Die Struktur kartesischer Produkte ganzer Zahlen modulo kartesische Produkte ganzer Zahlen.
Let be a Krull monoid with finite class group where every class contains some prime divisor. It is known that every set of lengths is an almost arithmetical multiprogression. We investigate which integers occur as differences of these progressions. In particular, we obtain upper bounds for the size of these differences. Then, we apply these results to show that, apart from one known exception, two elementary -groups have the same system of sets of lengths if and only if they are isomorphic.
Automorphic loops are loops in which all inner mappings are automorphisms. We study a generalization of the dihedral construction for groups. Namely, if is an abelian group, and , let be defined on by The resulting loop is automorphic if and only if or ( and is even). The case was introduced by Kinyon, Kunen, Phillips, and Vojtěchovský. We present several structural results about the automorphic dihedral loops in both cases.