Archimedan Prüfer Structures Alias Filter Archimedean d-Semigroups

Abstract

Inspired by the monograph of {Larsen/McCarthy} the author\break started a series of articles concerning abstract multiplicative ideal theory along the lines of [23]. In the present paper we turn to archimedean Pr\"ufer structures, that is to algebraic \(m\)-lattices satisfying the two implications \(\left( P \right)a_1 + ... + a_n \supseteqq B \Rightarrow a_1 + ... + a_n |B\) and \(\left( A \right)A^n \supseteqq B\left( {\forall n \in N} \right) \Rightarrow AB = B = BA\) . Since these properties imply commutativity we start from commutative structures without real loss of generality and give characterizations of arbitrary and also of special Archimedean Pr\"ufer structures.