On complemented, uniquely complemented and uniquely complemented nondistributive lattices (a historical and epistemological note about a mathematical mystery)

Abstract

Complemented lattices and uniquely complemented lattices are very important, not only in mathematics, but also in physics, biology, and even in social sciences. They have been investigated for a long time, especially by Huntington, Birkhoff, Dilworth and others. And yet, on some of these structures - namely, uniquely complemented nondistributive lattices -, despite the many existing articles concerning them, we basically know very little. In this article, we situate these lattest structures in the context of complemented and uniquely complemented lattices, offering a general overview of the links between these lattices and others, close to them, such as the orthocomplemented lattices of physics as well as various other partially ordered sets. We finally show how uniquely complemented nondistributive lattices have been constructed with the technique of free lattices.