Non-commutative Residuated Lattices*

Introduction and summary. In the theory of non-commutative rings certain distinguished subrings, one-sided and two-sided ideals, play the important roles. Ideals combine under crosscut, union and multiplication and hence are an instance of a lattice over which a non-commutative multiplication is defined.f The investigation of such lattices was begun by W. Krull (Krull [3]) who discussed decomposition into isolated component ideals. Our aim in this paper differs from that of Krull in that we shall be particularly interested in the lattice structure of these domains although certain related arithmetical questions are discussed. In Part I the properties of non-commutative multiplication and residuation over a lattice are developed. In particular it is shown that under certain general conditions each operation may be defined in terms of the other. The second division of the paper deals with the structure of non-commutative residuated lattices in the vicinity of the unit element. It is found that this structure may be characterized to a large extent in terms of special types of distributive lattices (arithmetical and semi-arithmetical lattices). The next division contains a discussion of the arithmetical properties of noncommutative residuated lattices. In particular decompositions into primary and semi-primary elements are discussed. Finally we investigate the case where both the ascending and descending chain conditions hold and prove some structure theorems which are analogous to the structure theorems of hypercomplex systems.