Graded Invariant Basis Number (grIBN)

• Graded Invariant Basis Number (grIBN) is a property for ℤ-graded rings that requires graded free modules to be isomorphic only when they have the same number of shifted summands.
• The talented monoid TU_E translates the graded module problem into combinatorial graph relations by linking graded vertex relations with module isomorphism conditions.
• A matrix-theoretic characterization of grIBN failure arises via unequal power sum tuples of the adjacency matrix, offering a concrete combinatorial test for the property.

Searching arXiv for the specified paper and closely related background to ground the article in published work. arxiv_search(query="(Phuc, 26 Mar 2026)", max_results=5, sort_by="submittedDate") arxiv_search(query="Leavitt path algebras having Graded Invariant Basis Number", max_results=10, sort_by="relevance") Graded Invariant Basis Number, abbreviated gr-IBN, is a basis-number property for unital $\mathbb Z$-graded rings that tests whether graded free modules can be isomorphic only when they have the same number of shifted summands. For a finite row-finite directed graph $E$, the Leavitt path algebra $L_K(E)$ carries its standard $\mathbb Z$-grading, and the gr-IBN problem asks when an isomorphism between finite direct sums of shifts of $L_K(E)$ forces equality of the numbers of summands. In the setting of finite graphs, the central result is a complete matrix-theoretic characterization of failure of gr-IBN, obtained through the talented monoid and its identification with the monoid of graded finitely generated projective $L_K(E)$-modules (Phuc, 26 Mar 2026).

1. Definition in the graded module category

Let $R=\bigoplus_{n\in\mathbb Z}R_n$ be a unital $\mathbb Z$-graded ring. For each shift $\alpha\in\mathbb Z$, the graded right module $R(\alpha)$ is defined by

$E$0

and a graded module isomorphism is required to respect degrees.

The ring $E$1 has the graded Invariant Basis Number property if, whenever

$E$2

as graded right $E$3-modules, it follows that $E$4. Equivalently,

$E$5

For Leavitt path algebras, this definition refines the ordinary IBN question by retaining the grading shifts as part of the module structure. The formulation is specifically adapted to the natural $E$6-grading of $E$7, where shifts of the regular module are fundamental graded free modules. In this setting, gr-IBN is a structural constraint on the graded module category rather than merely on ungraded free modules (Phuc, 26 Mar 2026).

2. The talented monoid and the covering graph

The principal tool is the talented monoid $E$8. In the ungraded setting one studies $E$9, the monoid of isomorphism classes of finitely generated projective modules. In the graded setting, this is replaced by $L_K(E)$0, the monoid of graded finitely generated projectives together with shifts.

For a row-finite graph $L_K(E)$1, the talented monoid $L_K(E)$2 is the commutative monoid with generators

$L_K(E)$3

subject to the graded vertex relations

$L_K(E)$4

There is a natural $L_K(E)$5-action by shifts, given by $L_K(E)$6.

The covering graph $L_K(E)$7 has vertex set $L_K(E)$8 and edge set $L_K(E)$9, where

$\mathbb Z$0

Its usual graph monoid $\mathbb Z$1 is naturally isomorphic, as a $\mathbb Z$2-monoid, to $\mathbb Z$3.

The key proposition is the canonical $\mathbb Z$4-equivariant isomorphism

$\mathbb Z$5

Consequently, graded module isomorphisms between finite direct sums of shifts of $\mathbb Z$6 become equalities in $\mathbb Z$7. This reduction from graded module theory to monoid computation is the decisive mechanism for testing gr-IBN. It means that the existence or failure of gr-IBN can be analyzed combinatorially through the graph and algebraically through the associated monoid relations (Phuc, 26 Mar 2026).

3. Matrix-theoretic characterization of failure

When $\mathbb Z$8 is a finite graph with no sinks, with $\mathbb Z$9, let $L_K(E)$0 be its adjacency matrix and write

$L_K(E)$1

The main criterion states that $L_K(E)$2 fails gr-IBN if and only if there exist two finite tuples of nonnegative integers

$L_K(E)$3

such that

$L_K(E)$4

The mechanism behind the criterion is monoidal. If gr-IBN fails, then there is a nontrivial graded module isomorphism

$L_K(E)$5

with $L_K(E)$6. In $L_K(E)$7, this becomes

$L_K(E)$8

Using confluence and repeated application of the relation (TE), all generators can be rewritten at a common level $L_K(E)$9. An elementary matrix computation then yields

$L_K(E)$0

which leads to the equality $L_K(E)$1. Conversely, any solution of $L_K(E)$2 gives a genuine equality in $L_K(E)$3, hence a graded module isomorphism between different finite sums of shifts.

This characterization is complete for finite graphs with no sinks. Its importance lies in turning the failure of gr-IBN into an explicit condition on powers of the adjacency matrix. The result thereby links graded basis-number phenomena to matrix identities derived from graph combinatorics (Phuc, 26 Mar 2026).

4. Graph classes for which gr-IBN is determined

Several graph classes admit explicit gr-IBN conclusions.

Graph class	Criterion stated in the paper	Consequence for $L_K(E)$4
Graphs with a sink	$L_K(E)$5 has at least one sink	Has gr-IBN
Graphs with a maximal sink or cycle having finitely many predecessors	Either condition holds	Has gr-IBN
Cayley graphs $L_K(E)$6	$L_K(E)$7	Has IBN and gr-IBN
Cayley graphs $L_K(E)$8	$L_K(E)$9	Fails gr-IBN
Hopf graphs $R=\bigoplus_{n\in\mathbb Z}R_n$0	$R=\bigoplus_{n\in\mathbb Z}R_n$1	Has IBN and gr-IBN
Hopf graphs $R=\bigoplus_{n\in\mathbb Z}R_n$2	common row/column sum $R=\bigoplus_{n\in\mathbb Z}R_n$3	Fails gr-IBN

For graphs with a sink, Proposition 3.3 states that $R=\bigoplus_{n\in\mathbb Z}R_n$4 has gr-IBN. The proof idea uses the fact that a sink $R=\bigoplus_{n\in\mathbb Z}R_n$5 yields a generator $R=\bigoplus_{n\in\mathbb Z}R_n$6 in $R=\bigoplus_{n\in\mathbb Z}R_n$7 that cannot be expanded by the relation (TE), and then compares lowest-level appearances on both sides of a candidate equality.

Corollary 2.7 gives another positive class: if $R=\bigoplus_{n\in\mathbb Z}R_n$8 has a maximal sink or a cycle having finitely many predecessors, then $R=\bigoplus_{n\in\mathbb Z}R_n$9 has gr-IBN, since these conditions imply genuine IBN.

For Cayley graphs, if $\mathbb Z$0 is finite and $\mathbb Z$1 is a generating set, Proposition 4.2 establishes the equivalence of the following statements: $\mathbb Z$2 has IBN; $\mathbb Z$3 has gr-IBN; $\mathbb Z$4; and $\mathbb Z$5 is a single cycle of length $\mathbb Z$6. Thus, the Cayley case exhibits exact agreement between IBN and gr-IBN.

For Hopf graphs $\mathbb Z$7 associated with a finite group $\mathbb Z$8 and a ramification datum $\mathbb Z$9, Lemma 4.4 shows that the adjacency matrix has constant row-sum and constant column-sum

$\alpha\in\mathbb Z$0

Proposition 4.5 then gives

$\alpha\in\mathbb Z$1

If the common row/column sum is at least $\alpha\in\mathbb Z$2, Corollary 3.10 applies to show failure of gr-IBN (Phuc, 26 Mar 2026).

5. Stability under quotients and Cartesian products

The paper also studies how gr-IBN behaves under standard graph-algebra constructions.

If $\alpha\in\mathbb Z$3 is hereditary-saturated, there is a $\alpha\in\mathbb Z$4-graded surjection

$\alpha\in\mathbb Z$5

Lemma 2.2 shows that a graded surjection from a ring of known gr-IBN forces gr-IBN on the domain. Equivalently, Proposition 4.7 states that whenever $\alpha\in\mathbb Z$6 has gr-IBN, then so does $\alpha\in\mathbb Z$7.

For Cartesian products, if $\alpha\in\mathbb Z$8 and $\alpha\in\mathbb Z$9 are finite graphs, then $R(\alpha)$0 has vertex set $R(\alpha)$1 and edges coming from horizontal copies of $R(\alpha)$2 and vertical copies of $R(\alpha)$3. Lemma 5.4 states that $R(\alpha)$4 has a sink if and only if both $R(\alpha)$5 and $R(\alpha)$6 have a sink. Hence, if both factors have sinks, then $R(\alpha)$7 has gr-IBN.

A further preservation statement is given by Proposition 5.7. Suppose that $R(\alpha)$8 has no sinks and $R(\alpha)$9 has gr-IBN, and that $E$00 is nontrivial and has at least one source. Then $E$01 has gr-IBN. The proof idea is to exhibit a hereditary-saturated subset

$E$02

such that

$E$03

The quotient result then lifts gr-IBN from $E$04 to $E$05.

As a corollary, the Cartesian product of an $E$06-cycle $E$07 and an $E$08-line graph $E$09 always has (gr-)IBN, because $E$10 has no sinks but has gr-IBN, and $E$11 has a source. This situates gr-IBN within a broader closure theory for graph operations, showing that the property can often be transferred from a simpler quotient or factor configuration (Phuc, 26 Mar 2026).

6. Examples and relation to ordinary IBN

The examples isolate the distinction between graded and ungraded basis-number behavior and show how the matrix criterion is used in practice.

Example 3.5 presents a graph

$E$12

In the ungraded monoid $E$13, one checks that

$E$14

so IBN fails. In the graded monoid $E$15, the example is given as one where gr-IBN holds. This demonstrates that ordinary IBN and gr-IBN do not coincide in general.

A contrasting example gives a graph

$E$16

with adjacency matrix

$E$17

Its column sums satisfy $E$18, and Corollary 3.11 applies to show that

$E$19

so $E$20 gives a solution of $E$21 and gr-IBN fails.

The Cayley and Hopf families provide systematic comparisons. The directed cycle $E$22, viewed as the Cayley graph of $E$23 by one generator, has gr-IBN; by contrast, the complete directed $E$24-generator Cayley graph of $E$25 has no gr-IBN. Likewise, among Hopf graphs, only those with total ramification

$E$26

have gr-IBN.

These examples rule out a common conflation of IBN with gr-IBN. In some families, notably Cayley graphs and Hopf graphs as treated here, the two properties are equivalent. In general, however, the graded setting retains enough information for the two notions to diverge, and the talented monoid is precisely the device that detects that divergence (Phuc, 26 Mar 2026).