The Structure and Arithmetical Theory of Non-Commutative Residuated Lattices

Citation

Dilworth, Robert Palmer (1939) The Structure and Arithmetical Theory of Non-Commutative Residuated Lattices. Dissertation (Ph.D.), California Institute of Technology. doi:10.7907/3JPC-3K49. https://resolver.caltech.edu/CaltechETD:etd-05222003-111959

Abstract

A survey of the field of non-commutative algebra and arithmetic indicates that a great many of the results are simply statements concerning a lattice the elements of which combine under an additional operation of multiplication. This fact suggests that the investigation of the algebraic and arithmetical properties of lattices with a non-commutative multiplication should simplify and correlate to a large extent a number of the important fields of modern algebra. Such an investigation is the subject of this thesis.

In chapter I the formal properties of lattices with a non-commutative multiplication and its associated residuation are treated in detail. It is shown that under certain general conditions each of these operations may be defined in terms of the other. This fact gives an easy method of extending the domain of definition of the operations and the properties of such extensions are discussed. Multiplications and residuations which are unchanged by the extension have particularly simple properties and are discussed in considerable detail. Finally, various special multiplications and residuations are investigated.

Chapter II treats lattices in which the multiplication is intimately connected with lattice division. It is shown that each element of a modular lattice in which suitable chain conditions hold, may be represented as a product of irreducibles; and if there are two such decompositions, the number of irreducibles is the same and they are similar in pairs. Applying these results to non-commutative semi-groups gives the following fundamental theorem:

The following three conditions are necessary and sufficient that a semi-group S with G.C.D. and L.C.M. operations have an arithmetic:

(i) ascending chain condition in S

(ii) descending chain condition for the factors of any element of S

(iii) modular condition in S.

Chapter III has three main divisions. In the first part the structure of ideal lattices in the vicinity of the unit element is characterized in terms of arithmetical and semi-arithmetical lattices. In the second division decompositions into primary and semi-primary elements are discussed. And finally in the third part, the structure of Archimedean residuated lattices is investigated. In particular structure theorems are proved which are analogous to the structure theorems of hypercomplex systems.

This investigation was undertaken at the suggestion of Professor Morgan Ward to whom I am indebted for constant encouragement and many timely suggestions.

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