arXiv subjects

Zhengpan Wang

Publications and source records attributed to Zhengpan Wang.

6 recordsLinked to original sources

arXiv2024

Graded isomorphisms of Leavitt path algebras and Leavitt inverse semigroups

Leavitt inverse semigroups of directed finite graphs are related to Leavitt graph algebras of (directed) graphs. Leavitt path algebras of graphs have the natural $\mathbb Z$-grading via the length of paths in graphs. We consider the $\mathbb Z$-grading on Leavitt inverse semigroups. For connected finite graphs having vertices out-degree at most $1$, we give a combinatorial sufficient and necessary condition on graphs to classify the corresponding Leavitt path algebras and Leavitt inverse semigroups up to graded isomorphisms. More precisely, the combinatorial condition on two graphs coincides if and only if the Leavitt path algebras of the two graphs are $\mathbb Z$-graded isomorphic if and only if the Leavitt inverse semigroups of the two graphs are $\mathbb Z$-graded isomorphic.

math.RA↗

arXiv2024

Some properties of congruence lattices of path semigroups

Each quiver corresponds to a path semigroup, and such a path semigroup also corresponds to an associative K-algebra over an algebraically closed field K. Let Q be a quiver and S_Q, KQ be its path semigroup, path algebra, respectively. In this paper, we study some properties of the congruence lattice of S_Q. First, we show that there is a one-to-one correspondence between congruences on S and certain algebraic ideals of KQ. Based on such a description, we consider acyclic quivers and show that the congruence latices of such path semigroups are strong upper semimodular but not necessarily lower semimodular. Moreover, we provide some equivalent conditions for the congruence lattices to be modular and distributive.

math.GR↗

arXiv2023

Distributivity in congruence lattices of graph inverse semigroups

Let Γ be a directed graph and Inv(Γ) be the graph inverse semigroup of Γ. Luo and Wang [7] showed that the congruence lattice C(Inv(Γ)) of any graph inverse semigroup Inv(Γ) is upper semimodular, but not lower semimodular in general. Anagnostopoulou-Merkouri, Mesyan and Mitchell characterized the directed graph Γ for which C(Inv(Γ)) is lower semimodular [2]. In the present paper, we show that the lower semimodularity, modularity and distributivity in the congruence lattice C(Inv(Γ)) of any graph inverse semigroup Inv(Γ) are equivalent.

math.GR↗

arXiv2020

On lattice of congruences on graph inverse semigroups

Congruences on a graph inverse semigroup were recently described in terms of the underline graph. Based on such descriptions, we show that the lattice of congruences on a graph inverse semigroup is upper semimodular but not lower semimodular.

math.GR↗

arXiv2019

Graph inverse semigroups and Leavitt path algebras

We study two classes of inverse semigroups built from directed graphs, namely graph inverse semigroups and a new class of semigroups that we refer to as Leavitt inverse semigroups. These semigroups are closely related to graph $C^*$-algebras and Leavitt path algebras. We provide a topological characterization of the universal groups of the local submonoids of these inverse semigroups. We study the relationship between the graph inverse semigroups of two graphs when there is a directed immersion between the graphs. We describe the structure of graphs that admit a directed cover or directed immersion into a circle and we provide structural information about graph inverse semigroups of finite graphs that admit a directed cover onto a bouquet of circles. We also find necessary and sufficient conditions for a homomorphic image of a graph inverse semigroup to be another graph inverse semigroup. We find a presentation for the Leavitt inverse semigroup of a graph in terms of generators and relations. We describe the structure of the Leavitt inverse semigroup and the Leavitt path algebra of a graph that admits a directed immersion into a circle. We show that two graphs that have isomorphic Leavitt inverse semigroups have isomorphic Leavitt path algebras and we classify graphs that have isomorphic Leavitt inverse semigroups. As a consequence, we show that Leavitt path algebras are $0$-retracts of certain matrix algebras.

math.GR↗

arXiv2019

Rigidity for the Hopf algebra of quasi-symmetric functions

We investigate the rigidity for the Hopf algebra ${\rm QSym}$ of quasisymmetric functions with respect to the monomial, the fundamental and the quasisymmetric Schur basis, respectively. By establishing some combinatorial properties of the posets of compositions arising from the analogous Pieri rules for quasisymmetric functions, we show that ${\rm QSym}$ is rigid as an algebra with respect to the quasisymmetric Schur basis, and rigid as a coalgebra with respect to the monomial and the quasisymmetric Schur basis, respectively. The natural actions of reversal, complement and transpose of the labelling compositions lead to some nontrivial graded (co)algebra automorphisms of ${\rm QSym}$. We prove that the linear maps induced by the three actions are precisely the only nontrivial graded algebra automorphisms that take the fundamental basis into itself. Furthermore, the complement map on the labels gives the unique nontrivial graded coalgebra automorphism preserving the fundamental basis, while the reversal map on the labels gives the unique nontrivial graded algebra automorphism preserving the monomial basis. Therefore, ${\rm QSym}$ is rigid as a Hopf algebra with respect to the monomial and the quasisymmetric Schur basis.

math.CO↗