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 of an -cycle by an -line has nontrivial hereditary saturated subsets even though the graphs and 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 of a graph and the set of hereditary saturated subsets of. This shows that the algebraic structure of the Leavitt path algebra of the Cartesian product is plentiful. We also prove that the invariant basis number property of can be derived from that of. More generally, we also show that the invariant basis number property of can be derived from that of if is a finite graph without sinks.