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

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.