Gary D. Crown

Publication Date

9-11-1967

Abstract

Croisot [1] was apparently first to examine some of the properties of residuated mappings. It is the purpose of this paper to show that certain constructions currently useful in homological algebra can be made in a suitable category of partially ordered sets with residuated mappings. In Section 1 we note some elementary properties of residuated mappings all of which are well known cf. Janowitz [4]. In Section 2 we notice some properties of various categories of partially ordered sets with residuated mappings. In Section 3 we prove that the category of complete lattices with residuated mappings has enough injectives and projectives and we obtain a characterization of these. In Section 4 we prove that has enough limits and colimits. In Section 5 we define the functors (-, M) and (M, -) and we observe that (M,-) preserves limits while (-,M) carries colimits to limits. In Section 6 we define exact sequences in and we observe that (M,-) and (-, M) are left-exact functors. The following is a list of basic definitions. If the reader is familiar with Mitchell [5] he may proceed to Section 1.

Degree Name

Mathematics

Level of Degree

Doctoral

Department Name

Mathematics & Statistics

First Committee Member (Chair)

Richard Clyde Metzler

Second Committee Member

Julius Rubin Blum

Third Committee Member

Lambert Herman Koopmans

Language

English

Document Type

Dissertation

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