Orthocomplemented weak tensor products Boris Ischi

Abstract

Let \({{\mathcal L}_1}\) and \({{\mathcal L}_2}\) be complete atomistic lattices. In a previous paper, we have defined a set S = S\({({\mathcal L}_1, {\mathcal L}_2)}\) of complete atomistic lattices, the elements of which are called weak tensor products of \({{\mathcal L}_1}\) and \({{\mathcal L}_2}\) . S is defined by means of three axioms, natural regarding the description of some compound systems in quantum logic. It has been proved that S is a complete lattice. The top element of S, denoted by Open image in new window, is the tensor product of Fraser whereas the bottom element, denoted by Open image in new window, is the box product of Gratzer and Wehrung. With some additional hypotheses on \({L_1}\) and \({L_2}\) (true for instance if \({{\mathcal L}_1}\) and \({{\mathcal L}_2}\) are moreover orthomodular with the covering property) we prove that S is a singleton if and only if \({{\mathcal L}_1}\) or \({{\mathcal L}_2}\) is distributive, if and only if Open image in new window has the covering property. Our main result reads: \({{\mathcal L} \in \mathsf {S}}\) admits an orthocomplementation if and only if Open image in new window. At the end, we construct an example Open image in new window in S which has the covering property.