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# Lattices of ideals in graph algebras: refining the join operation

thesis

posted on 2022-04-29, 04:12 authored by McDougall, PhoebeWe consider the ideal structure of Leavitt path algebras of graphs that satisfy Condition (K) over commutative rings with identity. The approach we use takes advantage of the relationship between Leavitt path algebras and Steinberg algebras of boundary path groupoids. We establish a lattice $\mathcal{F}'$ consisting of particular maps $\tau:E^0\to \LR$ and show that this lattice is isomorphic to the lattice of ideals of $L_R(E)$. The advantage to our approach over previous lattice isomorphisms, even in the case that $R$ is a field, is that we obtain convenient join and meet operations in $\mathcal{F}'$. Lastly, we provide three concrete examples.

## History

## Copyright Date

2022-04-29## Date of Award

2022-04-29## Publisher

Te Herenga Waka—Victoria University of Wellington## Rights License

CC BY-NC-SA 4.0## Degree Discipline

Mathematics## Degree Grantor

Te Herenga Waka—Victoria University of Wellington## Degree Level

Masters## Degree Name

Master of Science## ANZSRC Type Of Activity code

1 Pure basic research## Victoria University of Wellington Item Type

Awarded Research Masters Thesis## Language

en_NZ## Victoria University of Wellington School

School of Mathematics and Statistics## Advisors

Clark, Lisa## Usage metrics

## Keywords

Leavitt path algebras over arbitrary ringsIdealsLatticeSchool: School of Mathematics and Statistics010101 Algebra and Number Theory010107 Mathematical Logic, Set Theory, Lattices and Universal AlgebraDegree Discipline: MathematicsDegree Level: MastersDegree Name: Master of ScienceMathematical Logic, Set Theory, Lattices and Universal AlgebraAlgebra and Number Theory

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