Robinson lattices and their spectra (Q1337152)

The author continues his study [cf. the author, \textit{M. Insall} and \textit{K. Kaiser}, Z. Math. Logik Grundlagen Math. 36, No. 2, 123-131 (1990; Zbl 0711.03031)] of the Robinson-lattices (i.e. R-lattices). R-lattices are specialized distributive lattices \(D\). Since the lattices \(D\) are locally finite, using Robinson-styled nonstandard analysis appears very natural due to the fact that any standard set is contained in a hyperfinite set, assuming that the nonstandard structure is an enlargement. This fact allows for the generation of a hyperfinite lattice \(D_ \nu\) such that \(D\subset D_ \nu\subset {^* D}\). Further, the use of nonstandard models appears very important in obtaining various standard characterizations and properties for such lattices. As examples, the author shows that for such lattice \(D\) is an R-lattice iff each non- minimal prime filter is isolated in the strong topology. Every non- minimal prime filter of an R-lattice is principal. \(D\) is an R-lattice iff, for each non-minimal prime filter \(\rho\) of \(D\), there are elements \(x, y\in D\) such that \(\rho= [x)\), \((x]\backslash \{x\}= (y]\). Numerously many other interesting results are presented.