The lattice of algebraic closure operators

TL;DR

This paper explores the structure of algebraic closure operators on complete algebraic lattices, demonstrating that the set of such operators itself forms a complete algebraic lattice, extending the concept from finite to infinite contexts.

Contribution

It generalizes the notion of algebraic closure operators from finite sets to complete algebraic lattices, proving the set of these operators forms a complete algebraic lattice.

Findings

The set of algebraic closure operators forms a complete lattice.

This lattice is itself algebraic, extending finite set concepts to infinite lattices.

Provides a framework for understanding closure operators in more general algebraic structures.

Abstract

On an infinite set some closure operators are finitary (algebraic) while others are not. We can generalize this idea for a complete algebraic lattice letting the compact elements act as the finite sets. With this in mind, we will consider the set of algebraic closure operators on such a lattice. We will show this set forms a complete lattice that is also an algebraic lattice.