Publication Date

Spring 4-15-2024

Abstract

For any closure operation $\cl$ and interior operation $\ri$ on a class of $R$-modules, we develop the theory of $\cl$-prereductions and $\ri$-postexpansions. A pair operation is a generalization of closure and interior operations. Using Epstein, R.G. and Vassilev's duality \cite{ERGV-nonres}, we show that these notions are in fact dual to each other. We discuss the relationship between the core and hull and prereductions and postexpansions. We further the thematic notion of duality and seek to understand how it arises in the context of properties pair operations can be endowed with and focus on inner product spaces and properties demonstrated by the orthogonal complement. Finally, constructions of pair operations through ring extensions and collections will be explored with relation to how the new operation can preserve certain properties.

Degree Name

Mathematics

Level of Degree

Doctoral

Department Name

Mathematics & Statistics

First Committee Member (Chair)

Janet Vassilev

Second Committee Member

Alexandru Buium

Third Committee Member

Shuai Wei

Fourth Committee Member

Rebecca RG

Language

English

Keywords

algebra, commutative algebra, pair operations, closure operations

Document Type

Dissertation

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Algebra Commons

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