A non-capped tensor product of lattices. (Q485105)

Tensor product for join-semilattices was introduced by \textit{J. A. Anderson} and \textit{N. Kimura} [Semigroup Forum 16, 83-88 (1978; Zbl 0387.20044)], and by \textit{G. A. Fraser} [Trans. Am. Math. Soc. 217, 183-194 (1976; Zbl 0355.06013)]. If \(A\), \(B\) are lattices then their tensor product need not be a lattice but only a join-semilattice. However, if the tensor product of \(A\) and \(B\) is capped then it is a lattice. Whether the converse is true was an open problem posed by \textit{G. Grätzer} and \textit{F. Wehrung} [J. Pure Appl. Algebra 147, No. 3, 273-301 (2000; Zbl 0945.06003)]. The aim of this paper is to construct a tensor product of lattices \(A\) and \(B\) which is a lattice despite the fact that it is not capped. In this construction, \(A\) has length three and is generated by a nine-element set of atoms, while \(B\) is the dual of \(A\).