Abstract:
We provide some examples of irregular fully idempotent homomorphisms and study the pairs of abelian groups $A$ and $B$ for which the homomorphism group $\operatorname{Hom}(A,B)$ is fully idempotent. We show that if $B$ is a torsion group or a mixed split group and if at least one of the groups $A$ or $B$ is divisible then the full idempotence of the homomorphism group implies its regularity. If at least one of the groups $A$ or $B$ is a reduced torsion-free group and their homomorphism groups are nonzero then the group is not fully idempotent. The study of fully idempotent groups $\operatorname{Hom}(A,A)$ comes down to reduced mixed groups $A$ with dense elementary torsion part.
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