Leavitt Inverse Semigroup Overview

• Leavitt inverse semigroup is an inverse semigroup constructed from directed or separated graphs with relations that mimic Cuntz–Krieger constraints.
• It encodes combinatorial and grading invariants that classify associated Leavitt path algebras and facilitate concrete embeddings into matrix algebras.
• Its structure bridges graph theory, noncommutative algebra, and groupoid models, providing actionable insights on partial actions and Steinberg algebra constructions.

A Leavitt inverse semigroup is an inverse semigroup associated to a directed graph or a separated graph, equipped with relations reflecting Cuntz–Krieger-type constraints. It provides a semigroup-theoretic model that mediates between combinatorial graph theory, noncommutative algebra (Leavitt path algebras), and groupoid/Steinberg algebra constructions. The structural, combinatorial, and representation-theoretic properties of Leavitt inverse semigroups encode the essential features of the underlying graph and govern isomorphism and classification results for the corresponding Leavitt path algebras, with extensions to the context of separated graphs and their associated tame Leavitt path algebras.

1. Definition and Presentation

Given a finite directed graph $E = (E^0, E^1, s, r)$, the Leavitt inverse semigroup $S_E$ (or $\mathrm{LI}(E)$) is the inverse semigroup with zero, generated by:

• vertices $v \in E^0$,
• real edges $e \in E^1$,
• ghost edges $e^*$ for each $e \in E^1$,

subject to the following relations:

• (R1) $s(e) e = e = e r(e)$ and $e r(e) = e = e^* s(e)$,
• (R2) $u v = 0$ if $u, v \in E^0$ and $u \neq v$,
• (R3) $e f = 0$ if $e, f \in E^1$ and $e \neq f$,
• (R4) $e^* e = r(e)$,
• (R5) $v = e e^*$ for each $v \in E^0$ with $|s^{-1}(v)| = 1$.

Each nonzero element of $S_E$ can be uniquely written in reduced form either as $p q^*$ or as $p e e^* q^*$, where $p,q$ are paths in $E$, $r(p) = r(q)$, and, in the latter case, $s(e) = r(p) = r(q)$ has out-degree at least $2$ (Li et al., 12 Dec 2024, Meakin et al., 2019).

For a separated graph $(E, C)$, the Leavitt inverse semigroup $\mathrm{LI}(E, C)$ is constructed by quotienting the separated-graph inverse semigroup $\mathcal{S}(E, C)$ by Cuntz–Krieger relations at singleton separations, yielding a semigroup that canonically embeds into the tame Leavitt path algebra $\mathcal{L}_K^\mathrm{ab}(E, C)$ (Ara et al., 17 Dec 2025).

2. Structural Properties and Grading

Leavitt inverse semigroups encode the structure of finite paths in $E$, with the semilattice of idempotents being isomorphic to the prefix order on paths. Each nonzero idempotent corresponds to $p p^*$ for some directed path $p$. The natural partial order in $\mathrm{LI}(E)$ is given by $p p^* \leq q q^*$ if and only if $p = q t$ for some path $t$ (Meakin et al., 2019).

There is a natural $\mathbb{Z}$-grading on $S_E$, defined by $\varphi(p q^*) = |p| - |q|$, compatible with semigroup multiplication: $\varphi(xy) = \varphi(x) + \varphi(y)$ whenever $xy \neq 0$. This grading makes $S_E$ a graded inverse semigroup, and the associated Leavitt path algebra $L_K(E)$ inherits a compatible grading. When $E$ is a connected graph with out-degree at most $1$, the isomorphism type of the graded Leavitt inverse semigroup is determined by two combinatorial invariants: cycle length $s_E$ and the depth-profile $D_E$ (the multiset of relative depths modulo $s_E$) (Li et al., 12 Dec 2024).

For separated graphs, $\mathrm{LI}(E, C)$ admits a restricted semidirect product decomposition $$\mathrm{LI}(E, C) \cong \mathcal{E}(\mathrm{LI}(E, C)) \rtimes^r F(E^1)$$, where $F(E^1)$ is the free group on the edge set $E^1$ and $\mathcal{E}(\mathrm{LI}(E, C))$ is the semilattice of idempotents, realized through Leavitt–Munn trees (Ara et al., 17 Dec 2025).

3. Universal and Categorical Properties

Leavitt inverse semigroups are strongly $E^*$-unitary. For any strongly $E^*$-unitary inverse semigroup $S$, there exists a partial action on a generalized Boolean algebra (of compact open sets of tight filters in the idempotent semilattice), which canonically realizes its Steinberg algebra as a partial skew group ring. For graph inverse semigroups and their Cuntz–Krieger (Leavitt) completions, this partial action framework recovers Leavitt path algebras in a functorial manner (Zhang, 3 Mar 2025).

The Cuntz–Krieger semigroup $\mathrm{CK}_G$, obtained as a distributive completion of the graph inverse semigroup $S_G$, quotiented by the Lenz congruence, is the universal Boolean inverse $A$-semigroup satisfying the Cuntz–Krieger relations. Its semigroup algebra over a field coincides with the classical Leavitt path algebra $L_K(G)$, and its ample semigroup is that of the associated étale groupoid (the tight groupoid) (Jones et al., 2011).

4. Connections to Leavitt Path Algebras and Groupoid Models

The multiplicative subsemigroup of the Leavitt path algebra $L_K(E)$ generated by $E^0, E^1, (E^1)^*$ is isomorphic to the Leavitt inverse semigroup $\mathrm{LI}(E)$, and in fact generates $L_K(E)$ as a $K$-algebra. Moreover, an isomorphism $\mathrm{LI}(E) \cong \mathrm{LI}(F)$ as inverse semigroups induces an isomorphism $L_K(E) \cong L_K(F)$ as $K$-algebras (Meakin et al., 2019).

For separated graphs, the image of the generating partial isometries and vertices in the tame Leavitt path algebra $\mathcal{L}_K^\mathrm{ab}(E, C)$ yields an inverse semigroup isomorphic to $\mathrm{LI}(E, C)$; hence, $\mathrm{LI}(E, C)$ embeds as the semigroup generated by the canonical partial isometries in the algebra. The basis structure of the algebra is governed by the Leavitt–Munn tree model for the semigroup, which provides normal forms for elements of both the semigroup and the algebra (Ara et al., 17 Dec 2025). When $K$ is a field, $\mathcal{L}_K^\mathrm{ab}(E, C)$ is a Steinberg algebra associated to the tight groupoid of $(E, C)$ (Ara et al., 17 Dec 2025).

5. Classification and Invariants

Within the class of connected finite graphs with vertices of out-degree at most $1$, the graded isomorphism class of the Leavitt inverse semigroup (and hence of the Leavitt path algebra) is completely determined by the pair of invariants $(s_E, D_E)$:

• $s_E$: the length of the unique cycle (zero if the graph is a tree),
• $D_E$: the depth-profile recording, for each residue $d \pmod{s_E}$, the number of vertices at relative depth $d$.

Explicitly, two such graphs yield graded-isomorphic Leavitt inverse semigroups if and only if these invariants coincide. Examples demonstrate pairs of non-isomorphic graphs that are nevertheless graded-isomorphic in this sense, and the theorem fails if the depth profiles differ (Li et al., 12 Dec 2024). For broader classes, isomorphism corresponds to isomorphism of contracted graphs, suitably defined (Meakin et al., 2019).

For separated graphs, the structural theory is encoded through combinatorics of Leavitt–Munn trees and explicit computations in the tight groupoid, yielding a rich supply of invariants (including those for characterizing the socle and isolated spectrum points) (Ara et al., 17 Dec 2025).

6. Applications and Examples

Leavitt inverse semigroups provide a semigroup-theoretic underpinning for Leavitt path algebras, revealing explicit bases (via normal forms), providing classification results, and giving refined embeddings into matrix algebras via $0$-retracts (Meakin et al., 2019).

Notable examples include:

• For a bouquet of $n$ loops, $\mathrm{LI}(B_n) \cong P_n$, the polycyclic monoid; the corresponding algebra is $L_K(1,n)$.
• For connected graphs with vertices of out-degree $\leq 1$, trees yield Brandt semigroups (matrix algebras), while graphs with a unique cycle yield Brandt semigroups with $\mathbb{Z}$-maximal subgroups (matrix algebras over $K[x, x^{-1}]$) (Meakin et al., 2019).

In the context of separated graphs, the Leavitt inverse semigroup captures phenomena from both the traditional graph case and new settings (free inverse monoid, Cuntz separated graphs), with links to Cohn algebras, their kernels, and explicit socle decompositions (Ara et al., 17 Dec 2025).

7. Groupoid-Algebra Duality and Socle Structure

Under noncommutative Stone duality, the Boolean inverse $A$-semigroup structure on the Leavitt inverse semigroup mirrors the ample semigroup of the étale groupoid associated to a graph or separated graph. The tight groupoid model provides locally compact groupoids whose Steinberg algebras realize Leavitt path algebras and their labelled and separated variants, with socle and ideal structure determined by combinatorics of the underlying Leavitt–Munn trees and cycles without exits (Jones et al., 2011, Ara et al., 17 Dec 2025). The unique partial join operations in the semigroup context encode the additive and Cuntz–Krieger structure of the corresponding algebras.

Overall, the Leavitt inverse semigroup provides a universal and unifying object for the semigroup, algebraic, and groupoid-theoretic approaches to combinatorial and analytic invariants arising from directed and separated graphs (Li et al., 12 Dec 2024, Zhang, 3 Mar 2025, Ara et al., 17 Dec 2025, Jones et al., 2011, Meakin et al., 2019).