Finitely Aligned P-Graphs: Structure & Algebras

• Finitely aligned P-graphs are combinatorial structures that generalize directed and higher-rank graphs by grading paths with elements from a semigroup, ensuring finite minimal common extensions.
• They underpin advanced operator algebra constructions, enabling the development of universal C*-algebras and Kumjian–Pask algebras through robust factorization, exhaustiveness, and groupoid models.
• Their rich algebraic and topological properties facilitate the classification of Kirchberg algebras, providing deep insights into ideal structures, periodicity, and orbit equivalence.

A finitely aligned P-graph is a combinatorial structure generalizing directed and higher-rank graphs, where morphisms are paths graded by a semigroup $P$ that typically sits inside a quasi-lattice ordered group. This framework underpins advanced constructions in operator algebras, notably C*-algebras and Kumjian–Pask algebras associated to partial-isometric representations, and provides new insights into their ideal, orbit, and groupoid structures, with ramifications for the classification of Kirchberg algebras.

1. Definition and Structural Properties of Finitely Aligned P-Graphs

A P-graph $\Lambda$ is a small category equipped with a degree functor $d : \Lambda \to P$ satisfying the factorisation property: for any morphism $\lambda \in \Lambda$ and any factorization $d(\lambda) = pq$ (with $p, q \in P$), there exist unique $\mu, \nu \in \Lambda$ such that $\lambda = \mu \nu$, $d(\mu) = p$, $d(\nu) = q$. This replaces the length or multirank assignment in traditional graphs (with $P = \mathbb{N}$ or $P = \mathbb{N}^k$) by grading in a general semigroup.

Finite alignment, adapted from Raeburn and Sims, means that for any pair $\mu, \nu \in \Lambda$, the set of minimal common extensions,

$$\mathrm{MCE}(\mu,\nu) = \{ \lambda \in \Lambda : \lambda \text{ is a minimal common extension of } \mu \text{ and } \nu \},$$

is finite (possibly empty). An "exhaustive" finite subset $E \subset v\Lambda$ for $v \in \Lambda^0$ is one such that for every $\alpha \in v\Lambda$, some $\lambda \in E$ satisfies $\mathrm{MCE}(\alpha,\lambda) \ne \varnothing$ (Brownlowe et al., 2010). This ensures that intersections of extensions are sufficiently controlled for analytic constructions.

The definition encompasses both "globally" and "locally" finitely aligned graphs. In non-finitely aligned graphs, one often isolates a subcategory $\mathrm{FA}\Lambda$ consisting of those morphisms $\lambda$ such that every "local" intersection $(\mu\Lambda \cap \nu\Lambda)$ (for $\mu$ an extension of $\lambda$ and any $\nu$) decomposes as a finite union of principal right ideals. This subset forms a right ideal and has the structure of a right constellation, enabling relative combinatorial and algebraic constructions (Jones, 26 Mar 2025).

2. Topological and Groupoid Models: Path Space and Boundary-Path Space

For a finitely aligned P-graph, the path space $W$ is typically modeled using unions of finite or infinite paths, endowed with a locally compact, Hausdorff topology built from cylinder sets,

$$Z(\lambda \setminus G) = \{ x \in W : x(0, d(\lambda)) = \lambda \} \setminus \bigcup_{\mu \in G} Z(\lambda\mu),$$

for $\lambda \in \Lambda$ and $G$ a finite subset of $s(\lambda)\Lambda$ (Webster, 2011). The boundary-path space $\partial\Lambda$ comprises paths that cannot be extended in a way that violates exhaustiveness, and it is naturally identified with the spectrum of the commutative diagonal C*-subalgebra $D_\Lambda$ of $C^*(\Lambda)$.

In non-globally finitely aligned graphs, one restricts to filters meeting $\mathrm{FA}\Lambda$, constructing a locally compact path space $$F_p\Lambda := \{ x \in F\Lambda : x \cap \mathrm{FA}\Lambda \ne \varnothing \}$$, where $F\Lambda$ is the space of hereditary, directed subsets of $\Lambda$. Cylinder sets $F_p\Lambda(\lambda)$ (for $\lambda \in \mathrm{FA}\Lambda$) are then compact and form a base for the topology (Jones, 26 Mar 2025).

Both path and boundary-path spaces admit actions of $P$ by left shifting according to unique factorization, facilitating the construction of semidirect product groupoids $G(F_p\Lambda, P, T)$, which are ample, Hausdorff, and generalize Spielberg's groupoids; for boundary paths, one obtains an inverse semigroup model (Jones, 26 Mar 2025, Clark et al., 17 Sep 2025).

3. Groupoid Constructions: Filters, Graph Morphisms, and Conjugacy

Two main approaches for constructing path and boundary-path groupoids exist: the filter model (where elements are hereditary, directed subsets) and the graph morphism model (where elements are degree-preserving functors from P-path prototypes $\Omega_{P,(m_n)}$ to $\Lambda$). For finitely aligned P-graphs where $(Q, P)$ is a weakly quasi-lattice ordered group, these approaches yield isomorphic groupoids.

A crucial technical concept is the conjugacy of partial semigroup actions. Given actions $(X, P, T_X)$ and $(Y, P, T_Y)$, a homeomorphism $h:X \to Y$ is a conjugacy if it intertwines domains and actions for all $m \in P$. The main result is that conjugate semigroup actions yield isomorphic semidirect product groupoids (Clark et al., 17 Sep 2025).

This enables a unified theory where groupoids constructed via filters, graph morphisms, and Spielberg’s method are all isomorphic, and each provides a platform for analytic and algebraic modeling.

Approach	Path Space Model	Groupoid Presentation
Filter	Hereditary, directed subsets	Semidirect product
Graph morphism	Functors from $\Omega_{P,(m_n)}$	Semidirect product
Spielberg’s construction	Combinatorial "Spielfeld" sets	Ample groupoid
Inverse semigroup "tight"	Ultraproper filters	Tight groupoid

4. Operator Algebraic Structures and Co-universality

The operator algebra associated to a finitely aligned P-graph $\Lambda$ is constructed as a universal C*-algebra $C^*_{\min}(\Lambda)$, generated by partial isometries subject to factorization and generalized Cuntz–Krieger relations. The fundamental relation,

$$\prod_{\lambda \in E}(s_\mu s_\mu^* - s_{\mu\lambda} s_{\mu\lambda}^*) = 0,$$

for all $\mu \in \Lambda$ and finite exhaustive sets $E \subset s(\mu)\Lambda \setminus \{ s(\mu) \}$, ensures appropriate "gap filling" (Brownlowe et al., 2010).

The construction proceeds via a balanced algebra (the fixed-point algebra of a certain coaction), a co-universal property, and a canonical coaction $\delta$ of $G$:

$$\delta(s_\lambda) = s_\lambda \otimes U_{d(\lambda)}.$$

Any gauge-compatible representation by nonzero partial isometries with compatible coaction factors uniquely through $C^*_{\min}(\Lambda)$.

Injectivity of the induced homomorphism is characterized by: all range projections $t_\mu t_\mu^*$ are nonzero, and the exhaustive set relations above hold for all finite exhaustive $E$ (Brownlowe et al., 2010). Under these criteria, the balanced algebra is AF, enabling explicit analysis of ideals and representations.

A pivotal result shows that every Kirchberg algebra satisfying the UCT is Morita equivalent to $C^*_{\min}(\Lambda)$ for some finitely aligned $(\mathbb{N}^2 * \mathbb{N})$-graph $\Lambda$, with ideal structure and AF core compatible with classification invariants.

5. Kumjian–Pask Algebras, Steinberg Algebras, and Simplicity

Finitely aligned P-graphs yield Kumjian–Pask algebras $KP_R(\Lambda)$ over a commutative ring $R$, generalizing Leavitt path algebras to higher rank. The generators $s_\lambda$, $s_\mu^*$ (with $s(\lambda) = s(\mu)$) satisfy relations (KP1)–(KP4) encoding orthogonality, multiplication, minimal common extensions, and exhaustiveness.

There is a graded uniqueness theorem for representations that are nonzero on vertices and preserve the grading; any such representation is injective (Clark et al., 2015). The Cuntz–Krieger uniqueness theorem applies in aperiodic cases.

Importantly, for any finitely aligned higher-rank graph, $KP_R(\Lambda)$ is isomorphic to the Steinberg algebra $A_R(G_\Lambda)$ of the boundary path groupoid $G_\Lambda$. Simplicity and basic simplicity of $KP_R(\Lambda)$ are characterized by effectiveness (aperiodicity) and minimality (cofinality) of $G_\Lambda$. If $KP_R(\Lambda)$ is simple and every vertex is reached from a generalized cycle with an entrance, then it is purely infinite; otherwise, for the appropriate finiteness condition, a dichotomy holds: it is either purely infinite or locally matricial (Larki, 2016).

6. Periodicity, Pullbacks, Cycline MASA, and Ideal Structure

Periodicity in P-graphs is understood via the equivalence relation $\mu \sim \nu$ (if $\mu x = \nu x$ for all infinite $x \in s(\mu)\Lambda^\infty$), with the periodicity group $$\operatorname{Per}\Lambda = \{ d(\mu) - d(\nu) : \mu \sim \nu \} \subset G(P)$$ (Yang, 2014).

For appropriate P-graphs, there is an isomorphism $\Lambda \cong q^*(\Lambda/\sim)$ (pullback of the pushout), and embeddings of C*-algebras:

$$C^*(\Lambda) \hookrightarrow C^*(\Lambda/\sim) \otimes C^*(\operatorname{Per}\Lambda).$$

Cycline subalgebras generated by equivalent pairs form maximal abelian subalgebras (MASA) $$M \cong D_\Lambda \otimes C^*(\operatorname{Per}\Lambda)$$ in $C^*(\Lambda)$, supporting faithful conditional expectations.

Ideal structure for twisted relative Cuntz–Krieger algebras of finitely aligned P-graphs is governed by hereditary and saturated pairs in the vertex set and collections of finite exhaustive sets in the complement subgraph; the lattice of gauge-invariant ideals is completely described via such data (Whitehead, 2013).

7. Orbit Equivalence, Groupoid Isomorphism, and Classification

Continuous orbit equivalence between finitely aligned P-graphs is defined by a homeomorphism between boundary-path spaces that intertwines shift orbits via cocycles. Such equivalence is preserved if and only if the corresponding boundary path groupoids are isomorphic and this passes down to isomorphisms of associated C*-algebras and Kumjian–Pask algebras, with the diagonal subalgebras matched (Carlsen et al., 2019).

Examples indicate that homeomorphism of boundary spaces is insufficient—preservation of orbit structure and periodicity are essential for algebraic equivalence. These results are central for the dynamical classification of higher-rank graph C*-algebras.

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In summary, finitely aligned P-graphs provide comprehensive generalizations of directed and higher-rank graphs, enabling analytic and algebraic modeling through both filter and graph morphism constructions, supporting isomorphic groupoid presentations, and yielding operator algebras (C*-algebra, Kumjian–Pask, Steinberg) with deep structural properties. Their ideal structures, periodicity groups, co-universal representation theory, and orbit equivalence results profoundly influence the classification and analysis of noncommutative spaces generated by combinatorial data.