Lattice tensor products. II: Ideal lattices (Q1872917)

[For Part I see ibid. 95, 261-279 (2002; Zbl 0997.06002).] The concept of a lattice tensor product \(A\boxtimes B\) was recently introduced by \textit{G. Grätzer} and \textit{F. Wehrung} [J. Algebra 221, 315-344 (1999; Zbl 0961.06005)]. It is shown in this paper that if \(A\) is a finite lattice then \(A\boxtimes B\) is a subset of the direct power \(B^A\), and it provides an effective criterion for \(A\)-tuples of elements of \(B\) to occur in this representation. This enables the authors to prove the following result on ideal lattices: Theorem. Let \(A\), \(B\) be lattices. If \(A\) is finite then \(\text{Id}(A\boxtimes B)\cong A\boxtimes \text{Id}B\).