction property (in particular, finite or commutative) was given by \textit{E. J. Tully} jun. [J. Aust. Math. Soc. 9, 239--245 (1969; Zbl 0185.04802)].
scientific article; zbMATH DE number 1972503