How to extend closure and interior operations to more modules

TL;DR

This paper axiomatizes three methods to extend closure and interior operations to more modules, analyzing their properties and interactions with other desirable features like heredity and residuality.

Contribution

It introduces a unified axiomatic framework for extending closure and interior operations, incorporating examples like tight closure and integral closure.

Findings

Identifies three axiomatic methods for operation extension

Analyzes properties like hereditary, residual, and cofunctorial

Explores interactions with finitistic property

Abstract

There are several ways to convert a closure or interior operation to a different operation that has particular desirable properties. In this paper, we axiomatize 3 ways to do so, drawing on disparate examples from the literature, including tight closure, basically full closure, and various versions of integral closure. In doing so, we explore several such desirable properties, including *hereditary*, *residual*, and *cofunctorial*, and see how they interact with other properties such as the *finitistic* property.