We describe a recent program from the study of definable groups in certain o-minimal structures. A central notion of this program is that of a (geometric) lattice. We propose a definition of a lattice in an arbitrary first-order structure. We then use it to describe, uniformly, various structure theorems for o-minimal groups, each time recovering a lattice that captures some significant invariant of the group at hand. The analysis first goes through a local level, where a pertinent notion of pregeometry and generic elements is each time introduced.
"Non-standard lattices and o-minimal groups." Bull. Symbolic Logic 19 (1) 56 - 76, March 2013. https://doi.org/10.2178/bsl.1901020