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

TL;DR

This paper explores the historical and mathematical context of complemented and uniquely complemented nondistributive lattices, highlighting their significance across disciplines and discussing their construction via free lattices.

Contribution

It provides a comprehensive overview of these lattices, clarifies their relationships with related structures, and discusses their construction methods, addressing gaps in current understanding.

Findings

Connections between complemented lattices and physical orthocomplemented lattices

Construction of uniquely complemented nondistributive lattices using free lattices

Identification of gaps in the understanding of these lattices

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.