Hereditary Saturated Subsets and the Invariant Basis Number Property of the Leavitt Path Algebra of Cartesian Products

IF 0.6 4区数学 Q3 MATHEMATICS

Mathematical Notes Pub Date : 2024-07-05 DOI:10.1134/s0001434624030295

Min Li, Huanhuan Li, Yuquan Wen

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引用次数: 0

Abstract

In this note, first, we describe the (minimal) hereditary saturated subsets of finite acyclic graphs and finite graphs whose cycles have no exits. Then we show that the Cartesian product \(C_m\times L_n\) of an \(m\)-cycle \(C_m\) by an \(n\)-line \(L_n\) has nontrivial hereditary saturated subsets even though the graphs \(C_m\) and \(L_n\) themselves have only trivial hereditary saturated subsets. Tomforde (Theorem 5.7 in “Uniqueness theorems and ideal structure for Leavitt path algebras,” J. Algebra 318 (2007), 270–299) proved that there exists a one-to-one correspondence between the set of graded ideals of the Leavitt path algebra \(L(E)\) of a graph \(E\) and the set of hereditary saturated subsets of \(E^0\). This shows that the algebraic structure of the Leavitt path algebra \(L(C_m\times L_n)\) of the Cartesian product is plentiful. We also prove that the invariant basis number property of \(L(C_m\times L_n)\) can be derived from that of \(L(C_m)\). More generally, we also show that the invariant basis number property of \(L(E\times L_n)\) can be derived from that of \(L(E)\) if \(E\) is a finite graph without sinks.

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笛卡尔积的莱维特路径代数的遗传饱和子集和不变基数特性

摘要在这篇论文中，我们首先描述了有限无循环图和循环没有出口的有限图的（最小）遗传饱和子集。然后，我们证明了一个（m）循环（C_m\）与一个（n）线（L_n\）的笛卡儿积（Cartesian product \(C_m\times L_n\）具有非三维遗传饱和子集，即使图（C_m\）和（L_n\）本身只有三维遗传饱和子集。Tomforde（"Uniqueness theorems and ideal structure for Leavitt path algebras," J. Algebra 318 (2007), 270-299中的定理5.7）证明，图\(E\)的Leavitt路径代数\(L(E)\)的分级理想集和\(E^0\)的遗传饱和子集之间存在一一对应关系。这表明笛卡尔乘的 Leavitt 路径代数（L(C_m\times L_n)\）的代数结构是丰富的。我们还证明了 \(L(C_m\times L_n)\) 的不变基数性质可以从 \(L(C_m)\) 的不变基数性质推导出来。更一般地说，我们还证明了如果 \(E\) 是一个没有汇的有限图，那么 \(L(E\times L_n)\ 的不变基数性质可以从 \(L(E)\ 的不变基数性质推导出来。

本文章由计算机程序翻译，如有差异，请以英文原文为准。

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来源期刊

Mathematical Notes 数学-数学

CiteScore

0.90

自引率

16.70%

发文量

179

审稿时长

24 months

期刊介绍： Mathematical Notes is a journal that publishes research papers and review articles in modern algebra, geometry and number theory, functional analysis, logic, set and measure theory, topology, probability and stochastics, differential and noncommutative geometry, operator and group theory, asymptotic and approximation methods, mathematical finance, linear and nonlinear equations, ergodic and spectral theory, operator algebras, and other related theoretical fields. It also presents rigorous results in mathematical physics.