Lattice tensor products I. Coordinatization (Q1848175)

The lattice tensor product \(A\boxtimes B\) of lattices \(A\), \(B\) was defined by \textit{G. Grätzer} and \textit{F. Wehrung} [J. Algebra 221, 315-344 (1999; Zbl 0961.06005)]. The original definitions has one disadvantage: The lattice \(A\boxtimes B\) is defined as a set of certain subsets of \(A\times B\), but it is not easy to decide whether such a subset belongs to \(A\boxtimes B\). In this paper the tensor product \(A\boxtimes B\) is described by means of ``coordinatization'', i.e. by a certain representation as a subset of \(B^n\), where \(n\) is the number of joint-irreducible elements of \(A\). At the end it is proved that for a finite simple lattice \(A\) the lattice \(A\boxtimes B\) is a congruence-preserving extension of \(B\).