Аннотация:
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.
