The uniform dimension of a monoid with applications to graph algebras

TL;DR

This paper extends the concept of uniform dimension from module theory to $ ext{Gamma}$-monoids, applying it to graph algebras to analyze their structure and classify regular ideals, supporting the Graded Classification Conjecture.

Contribution

It introduces the uniform dimension for $ ext{Gamma}$-monoids, connects it to graph algebra structures, and relates regular ideals to graph properties and classification conjectures.

Findings

Uniform dimension measures graph branching complexity.

Orthogonality and regularity notions are developed for $ ext{Gamma}$-monoids.

Results unify studies on regular ideals in Leavitt path and graph $C^*$-algebras.

Abstract

We adapt Goldie's concept of uniform dimensions from module theory over rings to $\Gamma$-monoids. A $\Gamma$-monoid $M$ is said to have uniform dimension $n$ if $n$ is the largest number of pairwise incomparable nonzero $\Gamma$-order ideals contained in $M$. Specializing to the talented monoid of a graph, we show that the uniform dimension provides a rough measure of how the graph branches out. Since for any order ideal $I$, its orthogonal ideal $I^\perp$ is the largest ideal incomparable to $I$, we study the notions of orthogonality and regularity, particularly when $I^{\perp\perp}=I$. We show that the freeness of the action of $\mathbb Z$ on the talented monoid of a graph is preserved under quotienting by a regular ideal. Furthermore, we determine the underlying hereditary and saturated sets that generate these ideals. These results unify recent studies on regular ideals of the corresponding Leavitt path algebras and graph $C^*$-algebras. We conclude that for graphs $E$ and $F$, if there is a $\mathbb Z$-monoid isomorphism $T_E\cong T_F$, then there is a one-to-one correspondence between the regular ideals of the associated Leavitt path algebras $L_K(E)$ and $L_K(F)$ (and similarly, $C^*(E)$ and $C^*(F)$). Since the talented monoid $T_E$ is the positive cone of the graded Grothendieck group $K_0^{gr}(L_K(E))$, this provides further evidence supporting the Graded Classification Conjecture.