Separated Graph Inverse Semigroup Overview

• Separated graph inverse semigroups are universal inverse semigroups associated with separated graphs, extending classical graph theories by incorporating separation data.
• Their detailed idempotent structure and C-compatible trees yield explicit normal forms and enable restricted semidirect product decompositions via partial actions of free groups.
• Applications include canonical K-bases for Leavitt path algebras and C*-algebras, linking these structures to type semigroups and graph monoids.

A separated graph inverse semigroup is a universal inverse semigroup $\mathcal{S}(E,C)$ associated to a separated graph $(E, C)$, designed to generalize the classical graph inverse semigroup theory to capture the additional combinatorics and symmetries introduced by separation data. This structure is foundational in inverse semigroup approaches to tame graph algebras, Leavitt path algebras, and $C^*$-algebraic counterparts. The theory develops through detailed analysis of the idempotent structure via $C$-compatible trees, semidirect product decompositions involving partial actions of free groups, and connections with the tight groupoid and type semigroups, enabling explicit normal forms and canonical algebraic bases (Ara et al., 17 Dec 2025, Ara et al., 2024, Ara et al., 2019).

1. Separated Graphs and the Inverse Semigroup $\mathcal{S}(E,C)$

A separated graph is a pair $(E, C)$, where $E=(E^0, E^1, s, r)$ is a directed graph with vertex set $E^0$ and edge set $E^1$, and $C=\bigsqcup_{v\in E^0} C_v$ is a partition of the outgoing edges at each vertex: $C_v$ is a collection of pairwise disjoint, nonempty subsets partitioning $s^{-1}(v)$. Extremal cases include the trivial (unseparated) case where $C_v = \{ s^{-1}(v)\}$ for each $v$ and the free separation with $C_v = \{ \{e\} : e \in s^{-1}(v)\}$.

The separated graph inverse semigroup $\mathcal{S}(E,C)$ is universally generated by $E^0$, $E^1$, and their formal inverses $E^{1-1}$ (with involution $e\mapsto e^{-1}$, extended to paths), modulo the relations:

• $v w = \delta_{v, w} v$ for $v, w \in E^0$
• $s(x) x = x = x r(x)$ for $x \in E^1 \cup E^{1-1}$
• $e^{-1} f = \delta_{e,f} r(e)$ for $e, f \in X, X\in C$
• $x x^{-1}$ and $y y^{-1}$ commute for all $x, y \in \mathcal{S}(E,C)$.

Each $e\in E^1$ acts as a canonical partial isometry. This semigroup always contains zero $0$ and features a rich lattice of idempotents.

2. Idempotent Structure and $C$-Compatible Trees

Every element of $\mathcal{S}(E,C)$ can be uniquely written in Scheiblich normal form as $\alpha \beta^{-1}$, with $\alpha, \beta$ reduced $C$-separated paths in the double graph $\widehat E$ (i.e., no subword $x^{-1}y$ with $x, y\in X$ for some $X\in C$). Idempotents are elements of the form $\alpha \alpha^{-1}$.

The semilattice of idempotents $E(\mathcal{S}(E,C))$ is in bijection with the family $Y$ of all nonempty finite $C$-compatible $E$-trees—finite lower sets of $C$-separated paths at a base vertex, subject to compatibility (maximal path geodesic joins remain $C$-separated). For such a tree $T$:

$$e(T) = \prod_{\lambda\in\max(T)} \lambda \lambda^{-1}$$

Idempotents are partially ordered by inclusion of their associated trees: $e(T)\leq e(T')$ iff $T\supseteq T'$. This combinatorial structure encodes the underlying $C$-separation and is central to understanding both the inverse semigroup and associated algebras (Ara et al., 17 Dec 2025, Ara et al., 2024).

3. Strong $E^*$-Unitarity and (Restricted) Semidirect Product Decomposition

A defining structural result is that $\mathcal{S}(E,C)$ is strongly $E^*$-unitary. There exists an idempotent-pure partial homomorphism

$$\sigma: \mathcal{S}(E,C)^\times \to \mathbb{F}(E^1)$$

with $\sigma(v)=1$, $\sigma(e)=e$, $\sigma(e^{-1})=e^{-1}$ ($e\in E^1$), so that the kernel of $\sigma$ is the set of idempotents. This enables a presentation of $\mathcal{S}(E,C)$ as a restricted semidirect product

$$\mathcal{S}(E,C) \cong \mathcal{Y} \rtimes_\theta \mathbb{F}$$

where $\mathbb{F} = \mathbb{F}(E^1)$ is the free group on $E^1$ and $\mathcal{Y}$ is a quotient semilattice of finite $C$-compatible lower sets of paths. The partial action $\theta$ reflects pre- and post-concatenation dynamics, and the product is

$(I, g)\cdot (J, h) = (I \wedge \theta_g(J),\, gh)$

with inversion $(I, g)^{-1} = (\theta_{g^{-1}}(I),\, g^{-1})$, restricted to cases where domains are defined (Ara et al., 2024).

4. The Leavitt Inverse Semigroup and Quotient Structures

The Leavitt inverse semigroup $\mathcal{LI}(E,C)$ is a quotient of $\mathcal{S}(E,C)$ imposing the so-called single-edge Cuntz–Krieger relations $ee^{-1} = s(e)$ whenever $\{e\}\in C$. Algebraically, the associated tame Leavitt path algebra $\mathcal{L}_K^{\mathrm{ab}}(E,C)$ is obtained from the tame Cohn algebra $\mathcal{C}_K^{\mathrm{ab}}(E,C)$ by factoring out elements $q_X = v-\sum_{e\in X} ee^{-1}$.

A central theorem is that $\mathcal{LI}(E,C)$ remains strongly $E^*$-unitary and admits a restricted semidirect product decomposition:

$$\mathcal{LI}(E,C) \cong \mathcal{E}^L \rtimes^r \mathbb{F}$$

with $\mathcal{E}^L$ a suitable quotient semilattice and with canonical normal forms: every element $(T, g)$, with $T$ a Leavitt–Munn tree and $g$ a $C$-separated group element with left-special prefix in $T$ (Ara et al., 17 Dec 2025). This yields a linear basis for $\mathcal{L}_K^{\mathrm{ab}}(E,C)$ via pairs $(e(T), g)$, subject to reductions along finite $C$-edges.

5. Spectrum, Tight Groupoid, and Amenability

The spectrum of the semilattice of idempotents $\mathcal{Y}$, $\widehat{\mathcal{Y}}$, is comprised of nonzero characters, identified with filters, and equipped with a totally disconnected topology. The tight spectrum $\widehat{\mathcal{Y}}_{\mathrm{tight}}$ is the closure of ultrafilters, corresponding concretely to infinite $C$-separated paths in $\widehat{E}$ and vertices with infinite emission.

The universal groupoid $$\mathcal{G}(\mathcal{S}(E,C)) \cong \widehat{\mathcal{Y}} \rtimes_\theta \mathbb{F}$$ restricts to the tight groupoid $$\mathcal{G}_{\mathrm{tight}}(\mathcal{S}(E,C)) \cong \widehat{\mathcal{Y}}_{\mathrm{tight}} \rtimes_\theta \mathbb{F}$$ (Ara et al., 2024). For adaptable separated graphs, this groupoid is always Hausdorff, ample, and étale. Furthermore, it is amenable: the relevant examples reduce—over regular primes—to classical, amenable graph groupoids, and—over free primes—to transformation groupoids of abelian groups acting on Cantor-type spaces (Ara et al., 2019). This ensures that associated Steinberg and $C^*$-algebras are nuclear and satisfy the UCT.

6. Applications to Algebras and Type Semigroups

The combinatorial normal forms arising from the structure of $\mathcal{S}(E,C)$ and its quotient $\mathcal{LI}(E,C)$ yield canonical $K$-bases for tame Cohn and Leavitt path algebras, generalizing the well-known Zelmanov–Alahmadi bases in the unseparated case (Ara et al., 17 Dec 2025). Under mild assumptions (adaptability), the type semigroup $$\mathrm{Typ}(\mathcal{G}_{\mathrm{tight}}(\mathcal{S}(E,C)))$$ is isomorphic to the graph monoid $M(E, C)$:

$$M(E, C) = \langle a_v\ (v\in E^0)\ |\ a_v = \sum_{e\in X} a_{r(e)},\ X\in C_v \rangle$$

Every finitely generated conical refinement monoid arises this way. The direct connection to the realization problem for von Neumann regular rings and classification theory for Leavitt path algebras is a major motivation for the study (Ara et al., 2019).

Examples include:

• Ordinary graphs ($C_v$ trivial): recovers the classical theory.
• Cuntz separated graphs (free separation): $\mathcal{LI}(E,C)$ is a free group, idempotent semilattice is trivial.
• Realization of the free inverse monoid as a corner of a suitable separated graph inverse semigroup.
• Computation of the socle and isolated points of the spectrum, with a splitting as direct sums of full matrix algebras over $K$ for the tame Leavitt algebra.

7. Illustrative Examples and Further Directions

Several explicit constructions illustrate the theory:

• Zelmanov–Alahmadi basis: arises for the unseparated case and is recovered as a special case of the normal forms.
• Cuntz separated graphs: demonstrate how all idempotents can collapse, leaving the free group and group algebra structure.
• Two-fold refinement monoid: provides an example where separated graph monoids extend the range beyond classical row-finite graph monoids (Ara et al., 2019).
• Socle and kernel computations: normal-form bases describe both the socle of Leavitt path algebras and kernels of natural maps between associated tame algebras.

The unified framework of separated graph inverse semigroups and their quotients by Leavitt-type relations enables a canonical handling of tame Cohn and Leavitt algebras and their $C^*$-analogues, clarifies semidirect product decompositions, and supports combinatorial and spectral analysis essential to both algebraic and operator-algebraic applications (Ara et al., 17 Dec 2025, Ara et al., 2024).