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.

\r\n\r\nIn 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.

\r\n\r\nChapter 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:

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

\r\n\r\n(i) ascending chain condition in S

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

\r\n\r\n(iii) modular condition in S.

\r\n\r\nChapter 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.

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

", "date": "1939", "date_type": "degree", "id_number": "CaltechETD:etd-05222003-111959", "refereed": "FALSE", "official_url": "https://resolver.caltech.edu/CaltechETD:etd-05222003-111959", "rights": "No commercial reproduction, distribution, display or performance rights in this work are provided.", "collection": "CaltechTHESIS", "reviewer": "Kathy Johnson", "deposited_by": "Imported from ETD-db", "deposited_on": "2003-05-22", "doi": "10.7907/3JPC-3K49", "divisions": { "items": [ "div_pma" ] }, "institution": "California Institute of Technology", "thesis_type": "phd", "thesis_advisor": { "items": [ { "id": "Ward-M", "name": { "family": "Ward", "given": "Morgan" }, "role": "advisor" } ] }, "thesis_committee": { "items": [ { "name": { "family": "Unknown", "given": "Unknown" } } ] }, "thesis_degree": "PHD", "thesis_degree_grantor": "California Institute of Technology", "thesis_submitted_date": "2003-05-22", "thesis_defense_date": "1939-01-01", "thesis_approved_date": "2003-05-22", "review_status": "approved", "option_major": { "items": [ "math" ] }, "copyright_statement": "I hereby certify that, if appropriate, I have obtained a written permission statement from the owner(s) of each third party copyrighted matter to be included in my thesis, dissertation, or project report, allowing distribution as specified below. I certify that the version I submitted is the same as that approved by my advisory committee.\n\nI hereby grant to California Institute of Technology or its agents the non-exclusive\nlicense to archive and make accessible, under the conditions specified below,\nmy thesis, dissertation, or project report in whole or in part in all forms of media, now or hereafter known. I retain all other ownership rights to the copyright of the thesis, dissertation or project report. I also retain the right to use in future works (such as articles or books) all or part of this thesis, dissertation, or project report.", "resource_type": "thesis", "pub_year": "1939", "author_list": "Dilworth, Robert Palmer" } ]