The Leavitt inverse semigroup of a separated graph
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arXiv
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| Natura: | Preprint |
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2025
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| _version_ | 1866909968476667904 |
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| author | Ara, Pere Buss, Alcides Costa, Ado Dalla |
| author_facet | Ara, Pere Buss, Alcides Costa, Ado Dalla |
| contents | We introduce and study a new inverse semigroup associated to a separated graph $(E,C)$, which we call the \emph{Leavitt inverse semigroup}. This semigroup is obtained as a quotient of the separated graph inverse semigroup $\mathcal{S}(E,C)$, introduced in our previous paper [9], and it provides a canonical inverse semigroup model for the tame Leavitt path algebra $\mathcal{L}_K^\mathrm{ab}(E,C)$ over a commutative unital ring $K$.
Our first main result describes the Leavitt inverse semigroup $\mathcal{LI}(E,C)$ as a restricted semidirect product of the free group on the edges of $E$ acting partially on a certain semilattice, which is isomorphic to the semilattice of idempotents of $\mathcal{LI}(E,C)$. This description, given in terms of Leavitt--Munn trees, yields a normal form for the elements of $\mathcal{LI}(E,C)$.
We obtain a normal form for elements of $\mathcal{L}_K^\mathrm{ab}(E,C)$, leading to explicit linear bases for $\mathcal{L}_K^\mathrm{ab}(E,C)$. Building on this and on the structural properties of $\mathcal{LI} (E,C)$, we prove that the natural homomorphism from $\mathcal{LI}(E,C)$ to $\mathcal{L}_K^\mathrm{ab}(E,C)$ is injective, so that $\mathcal{LI}(E,C)$ embeds as the inverse semigroup generated by the canonical partial isometries in $\mathcal{L}_K^\mathrm{ab}(E,C)$. Further applications include the determination of natural bases of the kernel $\mathcal Q$ of the natural map from the tame Cohn algebra $\mathcal{C}_K^\mathrm{ab} (E,C)$ to the tame Leavitt path algebtra $\mathcal{L}_K^\mathrm{ab} (E,C)$, the computation of the socle, and a characterization of the isolated points of the spectrum. Several examples, such as the Cuntz separated graph and free separations, are discussed to illustrate the theory. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2512_16003 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | The Leavitt inverse semigroup of a separated graph Ara, Pere Buss, Alcides Costa, Ado Dalla Rings and Algebras Operator Algebras We introduce and study a new inverse semigroup associated to a separated graph $(E,C)$, which we call the \emph{Leavitt inverse semigroup}. This semigroup is obtained as a quotient of the separated graph inverse semigroup $\mathcal{S}(E,C)$, introduced in our previous paper [9], and it provides a canonical inverse semigroup model for the tame Leavitt path algebra $\mathcal{L}_K^\mathrm{ab}(E,C)$ over a commutative unital ring $K$. Our first main result describes the Leavitt inverse semigroup $\mathcal{LI}(E,C)$ as a restricted semidirect product of the free group on the edges of $E$ acting partially on a certain semilattice, which is isomorphic to the semilattice of idempotents of $\mathcal{LI}(E,C)$. This description, given in terms of Leavitt--Munn trees, yields a normal form for the elements of $\mathcal{LI}(E,C)$. We obtain a normal form for elements of $\mathcal{L}_K^\mathrm{ab}(E,C)$, leading to explicit linear bases for $\mathcal{L}_K^\mathrm{ab}(E,C)$. Building on this and on the structural properties of $\mathcal{LI} (E,C)$, we prove that the natural homomorphism from $\mathcal{LI}(E,C)$ to $\mathcal{L}_K^\mathrm{ab}(E,C)$ is injective, so that $\mathcal{LI}(E,C)$ embeds as the inverse semigroup generated by the canonical partial isometries in $\mathcal{L}_K^\mathrm{ab}(E,C)$. Further applications include the determination of natural bases of the kernel $\mathcal Q$ of the natural map from the tame Cohn algebra $\mathcal{C}_K^\mathrm{ab} (E,C)$ to the tame Leavitt path algebtra $\mathcal{L}_K^\mathrm{ab} (E,C)$, the computation of the socle, and a characterization of the isolated points of the spectrum. Several examples, such as the Cuntz separated graph and free separations, are discussed to illustrate the theory. |
| title | The Leavitt inverse semigroup of a separated graph |
| topic | Rings and Algebras Operator Algebras |
| url | https://arxiv.org/abs/2512.16003 |