Künzi, Hans-Peter; Zapata, Francisco; Kreinovich, Vladik When is the Busemann product a lattice? A relation between metric spaces and corresponding space-time models. (English) Zbl 1288.06017 Appl. Math. Sci., Ruse 6, No. 65-68, 3267-3276 (2012). Summary: The causality relation of special relativity is based on the assumption that the speed of all physical processes is limited by the speed of light. As a result, an event \((t, x)\) occurring at moment \(t\) at location \(x\) can influence an event \((y, s)\) if and only if s \(\geq t + \frac{d(x, y)}{c}\). We can simplify this formula if we use units of time and distance in which \(c = 1\) (e.g., by using a light second as a unit of distance). In this case, the above causality relation takes the form \(s \geq t + d(x, y)\). Since the actual space can be non-Euclidean, H. Busemann generalized this ordering relation to the case when points \(x, y\), etc. are taken from an arbitrary metric space \(X\). From the mathematical viewpoint, a natural question is: when is the resulting ordered space – called a Busemann product – a lattice? In this paper, we provide a necessary and sufficient condition for it being a lattice: it is a lattice if and only if \(X\) is a real tree, i.e., a metric space in which every two points are connected by exactly one arc, and this arc is geodesic (i.e., metrically isomorphic to an interval on a real line). MSC: 06B99 Lattices 54E35 Metric spaces, metrizability 83A05 Special relativity Keywords:space-time models; lattice; Busemann product; special relativity; real tree; causality relation PDFBibTeX XMLCite \textit{H.-P. Künzi} et al., Appl. Math. Sci., Ruse 6, No. 65--68, 3267--3276 (2012; Zbl 1288.06017) Full Text: Link